Polynomial Division
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Roots (Zeros) of Polynomials
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Partial Fractions
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- Introduction
- Basic Case I
- Basic Case II
- Basic Case III
- General Case
- General Case (continued)
- Worked Example
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18

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[{"Name":"Polynomial Division","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"5m 28s","ChapterTopicVideoID":5256,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5256.jpeg","UploadDate":"2020-09-30T13:25:13.7370000","DurationForVideoObject":"PT5M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this clip, I\u0027m going to talk about the division of polynomials."},{"Start":"00:04.410 ","End":"00:10.170","Text":"First off, I\u0027ll give an introduction and then a definition of what it means."},{"Start":"00:10.170 ","End":"00:16.740","Text":"Then we\u0027ll do some practical actual division of polynomials."},{"Start":"00:16.740 ","End":"00:19.900","Text":"I\u0027ll show you a technique of how to divide."},{"Start":"00:20.690 ","End":"00:23.100","Text":"I suppose that first of all,"},{"Start":"00:23.100 ","End":"00:25.370","Text":"I should make sure you all know what a polynomial is."},{"Start":"00:25.370 ","End":"00:27.525","Text":"You probably do, but in case not,"},{"Start":"00:27.525 ","End":"00:34.290","Text":"it\u0027s an expression of the form a plus"},{"Start":"00:34.290 ","End":"00:41.563","Text":"bx plus cx^2 plus dx^3 plus and so on,"},{"Start":"00:41.563 ","End":"00:50.755","Text":"finite number of terms up to some power n. Let\u0027s say the coefficient is kx^n,"},{"Start":"00:50.755 ","End":"00:56.765","Text":"where a, b and c and d and so on up to k are actual numbers that I put in."},{"Start":"00:56.765 ","End":"00:58.940","Text":"But the x, x^2,"},{"Start":"00:58.940 ","End":"01:00.320","Text":"they remain like that."},{"Start":"01:00.320 ","End":"01:02.875","Text":"Well, I\u0027ll give you some examples."},{"Start":"01:02.875 ","End":"01:11.490","Text":"One example would be 4 plus 5x plus 10x^2."},{"Start":"01:11.490 ","End":"01:18.100","Text":"Another example could be 1 minus x^7."},{"Start":"01:18.320 ","End":"01:21.990","Text":"Yet another example, x^2"},{"Start":"01:21.990 ","End":"01:31.960","Text":"plus 20x plus 50x^8."},{"Start":"01:31.960 ","End":"01:36.235","Text":"The several things I want to remark on, first of all,"},{"Start":"01:36.235 ","End":"01:41.995","Text":"in this definition, the n has to be positive."},{"Start":"01:41.995 ","End":"01:45.040","Text":"Well, it could be 0 because here it\u0027s 0."},{"Start":"01:45.040 ","End":"01:47.155","Text":"This is like ax^0,"},{"Start":"01:47.155 ","End":"01:50.515","Text":"but it has to be a whole number and non-negative."},{"Start":"01:50.515 ","End":"01:56.690","Text":"In other words, n is bigger or equal to 0 and n is a whole number."},{"Start":"01:56.690 ","End":"02:01.485","Text":"The other thing is that not all terms have to be present."},{"Start":"02:01.485 ","End":"02:04.150","Text":"For example, here we have x^7,"},{"Start":"02:04.150 ","End":"02:07.775","Text":"but we don\u0027t have any x^6 or x^5, there can be gaps."},{"Start":"02:07.775 ","End":"02:10.715","Text":"The other thing is it doesn\u0027t have to be in order, of course."},{"Start":"02:10.715 ","End":"02:14.010","Text":"I mean, I can put the x^2 term first,"},{"Start":"02:14.010 ","End":"02:15.180","Text":"the x^8 last,"},{"Start":"02:15.180 ","End":"02:19.625","Text":"it doesn\u0027t have to be in any order of the powers of x or anything."},{"Start":"02:19.625 ","End":"02:23.300","Text":"Now there is another definition that goes with this polynomial,"},{"Start":"02:23.300 ","End":"02:25.715","Text":"that is the degree of the polynomial."},{"Start":"02:25.715 ","End":"02:28.310","Text":"I want to talk about the degree."},{"Start":"02:28.310 ","End":"02:31.430","Text":"Degree is simply the highest power of x,"},{"Start":"02:31.430 ","End":"02:34.580","Text":"or rather the exponent in the highest power of x."},{"Start":"02:34.580 ","End":"02:37.715","Text":"For example, this polynomial,"},{"Start":"02:37.715 ","End":"02:42.605","Text":"the highest power of x is this."},{"Start":"02:42.605 ","End":"02:46.760","Text":"Here, the highest power of x is this."},{"Start":"02:46.760 ","End":"02:49.535","Text":"Here, the highest power,"},{"Start":"02:49.535 ","End":"02:54.325","Text":"it seems to always be the last one."},{"Start":"02:54.325 ","End":"02:57.975","Text":"Well, that\u0027s just coincidence, is this."},{"Start":"02:57.975 ","End":"03:00.680","Text":"Therefore, when we talk about the degrees,"},{"Start":"03:00.680 ","End":"03:03.740","Text":"then here it would be 2,"},{"Start":"03:03.740 ","End":"03:05.540","Text":"here it would be 7,"},{"Start":"03:05.540 ","End":"03:08.780","Text":"and here the degree is 8."},{"Start":"03:08.780 ","End":"03:10.670","Text":"Now the degree is,"},{"Start":"03:10.670 ","End":"03:13.400","Text":"of course, this n here."},{"Start":"03:13.400 ","End":"03:17.345","Text":"Before we talk about division of polynomials,"},{"Start":"03:17.345 ","End":"03:22.565","Text":"let\u0027s first talk about multiplication of polynomials."},{"Start":"03:22.565 ","End":"03:24.950","Text":"I\u0027ll do this by means of an example."},{"Start":"03:24.950 ","End":"03:35.505","Text":"Let\u0027s take x^2 plus x plus 1 times x^2 minus x plus 1."},{"Start":"03:35.505 ","End":"03:38.090","Text":"Now I\u0027m not going to do the actual computation."},{"Start":"03:38.090 ","End":"03:39.785","Text":"You will know how to do it."},{"Start":"03:39.785 ","End":"03:44.165","Text":"We basically multiply each term in here with each term in here and add them."},{"Start":"03:44.165 ","End":"03:46.354","Text":"I\u0027ll just give you the answer."},{"Start":"03:46.354 ","End":"03:52.960","Text":"The answer is x^4 plus x^2 plus 1."},{"Start":"03:52.960 ","End":"03:57.287","Text":"I want to use the concepts of multiplication to explain division,"},{"Start":"03:57.287 ","End":"03:58.985","Text":"just like in arithmetic."},{"Start":"03:58.985 ","End":"04:07.800","Text":"In arithmetic, we would say something like 4 times 3 equals 12."},{"Start":"04:07.800 ","End":"04:12.015","Text":"From this, I would conclude that"},{"Start":"04:12.015 ","End":"04:18.960","Text":"12 divided by 3 is equal to 4,"},{"Start":"04:18.960 ","End":"04:24.765","Text":"and also 12 over 4 is equal to 3 either way."},{"Start":"04:24.765 ","End":"04:29.000","Text":"This is the relationship in arithmetic between multiplication and division."},{"Start":"04:29.000 ","End":"04:30.320","Text":"If this times this is this,"},{"Start":"04:30.320 ","End":"04:32.585","Text":"then this over this is this and so on."},{"Start":"04:32.585 ","End":"04:35.090","Text":"I\u0027d like the same quality,"},{"Start":"04:35.090 ","End":"04:38.375","Text":"the same property to hold for polynomials."},{"Start":"04:38.375 ","End":"04:45.140","Text":"In other words, what I would expect is that if this times this is this,"},{"Start":"04:45.140 ","End":"04:55.370","Text":"I would like to expect that x^4 plus x^2 plus 1 divided by,"},{"Start":"04:55.370 ","End":"04:57.470","Text":"it doesn\u0027t matter, let\u0027s say this one,"},{"Start":"04:57.470 ","End":"05:01.090","Text":"x^2 minus x plus 1,"},{"Start":"05:01.090 ","End":"05:06.880","Text":"that this should equal x^2 plus x plus 1."},{"Start":"05:07.000 ","End":"05:10.550","Text":"When I teach you how to actually do the division,"},{"Start":"05:10.550 ","End":"05:13.560","Text":"I would expect the answer for this over this to come out this."},{"Start":"05:13.560 ","End":"05:15.755","Text":"This will be a check so that I\u0027ll know the answer."},{"Start":"05:15.755 ","End":"05:18.660","Text":"What we\u0027ll do next is learn a technique."},{"Start":"05:18.660 ","End":"05:21.440","Text":"This is something purely technical of how to divide,"},{"Start":"05:21.440 ","End":"05:24.725","Text":"say, this polynomial by this polynomial."},{"Start":"05:24.725 ","End":"05:27.065","Text":"As I say, we should get this."},{"Start":"05:27.065 ","End":"05:29.340","Text":"We\u0027re done with this part."}],"ID":5247},{"Watched":false,"Name":"Technique","Duration":"7m 2s","ChapterTopicVideoID":5257,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5257.jpeg","UploadDate":"2020-09-30T13:30:14.7870000","DurationForVideoObject":"PT7M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.230 ","End":"00:03.494","Text":"Continuing with polynomial division,"},{"Start":"00:03.494 ","End":"00:05.715","Text":"let\u0027s learn the actual technique."},{"Start":"00:05.715 ","End":"00:09.675","Text":"I will take the example we had just before,"},{"Start":"00:09.675 ","End":"00:13.875","Text":"which is this over this and we even"},{"Start":"00:13.875 ","End":"00:18.045","Text":"made a note of the answer we expect and I\u0027ll just write it down somewhere,"},{"Start":"00:18.045 ","End":"00:19.755","Text":"I\u0027ll hide it at the side somewhere."},{"Start":"00:19.755 ","End":"00:24.960","Text":"We expect to get the answer x^ 2 plus x plus 1,"},{"Start":"00:24.960 ","End":"00:29.160","Text":"forget about that for now at the end we\u0027ll check and see if we got it right."},{"Start":"00:29.160 ","End":"00:37.515","Text":"We set it up by writing like a right bracket and then as a roof over it,"},{"Start":"00:37.515 ","End":"00:43.110","Text":"I don\u0027t know how to describe this but here we put what\u0027s on the top"},{"Start":"00:43.110 ","End":"00:51.130","Text":"so x^4 plus x^2 plus 1."},{"Start":"00:51.140 ","End":"00:57.320","Text":"I have left some spaces to remind me there is a missing x cubed and the missing x here."},{"Start":"00:57.320 ","End":"01:00.650","Text":"Here, we put what\u0027s on the bottom,"},{"Start":"01:00.650 ","End":"01:04.730","Text":"which is x^2 minus x plus 1."},{"Start":"01:04.730 ","End":"01:14.540","Text":"Some people actually write in plus 0 x cubed plus 0 x and so on."},{"Start":"01:14.540 ","End":"01:18.610","Text":"It can, if you want, I don\u0027t find it necessary,"},{"Start":"01:18.610 ","End":"01:20.595","Text":"I\u0027ll leave it there for now."},{"Start":"01:20.595 ","End":"01:22.085","Text":"I forgot to say,"},{"Start":"01:22.085 ","End":"01:28.040","Text":"it\u0027s best really recommended to write these powers in decreasing order."},{"Start":"01:28.040 ","End":"01:32.345","Text":"What we do now is to take the highest power here,"},{"Start":"01:32.345 ","End":"01:35.030","Text":"which will be the left one if you\u0027ve done them in"},{"Start":"01:35.030 ","End":"01:38.405","Text":"order and the highest power here and ask,"},{"Start":"01:38.405 ","End":"01:41.330","Text":"how many times does this go into this?"},{"Start":"01:41.330 ","End":"01:43.625","Text":"Or in other words,"},{"Start":"01:43.625 ","End":"01:49.130","Text":"what is x^4 over x^2?"},{"Start":"01:49.130 ","End":"01:52.160","Text":"Well, the answer to this is straightforward,"},{"Start":"01:52.160 ","End":"01:58.590","Text":"it\u0027s just x squared and I take this x squared and I write it here."},{"Start":"01:59.530 ","End":"02:02.165","Text":"I\u0027d like to repeat that."},{"Start":"02:02.165 ","End":"02:06.740","Text":"We take the highest power in here which is this,"},{"Start":"02:06.740 ","End":"02:08.570","Text":"and the highest power here,"},{"Start":"02:08.570 ","End":"02:12.930","Text":"which is this and divide this over this."},{"Start":"02:13.090 ","End":"02:18.890","Text":"The answer, we put over here."},{"Start":"02:18.890 ","End":"02:21.845","Text":"Now the next step."},{"Start":"02:21.845 ","End":"02:25.955","Text":"I think I\u0027ll number these steps so we can refer to them."},{"Start":"02:25.955 ","End":"02:28.160","Text":"Let\u0027s say 1 and 2, yes."},{"Start":"02:28.160 ","End":"02:35.990","Text":"I was about to say, we take this x squared and multiply it by this whole polynomial."},{"Start":"02:35.990 ","End":"02:44.090","Text":"In other words, I take x squared and multiply it by x^2 minus x plus"},{"Start":"02:44.090 ","End":"02:54.685","Text":"1 and that gives me x^4 minus x^3 plus x^2."},{"Start":"02:54.685 ","End":"03:02.500","Text":"What I\u0027ll do is I\u0027ll take this polynomial and I\u0027m going to put it under here,"},{"Start":"03:02.500 ","End":"03:05.080","Text":"so let\u0027s see x^4 minus x^3"},{"Start":"03:05.080 ","End":"03:11.020","Text":"plus x^2,"},{"Start":"03:11.020 ","End":"03:15.820","Text":"here."},{"Start":"03:15.820 ","End":"03:21.070","Text":"Now we subtract this, this minus this."},{"Start":"03:21.070 ","End":"03:29.665","Text":"What I get is x^4, minus x^4 cancels."},{"Start":"03:29.665 ","End":"03:32.040","Text":"Nothing, minus,"},{"Start":"03:32.040 ","End":"03:35.175","Text":"minus x^3 gives me x^3,"},{"Start":"03:35.175 ","End":"03:40.410","Text":"x^2 minus x^2 is nothing,"},{"Start":"03:40.410 ","End":"03:49.215","Text":"nothing is just nothing and 1 minus nothing is 1, so plus 1."},{"Start":"03:49.215 ","End":"03:52.700","Text":"So x^3 plus 1 is what we have now,"},{"Start":"03:52.700 ","End":"03:54.890","Text":"let me just clean up a bit."},{"Start":"03:54.890 ","End":"03:59.555","Text":"Basically what we\u0027re going to do is just start over again from step 1,"},{"Start":"03:59.555 ","End":"04:04.430","Text":"say x^2 into x^3 goes how many times so like I\u0027m at step 1"},{"Start":"04:04.430 ","End":"04:10.070","Text":"again and I do x^3 over x^2 and this becomes x."},{"Start":"04:10.070 ","End":"04:19.130","Text":"Then I copy this x and I put it up here like we did before."},{"Start":"04:19.130 ","End":"04:25.730","Text":"Then the next step is to multiply this x by all this"},{"Start":"04:25.730 ","End":"04:33.320","Text":"so that step number 2 to take x and multiply it by x^2 minus x plus 1,"},{"Start":"04:33.320 ","End":"04:36.725","Text":"and the answer we write over here,"},{"Start":"04:36.725 ","End":"04:39.260","Text":"it\u0027s x^3 minus x^2 plus x,"},{"Start":"04:39.260 ","End":"04:49.140","Text":"so x^3 minus x^2 plus x. I highlight it last time,"},{"Start":"04:49.140 ","End":"04:52.600","Text":"let\u0027s just highlight it anyway."},{"Start":"04:57.740 ","End":"05:01.230","Text":"Again, we subtract."},{"Start":"05:01.230 ","End":"05:04.330","Text":"That\u0027s going to be the third step, this minus this."},{"Start":"05:04.330 ","End":"05:08.020","Text":"I\u0027m not going to write anything here because I don\u0027t think you need it."},{"Start":"05:08.020 ","End":"05:11.050","Text":"X^3 minus x^3 is nothing,"},{"Start":"05:11.050 ","End":"05:15.710","Text":"nothing minus minus x^2 is x^2,"},{"Start":"05:15.710 ","End":"05:21.550","Text":"1 minus x is like minus x plus 1."},{"Start":"05:26.100 ","End":"05:29.890","Text":"I got a bit crooked. I meant to keep x\u0027s over x\u0027s,"},{"Start":"05:29.890 ","End":"05:32.665","Text":"x^2 where it\u0027s neater."},{"Start":"05:32.665 ","End":"05:39.705","Text":"Then we start the circle again so we take the leading power here,"},{"Start":"05:39.705 ","End":"05:42.395","Text":"the leading term here,"},{"Start":"05:42.395 ","End":"05:45.530","Text":"the highest power and then again we say,"},{"Start":"05:45.530 ","End":"05:48.275","Text":"how many times does this go into this?"},{"Start":"05:48.275 ","End":"05:50.015","Text":"Or in other words,"},{"Start":"05:50.015 ","End":"05:53.390","Text":"what is x^2 over x^2?"},{"Start":"05:53.390 ","End":"06:00.540","Text":"That\u0027s like our step 1 and this time it just equals 1 so I put this 1"},{"Start":"06:00.540 ","End":"06:08.360","Text":"here and then I multiply 1 times x^2 minus x plus 1,"},{"Start":"06:08.360 ","End":"06:13.355","Text":"which of course is just x^2 minus x plus 1 and write it here."},{"Start":"06:13.355 ","End":"06:17.390","Text":"Then we again subtract and this time when we subtract,"},{"Start":"06:17.390 ","End":"06:22.160","Text":"we get nothing left because this is the same thing and that\u0027s how we know when to stop."},{"Start":"06:22.160 ","End":"06:24.395","Text":"If we don\u0027t get a 0 at the end,"},{"Start":"06:24.395 ","End":"06:27.530","Text":"we\u0027ve made a mistake and it doesn\u0027t divide into it,"},{"Start":"06:27.530 ","End":"06:30.000","Text":"but this time we knew it does."},{"Start":"06:30.220 ","End":"06:33.920","Text":"This is the answer and look,"},{"Start":"06:33.920 ","End":"06:37.455","Text":"we\u0027ve even got a confirmation or corroboration,"},{"Start":"06:37.455 ","End":"06:42.500","Text":"x squared plus x plus 1 is exactly what we were hoping to get and we got it,"},{"Start":"06:42.500 ","End":"06:44.975","Text":"so this is essentially the technique."},{"Start":"06:44.975 ","End":"06:49.670","Text":"This part here, which happened to be 0 this time is"},{"Start":"06:49.670 ","End":"06:54.560","Text":"called the remainder but we\u0027re not going to talk about division with remainder now,"},{"Start":"06:54.560 ","End":"06:56.780","Text":"this will be some time later."},{"Start":"06:56.780 ","End":"07:00.730","Text":"We expect meanwhile that this should be 0."},{"Start":"07:00.730 ","End":"07:03.310","Text":"Okay, done."}],"ID":5248},{"Watched":false,"Name":"Examples","Duration":"9m 34s","ChapterTopicVideoID":5248,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5248.jpeg","UploadDate":"2020-10-06T08:17:59.5400000","DurationForVideoObject":"PT9M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:04.050","Text":"The best way to learn polynomial division"},{"Start":"00:04.050 ","End":"00:07.125","Text":"is just to practice and I\u0027ll help you with that."},{"Start":"00:07.125 ","End":"00:09.915","Text":"I\u0027ll do 3 examples here."},{"Start":"00:09.915 ","End":"00:13.785","Text":"Let\u0027s get started with the first example."},{"Start":"00:13.785 ","End":"00:20.085","Text":"I see the highest power here is x^3 and the highest power here is x,"},{"Start":"00:20.085 ","End":"00:22.665","Text":"so I highlight that."},{"Start":"00:22.665 ","End":"00:25.965","Text":"How many times does x go into x^3?"},{"Start":"00:25.965 ","End":"00:27.480","Text":"We can do that in our heads."},{"Start":"00:27.480 ","End":"00:30.470","Text":"It\u0027s the same as saying x^3 over x,"},{"Start":"00:30.470 ","End":"00:33.375","Text":"so this goes in x^2 times."},{"Start":"00:33.375 ","End":"00:37.530","Text":"In other words, I take this highest power over the highest power and write it here."},{"Start":"00:37.530 ","End":"00:43.595","Text":"Next step is to multiply this by this and write it under here,"},{"Start":"00:43.595 ","End":"00:51.425","Text":"the x^2 I multiply it by the x minus 1 and get x^3 minus x^2."},{"Start":"00:51.425 ","End":"00:59.495","Text":"Now we subtract, dividing line and subtract x^3 minus x^3 is nothing."},{"Start":"00:59.495 ","End":"01:03.020","Text":"Minus x squared minus minus x squared is also nothing,"},{"Start":"01:03.020 ","End":"01:08.015","Text":"x minus nothing is just x and minus 1."},{"Start":"01:08.015 ","End":"01:14.150","Text":"Now, we\u0027re back to the beginning again, repeat the cycle."},{"Start":"01:14.150 ","End":"01:15.815","Text":"Highest power here is x,"},{"Start":"01:15.815 ","End":"01:17.210","Text":"highest power here is x,"},{"Start":"01:17.210 ","End":"01:20.660","Text":"x goes into x one time."},{"Start":"01:20.660 ","End":"01:22.835","Text":"Just wanted to just plus 1."},{"Start":"01:22.835 ","End":"01:26.570","Text":"The plus 1 that I just wrote,"},{"Start":"01:26.570 ","End":"01:34.130","Text":"I have to multiply by x minus 1 and I get x minus 1."},{"Start":"01:34.130 ","End":"01:40.880","Text":"Again, 1 times this here and I write it under here,"},{"Start":"01:40.880 ","End":"01:43.790","Text":"x minus x just retraction is nothing."},{"Start":"01:43.790 ","End":"01:46.085","Text":"This minus this is nothing so I get 0."},{"Start":"01:46.085 ","End":"01:49.220","Text":"When I get 0, then the process is done."},{"Start":"01:49.220 ","End":"01:51.920","Text":"This here at the top is the quotient."},{"Start":"01:51.920 ","End":"01:54.960","Text":"This is the answer, x^2 plus 1."},{"Start":"01:54.960 ","End":"01:57.695","Text":"If I want to express it another way,"},{"Start":"01:57.695 ","End":"02:05.090","Text":"I can write in the form x^3 minus x^2 plus x minus"},{"Start":"02:05.090 ","End":"02:13.845","Text":"1 divided by x minus 1 is equal to x^2 plus 1."},{"Start":"02:13.845 ","End":"02:18.545","Text":"That\u0027s just a way of rephrasing it and if you really want to be sure,"},{"Start":"02:18.545 ","End":"02:24.155","Text":"what you could do is take this multiply by this and see if you get this."},{"Start":"02:24.155 ","End":"02:28.170","Text":"If you have time leftover in the exam or something, you could do that."},{"Start":"02:28.170 ","End":"02:31.140","Text":"Let\u0027s go to the next question."},{"Start":"02:31.140 ","End":"02:33.435","Text":"Start right away."},{"Start":"02:33.435 ","End":"02:39.900","Text":"What we do first is to take the term with"},{"Start":"02:39.900 ","End":"02:47.100","Text":"the highest power that is and the term with the highest power here and then I ask,"},{"Start":"02:47.100 ","End":"02:51.435","Text":"how many times does x^2 go into 4x^4?"},{"Start":"02:51.435 ","End":"02:54.610","Text":"Well, the answer is 4x^2."},{"Start":"02:55.240 ","End":"02:58.415","Text":"Like this divided by this is this."},{"Start":"02:58.415 ","End":"03:00.140","Text":"Then I take what I just wrote,"},{"Start":"03:00.140 ","End":"03:03.140","Text":"multiply it out by this so I get"},{"Start":"03:03.140 ","End":"03:07.800","Text":"4x^4 minus"},{"Start":"03:07.940 ","End":"03:14.600","Text":"4x^3 plus 40x^2."},{"Start":"03:14.600 ","End":"03:16.460","Text":"Oh, I forgot the 2 here."},{"Start":"03:16.460 ","End":"03:19.225","Text":"Then we subtract."},{"Start":"03:19.225 ","End":"03:21.030","Text":"This has to disappear,"},{"Start":"03:21.030 ","End":"03:22.370","Text":"if it doesn\u0027t disappear,"},{"Start":"03:22.370 ","End":"03:23.885","Text":"then you made a mistake."},{"Start":"03:23.885 ","End":"03:25.985","Text":"This minus this is nothing,"},{"Start":"03:25.985 ","End":"03:30.945","Text":"6 takeaway minus 4 is plus 10x^3."},{"Start":"03:30.945 ","End":"03:38.900","Text":"This minus this is minus 9x^2 and I didn\u0027t draw the line all across,"},{"Start":"03:38.900 ","End":"03:42.080","Text":"but still subtracting from"},{"Start":"03:42.080 ","End":"03:49.898","Text":"the whole thing, 99x plus 10."},{"Start":"03:49.898 ","End":"03:55.340","Text":"Now, highest power term is here and over"},{"Start":"03:55.340 ","End":"04:01.415","Text":"here and now I ask how many times does x^2 go into 10 x^3?"},{"Start":"04:01.415 ","End":"04:07.680","Text":"So that goes in 10x times,"},{"Start":"04:08.980 ","End":"04:13.175","Text":"10x times all of this is"},{"Start":"04:13.175 ","End":"04:23.360","Text":"10x^3 minus 10x^2 plus 100x."},{"Start":"04:23.360 ","End":"04:27.075","Text":"Now subtract, this minus this disappears,"},{"Start":"04:27.075 ","End":"04:30.310","Text":"this minus this is plus 1,"},{"Start":"04:30.700 ","End":"04:38.820","Text":"x^2 minus x plus 10."},{"Start":"04:38.820 ","End":"04:42.290","Text":"Well, it\u0027s clear don\u0027t have to do the bit with the highest power."},{"Start":"04:42.290 ","End":"04:44.030","Text":"I can see x^2 squared minus x plus 10,"},{"Start":"04:44.030 ","End":"04:49.070","Text":"x^2 minus x plus 10 so goes in plus"},{"Start":"04:49.070 ","End":"04:54.990","Text":"1 time and the answer is 0,"},{"Start":"04:54.990 ","End":"05:01.959","Text":"which means that what I wrote here is the answer."},{"Start":"05:01.959 ","End":"05:04.300","Text":"That\u0027s the quotient."},{"Start":"05:04.300 ","End":"05:09.310","Text":"We can rewrite this as, let\u0027s see,"},{"Start":"05:09.310 ","End":"05:20.690","Text":"4x^4 plus 6x^3 plus 31x^2"},{"Start":"05:20.690 ","End":"05:27.080","Text":"plus 99x minus 10 over"},{"Start":"05:32.000 ","End":"05:39.420","Text":"x^2 minus x plus 10 is equal to"},{"Start":"05:39.420 ","End":"05:46.770","Text":"4x^2 plus 10x"},{"Start":"05:46.770 ","End":"05:50.850","Text":"plus 1."},{"Start":"05:50.850 ","End":"05:53.350","Text":"On to the next example."},{"Start":"05:53.350 ","End":"05:55.970","Text":"In the last example,"},{"Start":"05:55.970 ","End":"05:58.775","Text":"we\u0027re going to do division with remainder."},{"Start":"05:58.775 ","End":"06:01.835","Text":"Don\u0027t worry, pretty much the same."},{"Start":"06:01.835 ","End":"06:05.060","Text":"Let me just take you back to arithmetic."},{"Start":"06:05.060 ","End":"06:11.580","Text":"If we have something like 12 divided by 4,"},{"Start":"06:11.580 ","End":"06:15.490","Text":"we get a quotient of 3."},{"Start":"06:16.020 ","End":"06:22.570","Text":"But if we do a long division or otherwise of 12 divided by 5,"},{"Start":"06:22.570 ","End":"06:25.790","Text":"say in grade school,"},{"Start":"06:25.790 ","End":"06:28.425","Text":"how many times does 5 go into 12?"},{"Start":"06:28.425 ","End":"06:30.240","Text":"We say well it goes in twice,"},{"Start":"06:30.240 ","End":"06:31.845","Text":"but there\u0027s 2 leftover,"},{"Start":"06:31.845 ","End":"06:35.190","Text":"which means the division with remainder so we say it\u0027s 2,"},{"Start":"06:35.190 ","End":"06:42.330","Text":"but with remainder also we\u0027ve 2 goes in twice and 2 leftover."},{"Start":"06:42.330 ","End":"06:44.735","Text":"Now in this case,"},{"Start":"06:44.735 ","End":"06:51.550","Text":"what we say is that 12/5 is 2 and this remainder goes over"},{"Start":"06:51.550 ","End":"06:57.930","Text":"the original dividend or denominator so it\u0027s 2 and 2/5."},{"Start":"06:57.930 ","End":"06:59.180","Text":"If you get the idea,"},{"Start":"06:59.180 ","End":"07:02.285","Text":"I\u0027ll do another 1 of these in arithmetic if we have, say,"},{"Start":"07:02.285 ","End":"07:10.650","Text":"23/4 so 4 goes into 23 5 times."},{"Start":"07:10.650 ","End":"07:14.985","Text":"It goes in 20 and 3 leftover, remainder of 3."},{"Start":"07:14.985 ","End":"07:19.560","Text":"What we can say is that 23/4 is"},{"Start":"07:19.560 ","End":"07:25.770","Text":"5 and 3/4 because the remainder is still has to be divided by the 4."},{"Start":"07:25.770 ","End":"07:29.270","Text":"We\u0027re going to get the same thing in polynomials."},{"Start":"07:29.270 ","End":"07:31.220","Text":"We\u0027ll wait till we get to the end and then you\u0027ll see how"},{"Start":"07:31.220 ","End":"07:34.290","Text":"I put it together just like with arithmetic,"},{"Start":"07:34.780 ","End":"07:40.325","Text":"so x goes into 4x^2 4x times,"},{"Start":"07:40.325 ","End":"07:49.440","Text":"4x times x minus 2 is 4x^2 minus 8x and we subtract."},{"Start":"07:49.440 ","End":"07:52.180","Text":"This minus this is nothing, x minus,"},{"Start":"07:52.180 ","End":"07:55.290","Text":"minus 8x is 9x,"},{"Start":"07:55.290 ","End":"08:00.600","Text":"9x and minus 1 takeaway nothing is just minus 1,"},{"Start":"08:00.600 ","End":"08:04.200","Text":"x goes into 9x, 9 times,"},{"Start":"08:04.200 ","End":"08:14.680","Text":"9 times x minus 2 is 9x minus 18, I wrote 10."},{"Start":"08:15.650 ","End":"08:21.185","Text":"Finally we take this from this and we get 17,"},{"Start":"08:21.185 ","End":"08:25.835","Text":"which is not 0 and this we can\u0027t"},{"Start":"08:25.835 ","End":"08:31.550","Text":"go on dividing because x is already higher power than this."},{"Start":"08:31.550 ","End":"08:36.500","Text":"What we do is we just say that it doesn\u0027t divide evenly,"},{"Start":"08:36.500 ","End":"08:38.015","Text":"divides with a remainder."},{"Start":"08:38.015 ","End":"08:41.115","Text":"This is the quotient,"},{"Start":"08:41.115 ","End":"08:44.285","Text":"this is the remainder and what all we can do is write"},{"Start":"08:44.285 ","End":"08:53.540","Text":"that 4x^2 plus x minus 1 over x minus 2."},{"Start":"08:53.540 ","End":"08:55.175","Text":"In the previous exercises,"},{"Start":"08:55.175 ","End":"08:59.065","Text":"it was a polynomial here it\u0027s a polynomial but with the remainder,"},{"Start":"08:59.065 ","End":"09:07.940","Text":"so it\u0027s 4x plus 9 and the remainder 17 also has to be taken over the x minus 2 as with"},{"Start":"09:07.940 ","End":"09:11.450","Text":"the arithmetic so it\u0027s the remainder over"},{"Start":"09:11.450 ","End":"09:19.015","Text":"the original divisor, x minus 2."},{"Start":"09:19.015 ","End":"09:21.285","Text":"That\u0027s basically it."},{"Start":"09:21.285 ","End":"09:27.080","Text":"You can either say that this over this is this with remainder 17 or you can just"},{"Start":"09:27.080 ","End":"09:34.040","Text":"write it out fully like we did with fractions and we are done."}],"ID":5249},{"Watched":false,"Name":"Exercise 1","Duration":"14m 56s","ChapterTopicVideoID":5249,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5249.jpeg","UploadDate":"2016-03-07T08:35:59.9430000","DurationForVideoObject":"PT14M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this exercise, we have 2 parts, a and b,"},{"Start":"00:02.850 ","End":"00:07.740","Text":"and each of them we have to do long division of polynomials."},{"Start":"00:07.740 ","End":"00:10.350","Text":"Let\u0027s start with the first 1."},{"Start":"00:10.350 ","End":"00:12.870","Text":"What we do is,"},{"Start":"00:12.870 ","End":"00:15.180","Text":"first of all, we write this part."},{"Start":"00:15.180 ","End":"00:20.760","Text":"This is called the dividend and this one is the divisor,"},{"Start":"00:20.760 ","End":"00:22.470","Text":"but that doesn\u0027t matter anyway."},{"Start":"00:22.470 ","End":"00:24.795","Text":"The part that we\u0027re dividing,"},{"Start":"00:24.795 ","End":"00:30.223","Text":"put it under a division sign say"},{"Start":"00:30.223 ","End":"00:37.920","Text":"4x^3 plus 5x^2 minus x minus 11,"},{"Start":"00:37.920 ","End":"00:40.850","Text":"and the thing I\u0027m dividing by,"},{"Start":"00:40.850 ","End":"00:42.050","Text":"I put over here."},{"Start":"00:42.050 ","End":"00:44.210","Text":"What\u0027s that 4x^2 plus x minus 6,"},{"Start":"00:44.210 ","End":"00:48.215","Text":"so 4x^2 plus x minus 6."},{"Start":"00:48.215 ","End":"00:51.802","Text":"I don\u0027t know, I didn\u0027t leave enough room."},{"Start":"00:51.802 ","End":"00:55.845","Text":"That\u0027s better."},{"Start":"00:55.845 ","End":"00:59.345","Text":"What we do is,"},{"Start":"00:59.345 ","End":"01:03.530","Text":"first of all, verify this written in the correct order in decreasing powers,"},{"Start":"01:03.530 ","End":"01:05.930","Text":"which each one of them is."},{"Start":"01:05.930 ","End":"01:10.340","Text":"Then I take the leading term in each of them and say,"},{"Start":"01:10.340 ","End":"01:14.015","Text":"4x^2 goes into 4x^3."},{"Start":"01:14.015 ","End":"01:16.460","Text":"Let me just share so you see what I\u0027m talking about."},{"Start":"01:16.460 ","End":"01:20.980","Text":"This one goes into this one how many times?"},{"Start":"01:20.980 ","End":"01:31.145","Text":"The answer is x. I put that x above this and then I do x times this polynomial."},{"Start":"01:31.145 ","End":"01:41.250","Text":"X times this is 4x^3 plus x times x is x^2 minus 6x."},{"Start":"01:41.250 ","End":"01:43.725","Text":"Then I do a subtraction."},{"Start":"01:43.725 ","End":"01:48.405","Text":"The first term has to cancel otherwise you didn\u0027t do the division right."},{"Start":"01:48.405 ","End":"01:51.260","Text":"This minus this is nothing,"},{"Start":"01:51.260 ","End":"01:52.674","Text":"this minus this,"},{"Start":"01:52.674 ","End":"01:56.198","Text":"let me just put on maybe a minus here so you\u0027ll see I\u0027m subtracting,"},{"Start":"01:56.198 ","End":"02:05.520","Text":"is 4x^2 minus x minus minus 6x is plus 5x,"},{"Start":"02:06.500 ","End":"02:14.619","Text":"and each time I drop another term if necessary, minus 11."},{"Start":"02:14.619 ","End":"02:20.475","Text":"Now I ask 4x^2 goes into 4x^2."},{"Start":"02:20.475 ","End":"02:22.515","Text":"I\u0027ll just highlight that too."},{"Start":"02:22.515 ","End":"02:24.690","Text":"This into this goes how many times?"},{"Start":"02:24.690 ","End":"02:26.745","Text":"The answer is 1."},{"Start":"02:26.745 ","End":"02:29.700","Text":"I put a plus 1 here,"},{"Start":"02:29.700 ","End":"02:39.540","Text":"and then 1 times 4x^2 plus x minus 6 is just 4x^2 plus x minus 6."},{"Start":"02:39.540 ","End":"02:42.225","Text":"Then I do another subtraction,"},{"Start":"02:42.225 ","End":"02:45.630","Text":"and this with this cancels."},{"Start":"02:45.630 ","End":"02:49.930","Text":"5x minus x is 4x,"},{"Start":"02:49.970 ","End":"02:57.490","Text":"and minus 11 minus minus 6 is minus 5."},{"Start":"02:57.490 ","End":"03:00.260","Text":"At this point, we stopped dividing."},{"Start":"03:00.260 ","End":"03:05.870","Text":"When you get something here with a degree less than here, we can\u0027t continue."},{"Start":"03:05.870 ","End":"03:08.180","Text":"We can\u0027t say x^2 into x."},{"Start":"03:08.180 ","End":"03:10.655","Text":"That\u0027s where we stopped this process,"},{"Start":"03:10.655 ","End":"03:13.819","Text":"but the answer is written as follows."},{"Start":"03:13.819 ","End":"03:16.980","Text":"We write that 4x^3,"},{"Start":"03:16.980 ","End":"03:20.025","Text":"this is the dividend,"},{"Start":"03:20.025 ","End":"03:24.815","Text":"plus 5x^2 minus x minus 11."},{"Start":"03:24.815 ","End":"03:27.065","Text":"Well, that\u0027s what\u0027s under the division sign."},{"Start":"03:27.065 ","End":"03:30.510","Text":"Then over the other bit,"},{"Start":"03:30.510 ","End":"03:36.510","Text":"the divisor 4x^2 plus x minus 6 equals."},{"Start":"03:36.510 ","End":"03:39.875","Text":"This is actually called the quotient and this is the remainder."},{"Start":"03:39.875 ","End":"03:41.750","Text":"We write the quotient,"},{"Start":"03:41.750 ","End":"03:45.745","Text":"the bit on top, plus another remainder,"},{"Start":"03:45.745 ","End":"03:51.066","Text":"4x minus 5 goes over the divisor,"},{"Start":"03:51.066 ","End":"03:52.734","Text":"this bit here,"},{"Start":"03:52.734 ","End":"03:54.304","Text":"this one here,"},{"Start":"03:54.304 ","End":"04:00.085","Text":"4x^2 plus x minus 6."},{"Start":"04:00.085 ","End":"04:09.330","Text":"That\u0027s how we write the answer and maybe I\u0027ll highlight it."},{"Start":"04:09.330 ","End":"04:11.490","Text":"That\u0027s part a,"},{"Start":"04:11.490 ","End":"04:15.130","Text":"and now onto part b."},{"Start":"04:15.440 ","End":"04:18.225","Text":"Same idea."},{"Start":"04:18.225 ","End":"04:20.760","Text":"Do it a bit quicker this time."},{"Start":"04:20.760 ","End":"04:22.320","Text":"I\u0027ll leave enough room here."},{"Start":"04:22.320 ","End":"04:27.795","Text":"3x^4 plus 10x^3 plus"},{"Start":"04:27.795 ","End":"04:33.480","Text":"17x^2 plus 10x plus 24,"},{"Start":"04:33.480 ","End":"04:35.580","Text":"lot of writing here,"},{"Start":"04:35.580 ","End":"04:43.140","Text":"divided by 3x^2 plus"},{"Start":"04:43.140 ","End":"04:50.430","Text":"14x plus 8."},{"Start":"04:50.430 ","End":"04:54.945","Text":"I asked first, how many times does this go into this?"},{"Start":"04:54.945 ","End":"04:58.020","Text":"Answer, x^2."},{"Start":"04:58.020 ","End":"05:00.330","Text":"X^2 times this,"},{"Start":"05:00.330 ","End":"05:05.010","Text":"is 3x^4 plus 14x,"},{"Start":"05:05.010 ","End":"05:06.765","Text":"this will be cubed,"},{"Start":"05:06.765 ","End":"05:14.020","Text":"plus 8x^2 and then we subtract."},{"Start":"05:14.210 ","End":"05:17.990","Text":"This minus this is nothing because it should be,"},{"Start":"05:17.990 ","End":"05:23.030","Text":"10 minus 14 is minus 4x3,"},{"Start":"05:23.030 ","End":"05:33.060","Text":"17 minus 8 is 9x^3 and lower another term."},{"Start":"05:33.550 ","End":"05:39.860","Text":"This time I ask how many times does this go into this?"},{"Start":"05:39.860 ","End":"05:41.990","Text":"It\u0027s the leading terms always."},{"Start":"05:41.990 ","End":"05:45.140","Text":"Should have mentioned that to verify that really everything"},{"Start":"05:45.140 ","End":"05:48.266","Text":"is in decreasing order of exponent 4,"},{"Start":"05:48.266 ","End":"05:50.460","Text":"3, 2, and so on."},{"Start":"05:50.800 ","End":"05:54.360","Text":"This goes into this,"},{"Start":"05:54.770 ","End":"05:57.630","Text":"not a whole number."},{"Start":"05:57.630 ","End":"06:01.685","Text":"I mean, the x^2 goes into x^3 x times,"},{"Start":"06:01.685 ","End":"06:05.075","Text":"but 3 into 4 goes 4/3,"},{"Start":"06:05.075 ","End":"06:08.060","Text":"or rather minus 4/3 times,"},{"Start":"06:08.060 ","End":"06:10.715","Text":"so we write here,"},{"Start":"06:10.715 ","End":"06:15.385","Text":"minus 4/3 times x."},{"Start":"06:15.385 ","End":"06:18.450","Text":"Fraction are okay in the number part."},{"Start":"06:18.450 ","End":"06:22.350","Text":"The x\u0027s go in evenly and that can happen."},{"Start":"06:22.350 ","End":"06:27.030","Text":"Then multiply this by this,"},{"Start":"06:27.030 ","End":"06:33.690","Text":"so minus 4/3x times 3x^2 is minus 4x^3,"},{"Start":"06:33.690 ","End":"06:38.280","Text":"it has to be, the 3 cancels and it\u0027s minus 4."},{"Start":"06:38.280 ","End":"06:41.545","Text":"This times this,"},{"Start":"06:41.545 ","End":"06:45.785","Text":"I\u0027ll do that bit at the side just the number part."},{"Start":"06:45.785 ","End":"06:50.845","Text":"14 times 4/3,"},{"Start":"06:50.845 ","End":"06:52.925","Text":"let\u0027s even leave the minus out of this,"},{"Start":"06:52.925 ","End":"07:03.430","Text":"this would equal 14 times 4 is 56/3,"},{"Start":"07:03.430 ","End":"07:08.965","Text":"and it would be easier if this was a mixed number."},{"Start":"07:08.965 ","End":"07:15.105","Text":"3 into 56 goes 18 times and a remainder 2,"},{"Start":"07:15.105 ","End":"07:24.090","Text":"so what I get here is minus 18 and 2/3,"},{"Start":"07:24.090 ","End":"07:27.410","Text":"and x times x is x^2."},{"Start":"07:27.410 ","End":"07:31.975","Text":"Then minus 4/3 times 8."},{"Start":"07:31.975 ","End":"07:37.885","Text":"8 times 4/3 is 32 over 3,"},{"Start":"07:37.885 ","End":"07:40.585","Text":"which is 10 and 2/3."},{"Start":"07:40.585 ","End":"07:46.810","Text":"So this time it\u0027s"},{"Start":"07:46.810 ","End":"07:53.635","Text":"minus 10 and 2/3x."},{"Start":"07:53.635 ","End":"07:56.740","Text":"Again, subtract and see what we get."},{"Start":"07:56.740 ","End":"08:02.025","Text":"The first one always cancels."},{"Start":"08:02.025 ","End":"08:04.600","Text":"Minus minus is plus."},{"Start":"08:04.600 ","End":"08:11.020","Text":"So 27 and 2/3x^2."},{"Start":"08:11.020 ","End":"08:13.570","Text":"This takeaway this is a plus,"},{"Start":"08:13.570 ","End":"08:18.835","Text":"plus 20 and 2/3x."},{"Start":"08:18.835 ","End":"08:21.460","Text":"Let\u0027s drop the next term,"},{"Start":"08:21.460 ","End":"08:23.635","Text":"which is the last one also."},{"Start":"08:23.635 ","End":"08:28.195","Text":"Then we ask 3x^2"},{"Start":"08:28.195 ","End":"08:35.050","Text":"into the leading term here, 27 and 2/3x^2."},{"Start":"08:35.050 ","End":"08:38.470","Text":"Again, l need a side exercise,"},{"Start":"08:38.470 ","End":"08:45.205","Text":"27 and 2/3 over 3."},{"Start":"08:45.205 ","End":"08:47.920","Text":"The 27 and 2/3,"},{"Start":"08:47.920 ","End":"08:54.475","Text":"I can write as 27 times 3 is 81 plus 2 is 83 over 3."},{"Start":"08:54.475 ","End":"08:59.095","Text":"Over 3 means 83 over 9,"},{"Start":"08:59.095 ","End":"09:06.670","Text":"which means 9 into 83 goes 9 remainder 2, 9 and 2/9."},{"Start":"09:06.670 ","End":"09:08.980","Text":"It\u0027s not very nice numbers I must admit."},{"Start":"09:08.980 ","End":"09:17.200","Text":"Plus 9 and 2/9,"},{"Start":"09:17.200 ","End":"09:21.625","Text":"and this doesn\u0027t have any x with it because x^2 goes into x^2 evenly."},{"Start":"09:21.625 ","End":"09:25.690","Text":"When I multiply this by this,"},{"Start":"09:25.690 ","End":"09:27.790","Text":"I\u0027m not even going to check it."},{"Start":"09:27.790 ","End":"09:30.190","Text":"I\u0027m just going to assume that it\u0027s okay."},{"Start":"09:30.190 ","End":"09:33.920","Text":"It should come out to 27 and 2/3x^2."},{"Start":"09:36.600 ","End":"09:40.735","Text":"Actually, it makes sense because 9 times 3 is 27,"},{"Start":"09:40.735 ","End":"09:44.500","Text":"and 2/9 times 3 is 2/3, yeah, it looks right."},{"Start":"09:44.500 ","End":"09:47.950","Text":"Then 9 and 2/9 times 14,"},{"Start":"09:47.950 ","End":"09:50.515","Text":"a lot of fractions here,"},{"Start":"09:50.515 ","End":"09:56.480","Text":"9 and 2/9 is 83 over 9,"},{"Start":"09:56.580 ","End":"10:03.229","Text":"and times 14 is equal to,"},{"Start":"10:03.690 ","End":"10:07.120","Text":"well, I use the calculator."},{"Start":"10:07.120 ","End":"10:10.000","Text":"I did this times this over this,"},{"Start":"10:10.000 ","End":"10:13.810","Text":"and I got a 129.11111,"},{"Start":"10:13.810 ","End":"10:16.240","Text":"which means and a 9."},{"Start":"10:16.240 ","End":"10:20.544","Text":"It\u0027s a bit messy this exercise I admit,"},{"Start":"10:20.544 ","End":"10:22.735","Text":"but you could get such a thing."},{"Start":"10:22.735 ","End":"10:27.070","Text":"So that\u0027s this times this,"},{"Start":"10:27.070 ","End":"10:29.350","Text":"and that\u0027s how many x we have,"},{"Start":"10:29.350 ","End":"10:33.740","Text":"129 and 1/9x,"},{"Start":"10:34.800 ","End":"10:38.770","Text":"and 9 and 2/9 times 8."},{"Start":"10:38.770 ","End":"10:41.090","Text":"Let\u0027s see."},{"Start":"10:44.490 ","End":"10:50.960","Text":"Maybe it\u0027s easier to just multiply 9 times 8 is 72,"},{"Start":"10:51.300 ","End":"10:57.880","Text":"2/9 times 8 is 60 over 9,"},{"Start":"10:57.880 ","End":"11:00.805","Text":"but this is 1 and 7/9."},{"Start":"11:00.805 ","End":"11:04.870","Text":"So it\u0027s 73 and 7/9,"},{"Start":"11:04.870 ","End":"11:07.960","Text":"whichever way you want to do it."},{"Start":"11:07.960 ","End":"11:11.600","Text":"This is the ugly part of the exercise."},{"Start":"11:12.000 ","End":"11:17.380","Text":"Plus 73 and 7/9,"},{"Start":"11:17.380 ","End":"11:21.835","Text":"and hopefully I haven\u0027t made an arithmetical error somewhere."},{"Start":"11:21.835 ","End":"11:25.825","Text":"The general method is fine, but who knows?"},{"Start":"11:25.825 ","End":"11:27.910","Text":"Let\u0027s be optimistic."},{"Start":"11:27.910 ","End":"11:30.800","Text":"This into this."},{"Start":"11:31.170 ","End":"11:34.795","Text":"Sorry, we\u0027re subtracting."},{"Start":"11:34.795 ","End":"11:37.225","Text":"This minus this."},{"Start":"11:37.225 ","End":"11:39.595","Text":"This first part cancels,"},{"Start":"11:39.595 ","End":"11:47.440","Text":"and then we need 20 and 2/3 minus 129 and a 1/9. Let\u0027s see, 129."},{"Start":"11:47.440 ","End":"11:48.640","Text":"Let\u0027s subtract it the other way,"},{"Start":"11:48.640 ","End":"11:49.660","Text":"and then make it minus."},{"Start":"11:49.660 ","End":"11:57.520","Text":"129 and 1/9 minus 20 and 2/3."},{"Start":"11:57.520 ","End":"11:59.140","Text":"Here\u0027s a creative way of doing it."},{"Start":"11:59.140 ","End":"12:05.455","Text":"Let me write this as 128 and 10/9,"},{"Start":"12:05.455 ","End":"12:08.710","Text":"because it\u0027ll borrow one from here,"},{"Start":"12:08.710 ","End":"12:11.890","Text":"minus 20 and 2/3."},{"Start":"12:11.890 ","End":"12:13.960","Text":"I wanted the fraction part to be bigger."},{"Start":"12:13.960 ","End":"12:18.250","Text":"So 128 minus 20 is 108."},{"Start":"12:18.250 ","End":"12:20.860","Text":"Now all I need is 10/9 minus 2/3."},{"Start":"12:20.860 ","End":"12:23.815","Text":"But 2/3 is 6/9,"},{"Start":"12:23.815 ","End":"12:27.530","Text":"so I get 4/9, hopefully."},{"Start":"12:28.890 ","End":"12:31.975","Text":"That makes that."},{"Start":"12:31.975 ","End":"12:35.050","Text":"Now minus 108"},{"Start":"12:35.050 ","End":"12:42.685","Text":"and 4/9x."},{"Start":"12:42.685 ","End":"12:46.280","Text":"Then this minus this,"},{"Start":"12:46.830 ","End":"12:50.815","Text":"I\u0027ll do the subtraction the other way round."},{"Start":"12:50.815 ","End":"12:58.195","Text":"Let\u0027s see, 73 minus 24 is 49."},{"Start":"12:58.195 ","End":"13:05.650","Text":"This is the 7/9."},{"Start":"13:05.650 ","End":"13:06.970","Text":"I guess, If we haven\u0027t made a mistake,"},{"Start":"13:06.970 ","End":"13:11.530","Text":"this is the point at which we stop because when I say this into this,"},{"Start":"13:11.530 ","End":"13:15.850","Text":"it won\u0027t go anymore because this is a power of x."},{"Start":"13:15.850 ","End":"13:17.215","Text":"It\u0027s lower than this,"},{"Start":"13:17.215 ","End":"13:18.790","Text":"so we stop,"},{"Start":"13:18.790 ","End":"13:24.850","Text":"and then we just write the answer as just,"},{"Start":"13:24.850 ","End":"13:26.110","Text":"let me scroll down first,"},{"Start":"13:26.110 ","End":"13:27.445","Text":"get some more room."},{"Start":"13:27.445 ","End":"13:29.635","Text":"We write the answer as,"},{"Start":"13:29.635 ","End":"13:32.560","Text":"I did a copy, and here\u0027s the paste."},{"Start":"13:32.560 ","End":"13:36.325","Text":"I didn\u0027t want to waste time writing these numbers."},{"Start":"13:36.325 ","End":"13:40.330","Text":"This over this is equal to,"},{"Start":"13:40.330 ","End":"13:46.610","Text":"and now I\u0027m going to copy this bit here."},{"Start":"13:46.650 ","End":"13:52.670","Text":"Then what we want to do is write plus,"},{"Start":"13:53.010 ","End":"13:56.800","Text":"then the remainder, which is this,"},{"Start":"13:56.800 ","End":"13:59.210","Text":"l can do this manually."},{"Start":"13:59.790 ","End":"14:03.400","Text":"Well, since they\u0027re both minus,"},{"Start":"14:03.400 ","End":"14:05.800","Text":"I\u0027ll just change this to a minus,"},{"Start":"14:05.800 ","End":"14:07.870","Text":"then I can write these both as plus."},{"Start":"14:07.870 ","End":"14:13.000","Text":"It\u0027s a 108 and 4/9x,"},{"Start":"14:13.000 ","End":"14:22.375","Text":"better just separate the scratch work, back here."},{"Start":"14:22.375 ","End":"14:30.680","Text":"Minus is plus 49 and 7/9,"},{"Start":"14:30.680 ","End":"14:32.695","Text":"and 7/9 over,"},{"Start":"14:32.695 ","End":"14:35.930","Text":"and then this bit again."},{"Start":"14:36.390 ","End":"14:43.675","Text":"There we go. This is the answer,"},{"Start":"14:43.675 ","End":"14:47.419","Text":"and I\u0027ll highlight it."},{"Start":"14:47.520 ","End":"14:54.560","Text":"There we go. We are done."}],"ID":5250},{"Watched":false,"Name":"Exercise 2","Duration":"10m 33s","ChapterTopicVideoID":5250,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5250.jpeg","UploadDate":"2016-03-07T08:37:43.0430000","DurationForVideoObject":"PT10M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.745","Text":"In this exercise, we have a couple of long division problems,"},{"Start":"00:05.745 ","End":"00:08.520","Text":"long division of polynomials, that is,"},{"Start":"00:08.520 ","End":"00:13.540","Text":"and let\u0027s start with the first one."},{"Start":"00:13.640 ","End":"00:19.305","Text":"The bit on the top goes under the division sign."},{"Start":"00:19.305 ","End":"00:25.050","Text":"Let us first of all write x^5 minus 8x^4 plus"},{"Start":"00:25.050 ","End":"00:33.210","Text":"15x^3 plus 20x^2 minus 76x plus 48,"},{"Start":"00:33.210 ","End":"00:38.955","Text":"long one, divided by this,"},{"Start":"00:38.955 ","End":"00:43.260","Text":"x^2 minus x minus 6."},{"Start":"00:43.260 ","End":"00:47.720","Text":"Notice that both of them have been arranged in"},{"Start":"00:47.720 ","End":"00:51.800","Text":"decreasing order of exponents of powers of x from 5,"},{"Start":"00:51.800 ","End":"00:53.585","Text":"4, 3, 2, and so on."},{"Start":"00:53.585 ","End":"00:56.420","Text":"If not, we\u0027d have to rearrange it."},{"Start":"00:56.420 ","End":"01:02.515","Text":"We ask, how many times does x^2 go into x^5?"},{"Start":"01:02.515 ","End":"01:05.565","Text":"The answer, of course, is x^3,"},{"Start":"01:05.565 ","End":"01:07.560","Text":"5 minus 2 is 3."},{"Start":"01:07.560 ","End":"01:11.550","Text":"We multiply the x^3 by this and we"},{"Start":"01:11.550 ","End":"01:18.765","Text":"get x^5 minus x times x^3 is x^4,"},{"Start":"01:18.765 ","End":"01:22.845","Text":"and here minus 6x^3."},{"Start":"01:22.845 ","End":"01:25.550","Text":"Then we do a little subtraction here."},{"Start":"01:25.550 ","End":"01:29.520","Text":"I\u0027ll put a minus to remind me."},{"Start":"01:29.590 ","End":"01:31.970","Text":"When we do the subtraction,"},{"Start":"01:31.970 ","End":"01:34.370","Text":"we expect the first term to cancel,"},{"Start":"01:34.370 ","End":"01:36.875","Text":"otherwise we\u0027ve done something wrong."},{"Start":"01:36.875 ","End":"01:42.915","Text":"The next term is minus 8 take away minus 1,"},{"Start":"01:42.915 ","End":"01:47.190","Text":"which is minus 8 plus 1,"},{"Start":"01:47.190 ","End":"01:50.700","Text":"so it\u0027s minus 7x^4."},{"Start":"01:50.700 ","End":"01:53.145","Text":"We\u0027re doing a subtraction."},{"Start":"01:53.145 ","End":"01:59.810","Text":"15 subtract minus 6 gives 15 plus 6,"},{"Start":"01:59.810 ","End":"02:07.535","Text":"which is 21x^3 and drop another term, 20x^2."},{"Start":"02:07.535 ","End":"02:13.970","Text":"Once again we ask how many times does this go into this?"},{"Start":"02:13.970 ","End":"02:17.900","Text":"The answer here is minus 7x^2,"},{"Start":"02:17.900 ","End":"02:20.370","Text":"fairly clear to see."},{"Start":"02:23.320 ","End":"02:28.835","Text":"This part is always given a check because when you multiply out,"},{"Start":"02:28.835 ","End":"02:31.670","Text":"then we see if we made a mistake or not."},{"Start":"02:31.670 ","End":"02:34.490","Text":"What\u0027s minus 7x^2 times x^2?"},{"Start":"02:34.490 ","End":"02:37.205","Text":"Minus 7x^2 times x^2 is x^4."},{"Start":"02:37.205 ","End":"02:39.215","Text":"So minus 7x^4."},{"Start":"02:39.215 ","End":"02:41.645","Text":"That\u0027s okay and we continue,"},{"Start":"02:41.645 ","End":"02:48.375","Text":"minus 7x^2 times minus x is plus 7x^3."},{"Start":"02:48.375 ","End":"02:50.630","Text":"This and this also minus with minus,"},{"Start":"02:50.630 ","End":"02:53.970","Text":"so it\u0027s plus 42x^2."},{"Start":"02:54.140 ","End":"03:01.280","Text":"Again, I subtract and that\u0027s a line here."},{"Start":"03:01.280 ","End":"03:03.583","Text":"This minus this,"},{"Start":"03:03.583 ","End":"03:06.245","Text":"nothing, this minus this,"},{"Start":"03:06.245 ","End":"03:13.080","Text":"14x^3, this minus this minus 22x^2."},{"Start":"03:13.080 ","End":"03:17.520","Text":"Drop the next term, 76x."},{"Start":"03:17.520 ","End":"03:23.840","Text":"Now x^2 into 14x^3 Well,"},{"Start":"03:23.840 ","End":"03:26.030","Text":"x^2 into x^3 goes x times,"},{"Start":"03:26.030 ","End":"03:32.450","Text":"so what we need is plus 14x."},{"Start":"03:32.450 ","End":"03:34.385","Text":"Then, this times this."},{"Start":"03:34.385 ","End":"03:38.630","Text":"So 14x times x^2 is 14x^3."},{"Start":"03:38.630 ","End":"03:44.150","Text":"14x times minus x minus 14x^2."},{"Start":"03:44.150 ","End":"03:51.740","Text":"Then with this, 14 times 6 is 84."},{"Start":"03:51.740 ","End":"03:54.630","Text":"So minus 84x."},{"Start":"03:57.230 ","End":"04:00.075","Text":"Now do the subtraction,"},{"Start":"04:00.075 ","End":"04:02.295","Text":"this minus this, nothing,"},{"Start":"04:02.295 ","End":"04:10.255","Text":"this minus this, minus 8x^2."},{"Start":"04:10.255 ","End":"04:19.175","Text":"Then this minus this minus 76 plus 84 is plus 8x,"},{"Start":"04:19.175 ","End":"04:23.705","Text":"and then there\u0027s a plus 48."},{"Start":"04:23.705 ","End":"04:30.295","Text":"Now we ask how many times does x^2 go into minus 8x^2?"},{"Start":"04:30.295 ","End":"04:32.880","Text":"That\u0027s just minus 8,"},{"Start":"04:32.880 ","End":"04:34.970","Text":"the x^2 is already there."},{"Start":"04:34.970 ","End":"04:41.450","Text":"So this times this minus 8x^3 plus"},{"Start":"04:41.450 ","End":"04:50.115","Text":"8x minus 8 times minus 6 is plus 48."},{"Start":"04:50.115 ","End":"04:56.560","Text":"When we subtract nothing left, just zero."},{"Start":"04:56.560 ","End":"05:00.320","Text":"When this happens, that means that there is"},{"Start":"05:00.320 ","End":"05:05.375","Text":"no remainder and that this goes into this evenly,"},{"Start":"05:05.375 ","End":"05:12.230","Text":"and so the answer here is precisely what is written up here is"},{"Start":"05:12.230 ","End":"05:20.030","Text":"x^3 minus 7x^2 plus 14x minus 8, and no remainder."},{"Start":"05:20.030 ","End":"05:21.050","Text":"If there was a remainder,"},{"Start":"05:21.050 ","End":"05:27.060","Text":"I would write plus the remainder over x^2 minus x minus 6. But there isn\u0027t."},{"Start":"05:27.410 ","End":"05:31.505","Text":"This is the answer to part a."},{"Start":"05:31.505 ","End":"05:33.365","Text":"We still have part b."},{"Start":"05:33.365 ","End":"05:41.685","Text":"Let\u0027s scroll down to it, another division."},{"Start":"05:41.685 ","End":"05:43.970","Text":"This is already been arranged,"},{"Start":"05:43.970 ","End":"05:50.345","Text":"but we just have to note that it is decreasing order of powers of x,"},{"Start":"05:50.345 ","End":"05:52.189","Text":"and no missing terms."},{"Start":"05:52.189 ","End":"05:53.480","Text":"If there were missing terms,"},{"Start":"05:53.480 ","End":"05:57.380","Text":"we put in zero where as placeholders,"},{"Start":"05:57.380 ","End":"06:03.255","Text":"but so far we\u0027ve only had everything in the right order and nothing missing."},{"Start":"06:03.255 ","End":"06:06.095","Text":"Let\u0027s do this as a long division."},{"Start":"06:06.095 ","End":"06:17.405","Text":"Long division symbol is line like this and then horizontal line like so."},{"Start":"06:17.405 ","End":"06:19.775","Text":"I think I didn\u0027t leave enough room."},{"Start":"06:19.775 ","End":"06:23.040","Text":"Just move it over a bit."},{"Start":"06:23.650 ","End":"06:26.940","Text":"Now let\u0027s start copying,"},{"Start":"06:28.210 ","End":"06:33.175","Text":"x^5 minus 8x^4 plus 15x^3"},{"Start":"06:33.175 ","End":"06:41.070","Text":"plus 20x^2 minus 76 plus 48."},{"Start":"06:41.070 ","End":"06:46.010","Text":"Let\u0027s see, there\u0027s a minus and a minus sometimes."},{"Start":"06:46.010 ","End":"06:48.935","Text":"Get that wrong. No, it\u0027s fine."},{"Start":"06:48.935 ","End":"06:58.350","Text":"Over here, x^3 minus 2x^2 minus 5x plus 6."},{"Start":"06:58.350 ","End":"07:01.210","Text":"Check yet, copied."},{"Start":"07:02.710 ","End":"07:05.615","Text":"Look at the leading terms."},{"Start":"07:05.615 ","End":"07:08.080","Text":"This goes into this,"},{"Start":"07:08.080 ","End":"07:11.785","Text":"x^2 times 5 minus 3 is 2,"},{"Start":"07:11.785 ","End":"07:13.925","Text":"you know your rules of exponents."},{"Start":"07:13.925 ","End":"07:20.330","Text":"This times this is x^5 minus"},{"Start":"07:20.330 ","End":"07:24.965","Text":"2x^4 minus 5x^3"},{"Start":"07:24.965 ","End":"07:31.505","Text":"plus 6x^2,"},{"Start":"07:31.505 ","End":"07:41.430","Text":"subtract, separator line or whatever you call it, the equals line."},{"Start":"07:41.480 ","End":"07:44.625","Text":"This minus this is nothing,"},{"Start":"07:44.625 ","End":"07:46.050","Text":"minus 8 minus,"},{"Start":"07:46.050 ","End":"07:51.030","Text":"minus 2 is minus 6,15 minus,"},{"Start":"07:51.030 ","End":"07:54.690","Text":"minus 5 is 20."},{"Start":"07:54.690 ","End":"08:00.900","Text":"That\u0027s for x^3 and 20 minus 6 is 14x^2."},{"Start":"08:00.900 ","End":"08:03.210","Text":"I did make a mistake in the copying,"},{"Start":"08:03.210 ","End":"08:05.530","Text":"I left the x out."},{"Start":"08:06.400 ","End":"08:11.165","Text":"Sorry about that, but no harm done, not too late."},{"Start":"08:11.165 ","End":"08:16.380","Text":"Yeah, we dropped the minus 76x."},{"Start":"08:17.440 ","End":"08:21.620","Text":"Then again, I ask how many times does x^3"},{"Start":"08:21.620 ","End":"08:28.580","Text":"go into minus 6x^4?"},{"Start":"08:28.580 ","End":"08:36.330","Text":"That would be minus 6x times because x^3 into x^4 goes x."},{"Start":"08:37.030 ","End":"08:42.440","Text":"Now we multiply minus 6x times all of this."},{"Start":"08:42.440 ","End":"08:51.005","Text":"Minus 6x^4 minus 6 times minus 2 is 12x^3,"},{"Start":"08:51.005 ","End":"08:58.160","Text":"and then we get plus 30x^2 and"},{"Start":"08:58.160 ","End":"09:07.430","Text":"minus 36x, subtract."},{"Start":"09:07.430 ","End":"09:09.350","Text":"Let\u0027s see what we get."},{"Start":"09:09.350 ","End":"09:11.255","Text":"This minus this, nothing,"},{"Start":"09:11.255 ","End":"09:14.900","Text":"this minus this 8x^3,"},{"Start":"09:14.900 ","End":"09:18.075","Text":"14 minus 30 is"},{"Start":"09:18.075 ","End":"09:26.135","Text":"minus 16x^2,"},{"Start":"09:26.135 ","End":"09:36.120","Text":"and this minus 76 plus 36 is minus 40x."},{"Start":"09:39.800 ","End":"09:44.490","Text":"I almost forgot to drop the 48."},{"Start":"09:44.490 ","End":"09:53.190","Text":"Now, x^3(8x^3) is precisely eight times,"},{"Start":"09:53.190 ","End":"09:58.845","Text":"I put plus 8 here and then multiply out and get 8x^3,"},{"Start":"09:58.845 ","End":"10:02.700","Text":"8 times minus 2 is minus 16x^2,"},{"Start":"10:02.700 ","End":"10:06.209","Text":"8 times minus 5 is minus 40."},{"Start":"10:06.209 ","End":"10:10.390","Text":"You can see it, it comes out exactly the same as this."},{"Start":"10:10.390 ","End":"10:12.190","Text":"There is no remainder,"},{"Start":"10:12.190 ","End":"10:17.860","Text":"or a remainder of zero if you want and that means that this divides exactly."},{"Start":"10:17.860 ","End":"10:21.450","Text":"What we get is what we wrote here,"},{"Start":"10:21.450 ","End":"10:27.315","Text":"is x^2 minus 6x plus 8."},{"Start":"10:27.315 ","End":"10:30.920","Text":"This is the answer and this is part b,"},{"Start":"10:30.920 ","End":"10:33.600","Text":"so we are done."}],"ID":5251},{"Watched":false,"Name":"Exercise 3","Duration":"10m 6s","ChapterTopicVideoID":5251,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5251.jpeg","UploadDate":"2016-03-07T08:39:26.7570000","DurationForVideoObject":"PT10M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we have a couple of long division problems,"},{"Start":"00:04.500 ","End":"00:06.525","Text":"long division of polynomials, that is. Let\u0027s start with the first."},{"Start":"00:06.525 ","End":"00:14.320","Text":"Let\u0027s start by drawing the division sign."},{"Start":"00:14.360 ","End":"00:18.543","Text":"This part is the dividend,"},{"Start":"00:18.543 ","End":"00:20.115","Text":"and it goes here,"},{"Start":"00:20.115 ","End":"00:25.170","Text":"3x^3 plus 3x^2 plus 3x plus 1,"},{"Start":"00:25.170 ","End":"00:30.555","Text":"and this part called the divisor goes here, x plus 6."},{"Start":"00:30.555 ","End":"00:35.910","Text":"Then we are asked how many times does this go into this?"},{"Start":"00:35.910 ","End":"00:39.795","Text":"The answer is 3x^2."},{"Start":"00:39.795 ","End":"00:45.110","Text":"We also get to check on that when we multiply this by this or this whole thing,"},{"Start":"00:45.110 ","End":"00:48.650","Text":"we get 3x^2 times x is 3x^3,"},{"Start":"00:48.650 ","End":"00:51.710","Text":"which is supposed to come out like this."},{"Start":"00:51.710 ","End":"00:55.460","Text":"Then plus 6 times 3 is 18x^2,"},{"Start":"00:55.460 ","End":"00:59.150","Text":"and then we do a subtraction."},{"Start":"00:59.150 ","End":"01:02.760","Text":"This has to drop out, to cancel."},{"Start":"01:02.760 ","End":"01:07.950","Text":"3 minus 18 is minus 15x^2."},{"Start":"01:07.950 ","End":"01:11.860","Text":"Drop another term, 3x."},{"Start":"01:11.860 ","End":"01:18.625","Text":"Next we\u0027re asked how many times does x go into minus 15x^2."},{"Start":"01:18.625 ","End":"01:23.835","Text":"That would be minus 15 and then x,"},{"Start":"01:23.835 ","End":"01:27.560","Text":"and if we multiply this by the divisor,"},{"Start":"01:27.560 ","End":"01:36.810","Text":"we get minus 15x^2 minus 90x."},{"Start":"01:36.810 ","End":"01:38.760","Text":"We do a subtraction,"},{"Start":"01:38.760 ","End":"01:40.665","Text":"this minus this is nothing."},{"Start":"01:40.665 ","End":"01:45.375","Text":"3 minus minus 90 is 93x."},{"Start":"01:45.375 ","End":"01:47.680","Text":"Drop another term."},{"Start":"01:47.680 ","End":"01:53.240","Text":"Then ask how many times does x go into 93x?"},{"Start":"01:53.240 ","End":"01:57.425","Text":"The answer to that is 93 times."},{"Start":"01:57.425 ","End":"02:01.220","Text":"So multiplying out, we"},{"Start":"02:01.220 ","End":"02:07.590","Text":"get 93x plus,"},{"Start":"02:07.590 ","End":"02:09.570","Text":"I don\u0027t have my calculator with me, let\u0027s see,"},{"Start":"02:09.570 ","End":"02:11.805","Text":"93 times 6,"},{"Start":"02:11.805 ","End":"02:13.740","Text":"3 times 6 is 18,"},{"Start":"02:13.740 ","End":"02:14.955","Text":"8 carry 1,"},{"Start":"02:14.955 ","End":"02:18.465","Text":"9 times 6 is 54, so that\u0027s 55."},{"Start":"02:18.465 ","End":"02:26.200","Text":"So here we have 558, and then subtract."},{"Start":"02:26.890 ","End":"02:31.250","Text":"We also practice the multiplication without calculators."},{"Start":"02:31.250 ","End":"02:43.055","Text":"Good. Subtracting this minus this is nothing and this minus this is minus 557."},{"Start":"02:43.055 ","End":"02:45.905","Text":"That\u0027s known as the remainder."},{"Start":"02:45.905 ","End":"02:49.830","Text":"But when we write the answer,"},{"Start":"02:50.650 ","End":"02:53.165","Text":"we write it as,"},{"Start":"02:53.165 ","End":"02:54.770","Text":"first of all this,"},{"Start":"02:54.770 ","End":"03:04.920","Text":"which is 3x^2 minus 15x plus 93,"},{"Start":"03:04.920 ","End":"03:06.945","Text":"that\u0027s called the quotient by the way,"},{"Start":"03:06.945 ","End":"03:14.355","Text":"and then the remainder over the divisor, this over this."},{"Start":"03:14.355 ","End":"03:17.205","Text":"Instead of putting a plus I\u0027ll just put minus,"},{"Start":"03:17.205 ","End":"03:24.690","Text":"and then I\u0027ll put a positive 557 over x plus 6."},{"Start":"03:24.690 ","End":"03:27.965","Text":"This divided by this equals this."},{"Start":"03:27.965 ","End":"03:35.490","Text":"Now I\u0027ll just highlight that and then we can move on to part B."},{"Start":"03:36.910 ","End":"03:41.280","Text":"We begin with the division sign."},{"Start":"03:42.190 ","End":"03:51.150","Text":"This part goes underneath x^3 plus 12x^2 plus 10x plus 8,"},{"Start":"03:51.150 ","End":"03:53.830","Text":"and this part goes over here."},{"Start":"03:53.900 ","End":"03:58.930","Text":"4x^2 plus 3x plus 1."},{"Start":"03:58.930 ","End":"04:04.530","Text":"I want to know how many times this goes into this."},{"Start":"04:04.540 ","End":"04:08.535","Text":"X^2 goes into x^3, x times,"},{"Start":"04:08.535 ","End":"04:14.969","Text":"and 4 goes into 1, 1 1/4 times."},{"Start":"04:14.969 ","End":"04:16.360","Text":"We start with this,"},{"Start":"04:16.360 ","End":"04:19.910","Text":"then multiply this by this,"},{"Start":"04:19.910 ","End":"04:21.485","Text":"that\u0027s the divisor,"},{"Start":"04:21.485 ","End":"04:28.590","Text":"and we get a 1/4x times 4x^2 gives us x^3."},{"Start":"04:28.590 ","End":"04:31.930","Text":"I could put the 1 here just to make it match."},{"Start":"04:32.380 ","End":"04:39.860","Text":"A 1/4x times 3x is 3/4x^2,"},{"Start":"04:39.860 ","End":"04:45.120","Text":"and 1/4x times 1 is just a 1/4x."},{"Start":"04:45.120 ","End":"04:49.420","Text":"Now subtract, and let\u0027s see what we get."},{"Start":"04:49.420 ","End":"04:53.450","Text":"This minus this is nothing and that\u0027s how it\u0027s supposed to be."},{"Start":"04:53.450 ","End":"05:00.320","Text":"12 minus 3/4 is 11 1/4x^2."},{"Start":"05:00.320 ","End":"05:07.820","Text":"10 minus a 1/4 is 9 and 3/4x,"},{"Start":"05:07.820 ","End":"05:11.195","Text":"and then we drop the last term, 8."},{"Start":"05:11.195 ","End":"05:19.370","Text":"Now we ask how many times does 4x^2 go into 11 and 1/4x^2."},{"Start":"05:19.370 ","End":"05:21.769","Text":"Just need to do a fraction at the side."},{"Start":"05:21.769 ","End":"05:27.790","Text":"Let\u0027s see, 11 1/4 divided by 4."},{"Start":"05:27.790 ","End":"05:31.715","Text":"Let\u0027s multiply top and bottom by 4."},{"Start":"05:31.715 ","End":"05:39.375","Text":"If I multiply this by 4 I get 44 and 4 quarters, which is 45,"},{"Start":"05:39.375 ","End":"05:42.915","Text":"and multiply this by 4 is 16,"},{"Start":"05:42.915 ","End":"05:49.175","Text":"16 goes into 45 twice, that\u0027s 32,"},{"Start":"05:49.175 ","End":"05:51.725","Text":"and this 13 leftover,"},{"Start":"05:51.725 ","End":"05:54.810","Text":"it\u0027s 13 over 16,"},{"Start":"05:54.810 ","End":"06:05.040","Text":"so that\u0027s what I write here, 2 13/16."},{"Start":"06:05.040 ","End":"06:07.395","Text":"Then I multiply out."},{"Start":"06:07.395 ","End":"06:11.785","Text":"This is supposed to come out 11 1/4 actually,"},{"Start":"06:11.785 ","End":"06:13.220","Text":"didn\u0027t even check that."},{"Start":"06:13.220 ","End":"06:15.755","Text":"But let\u0027s see, just mentally,"},{"Start":"06:15.755 ","End":"06:17.510","Text":"2 times 16 is 32,"},{"Start":"06:17.510 ","End":"06:24.975","Text":"plus 13 is 45 over 16,"},{"Start":"06:24.975 ","End":"06:29.655","Text":"multiplied by 4 it\u0027s 45 over Fourier 11 1/4, that\u0027s right."},{"Start":"06:29.655 ","End":"06:35.835","Text":"Then 2 13/16 times 3."},{"Start":"06:35.835 ","End":"06:38.430","Text":"We\u0027ll take it from this form,"},{"Start":"06:38.430 ","End":"06:43.305","Text":"and then 45 over 16"},{"Start":"06:43.305 ","End":"06:51.030","Text":"times 3 is 45 times 3 is 135 over 16,"},{"Start":"06:51.030 ","End":"06:58.350","Text":"which is 16 goes into 128, 8 times,"},{"Start":"06:58.350 ","End":"07:00.630","Text":"and then the 7 leftover,"},{"Start":"07:00.630 ","End":"07:06.880","Text":"so plus 8 7/16x."},{"Start":"07:11.180 ","End":"07:22.200","Text":"Then this times this is just 2 13/16, subtract."},{"Start":"07:22.200 ","End":"07:26.730","Text":"What we get is this minus this is nothing."},{"Start":"07:26.730 ","End":"07:29.580","Text":"This minus this, let\u0027s see,"},{"Start":"07:29.580 ","End":"07:36.570","Text":"9 3/4 minus 8 7/16."},{"Start":"07:36.570 ","End":"07:41.570","Text":"1 way to do it is just to write this in terms of,"},{"Start":"07:41.570 ","End":"07:43.640","Text":"the fraction part is over 16,"},{"Start":"07:43.640 ","End":"07:50.510","Text":"so it\u0027s 9 12/16 minus 8 7/16."},{"Start":"07:50.510 ","End":"07:53.110","Text":"9 minus 8 is 1,"},{"Start":"07:53.110 ","End":"07:56.850","Text":"12 minus 7 is 5 over 16,"},{"Start":"07:56.850 ","End":"08:06.065","Text":"so that is 1 5/16x plus,"},{"Start":"08:06.065 ","End":"08:08.315","Text":"let\u0027s see, this minus this,"},{"Start":"08:08.315 ","End":"08:10.920","Text":"this you can do mentally."},{"Start":"08:11.830 ","End":"08:21.190","Text":"I could do 8 minus 2 is 6 and then subtract 13/16,"},{"Start":"08:21.190 ","End":"08:26.160","Text":"so it\u0027s 5 3/16."},{"Start":"08:26.160 ","End":"08:31.935","Text":"I subtracted 3 and got 5 and then added the 3 over 16 that was missing,"},{"Start":"08:31.935 ","End":"08:36.940","Text":"and that is the remainder."},{"Start":"08:37.430 ","End":"08:42.650","Text":"It\u0027s probably more customary to write"},{"Start":"08:42.650 ","End":"08:48.065","Text":"it in terms of improper fractions rather than mixed numbers."},{"Start":"08:48.065 ","End":"08:53.935","Text":"1 times 16 plus 5 is 21,"},{"Start":"08:53.935 ","End":"08:57.750","Text":"over 16 times x,"},{"Start":"08:57.750 ","End":"09:04.465","Text":"5 times 16 is 80 plus 3 is 83 over 16."},{"Start":"09:04.465 ","End":"09:07.040","Text":"You know what? I\u0027ll write the answer up here"},{"Start":"09:07.040 ","End":"09:16.680","Text":"as 4x^2 plus 3x plus 1,"},{"Start":"09:19.280 ","End":"09:24.230","Text":"plus the remainder over this."},{"Start":"09:24.230 ","End":"09:30.840","Text":"Let me rewrite this as 21x plus 83 over 16,"},{"Start":"09:30.840 ","End":"09:34.820","Text":"because then this over this is easier to write as"},{"Start":"09:34.820 ","End":"09:41.345","Text":"21x plus 83 over 16 over this,"},{"Start":"09:41.345 ","End":"09:51.720","Text":"so I write it over 16 times 4x^2 plus 3x plus 1."},{"Start":"09:52.210 ","End":"09:55.805","Text":"A bit messy but there it is."},{"Start":"09:55.805 ","End":"09:59.640","Text":"I\u0027ll highlight this answer."},{"Start":"10:02.770 ","End":"10:05.970","Text":"That\u0027s it, we\u0027re done."}],"ID":5252},{"Watched":false,"Name":"Exercise 4","Duration":"8m 13s","ChapterTopicVideoID":5252,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5252.jpeg","UploadDate":"2016-03-07T08:40:47.8270000","DurationForVideoObject":"PT8M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.565","Text":"In this exercise, we have a couple of long division of polynomials."},{"Start":"00:05.565 ","End":"00:09.315","Text":"Let\u0027s start with the first one of course."},{"Start":"00:09.315 ","End":"00:17.380","Text":"I start it by just writing the symbol for the long division."},{"Start":"00:18.080 ","End":"00:22.125","Text":"This pound, the divisor goes here,"},{"Start":"00:22.125 ","End":"00:25.320","Text":"x^2 plus 10x minus 9."},{"Start":"00:25.320 ","End":"00:28.245","Text":"This part called the dividend goes underneath here."},{"Start":"00:28.245 ","End":"00:37.920","Text":"X^4 plus 4x^3 plus 3x^2 minus 4x minus 3."},{"Start":"00:37.920 ","End":"00:41.050","Text":"We start by asking,"},{"Start":"00:41.050 ","End":"00:46.270","Text":"how many times does this leading term go into this leading term?"},{"Start":"00:46.270 ","End":"00:49.470","Text":"Through straightforward it\u0027s x^2."},{"Start":"00:49.470 ","End":"00:53.610","Text":"The x^2 here then multiply x^2 by the divisor."},{"Start":"00:53.610 ","End":"00:58.185","Text":"We get x^4 plus 10x times x^2"},{"Start":"00:58.185 ","End":"01:04.875","Text":"is 10x^3, then minus 9x^2."},{"Start":"01:04.875 ","End":"01:08.140","Text":"Then we do a subtraction."},{"Start":"01:08.290 ","End":"01:12.260","Text":"I\u0027ll put a minus here to show its subtraction."},{"Start":"01:12.260 ","End":"01:14.930","Text":"I could put also brackets,"},{"Start":"01:14.930 ","End":"01:17.470","Text":"but we know it\u0027s subtraction."},{"Start":"01:17.470 ","End":"01:20.559","Text":"X^4 minus x^4, nothing."},{"Start":"01:20.559 ","End":"01:22.595","Text":"That\u0027s how it should be."},{"Start":"01:22.595 ","End":"01:28.670","Text":"4x^3 minus 10x^3 is minus 6x^3."},{"Start":"01:28.670 ","End":"01:29.900","Text":"Here we have three minus,"},{"Start":"01:29.900 ","End":"01:34.980","Text":"minus 9 is plus 12x^2,"},{"Start":"01:34.980 ","End":"01:37.830","Text":"then we drop another term."},{"Start":"01:37.830 ","End":"01:42.170","Text":"Now again, we look at the leading coefficients and ask"},{"Start":"01:42.170 ","End":"01:46.860","Text":"how many times does x^2 go into minus 6x^3?"},{"Start":"01:47.150 ","End":"01:51.465","Text":"That would be minus 6x times."},{"Start":"01:51.465 ","End":"02:00.660","Text":"Again, we multiply minus 6x^3 minus 60x^2,"},{"Start":"02:00.660 ","End":"02:08.670","Text":"and then plus 54x"},{"Start":"02:08.670 ","End":"02:12.300","Text":"from the 6 times 9x here."},{"Start":"02:12.300 ","End":"02:14.655","Text":"Again a subtraction,"},{"Start":"02:14.655 ","End":"02:18.600","Text":"so we get here nothing."},{"Start":"02:18.600 ","End":"02:28.860","Text":"12 minus, minus 60 is 72x^2 minus 4,"},{"Start":"02:28.860 ","End":"02:32.650","Text":"minus 54, minus 58x."},{"Start":"02:32.650 ","End":"02:37.195","Text":"Now time to drop the 3 minus 3."},{"Start":"02:37.195 ","End":"02:44.200","Text":"Then x^2 into 72 x^2 goes just 72 times."},{"Start":"02:44.240 ","End":"02:47.400","Text":"Plus 72, and then we multiply"},{"Start":"02:47.400 ","End":"02:55.735","Text":"out 72x^2 plus"},{"Start":"02:55.735 ","End":"02:59.260","Text":"720x,"},{"Start":"02:59.260 ","End":"03:02.895","Text":"and minus 9 times 72."},{"Start":"03:02.895 ","End":"03:04.890","Text":"Just a second."},{"Start":"03:04.890 ","End":"03:10.590","Text":"72 times 9 is 188."},{"Start":"03:10.590 ","End":"03:13.305","Text":"Carry one 79, 63."},{"Start":"03:13.305 ","End":"03:22.370","Text":"That makes it 64. I make it that it\u0027s minus 648."},{"Start":"03:22.370 ","End":"03:29.885","Text":"Then subtract and we get this minus, this is nothing."},{"Start":"03:29.885 ","End":"03:33.620","Text":"This minus this would be"},{"Start":"03:33.620 ","End":"03:41.400","Text":"minus 778."},{"Start":"03:41.400 ","End":"03:43.850","Text":"X minus 3, minus,"},{"Start":"03:43.850 ","End":"03:50.470","Text":"minus 648 makes it plus 645."},{"Start":"03:50.470 ","End":"03:53.445","Text":"Then that\u0027s the remainder."},{"Start":"03:53.445 ","End":"04:02.310","Text":"The answer to this question is the quotient which is this bit."},{"Start":"04:02.310 ","End":"04:09.390","Text":"It\u0027s x^2 minus 6x plus 72."},{"Start":"04:09.390 ","End":"04:14.000","Text":"Then we add the remainder over this."},{"Start":"04:14.000 ","End":"04:18.525","Text":"Plus is a dividing line,"},{"Start":"04:18.525 ","End":"04:22.390","Text":"x^2 plus 10x minus 9 here."},{"Start":"04:23.030 ","End":"04:27.640","Text":"This minus 778x"},{"Start":"04:29.330 ","End":"04:37.950","Text":"plus 645."},{"Start":"04:37.950 ","End":"04:39.975","Text":"This is the answer."},{"Start":"04:39.975 ","End":"04:47.250","Text":"Then on to Part B. In Part B, something similar."},{"Start":"04:47.620 ","End":"04:49.910","Text":"In Part B, once again,"},{"Start":"04:49.910 ","End":"04:56.150","Text":"let\u0027s start with the long division sign."},{"Start":"04:59.450 ","End":"05:02.460","Text":"This part goes here."},{"Start":"05:02.460 ","End":"05:04.810","Text":"Thus, this is the dividend,"},{"Start":"05:04.810 ","End":"05:06.415","Text":"this is the divisor."},{"Start":"05:06.415 ","End":"05:12.390","Text":"We are asked, how many times does this leading term go into this leading term?"},{"Start":"05:12.390 ","End":"05:15.380","Text":"Clearly, the answer is 3x."},{"Start":"05:15.380 ","End":"05:22.045","Text":"I write that above this term and then multiply 3x by the divisor."},{"Start":"05:22.045 ","End":"05:28.285","Text":"We get 3x^4, then 3x^3,"},{"Start":"05:28.285 ","End":"05:37.055","Text":"21x^2, and then 3x."},{"Start":"05:37.055 ","End":"05:40.570","Text":"Then we do a subtraction."},{"Start":"05:41.650 ","End":"05:44.855","Text":"The first one always cancels."},{"Start":"05:44.855 ","End":"05:47.375","Text":"If it doesn\u0027t, then you\u0027ve done something wrong."},{"Start":"05:47.375 ","End":"05:53.480","Text":"Minus 12 minus 3 is minus 15x^3."},{"Start":"05:53.480 ","End":"06:01.100","Text":"Minus 21 is minus 13x^2."},{"Start":"06:01.100 ","End":"06:04.790","Text":"Minus 10 minus 3 is minus 13x,"},{"Start":"06:04.790 ","End":"06:10.835","Text":"and then drop the last one down. Once again."},{"Start":"06:10.835 ","End":"06:14.675","Text":"If we look at the leading terms, this and this,"},{"Start":"06:14.675 ","End":"06:23.040","Text":"and it goes in exactly minus 15 times already the next cubed just need the constant."},{"Start":"06:23.040 ","End":"06:31.230","Text":"Then minus 15 times this is minus 15x^3."},{"Start":"06:31.230 ","End":"06:34.090","Text":"Minus 15x^2."},{"Start":"06:34.540 ","End":"06:42.270","Text":"Minus 15 times 7 is minus 105x."},{"Start":"06:43.700 ","End":"06:50.295","Text":"Then just minus 15, again a subtraction."},{"Start":"06:50.295 ","End":"06:53.644","Text":"We get this minus, this is nothing."},{"Start":"06:53.644 ","End":"06:58.680","Text":"This minus, this is plus 15, so it\u0027s 2x^2."},{"Start":"07:00.020 ","End":"07:08.145","Text":"Then minus 13 plus 105 is plus 92x."},{"Start":"07:08.145 ","End":"07:12.300","Text":"Then 2 minus, minus 15 is 17."},{"Start":"07:12.300 ","End":"07:14.295","Text":"This is the remainder."},{"Start":"07:14.295 ","End":"07:20.250","Text":"We don\u0027t continue because the degree here is already smaller than the degree here."},{"Start":"07:20.450 ","End":"07:23.610","Text":"Here we have an x^2 and here we have an x^3,"},{"Start":"07:23.610 ","End":"07:26.595","Text":"so it won\u0027t possibly go in. That\u0027s where we stop."},{"Start":"07:26.595 ","End":"07:28.889","Text":"Then we say that the answer,"},{"Start":"07:28.889 ","End":"07:33.885","Text":"we start off with just copying the quotient."},{"Start":"07:33.885 ","End":"07:38.535","Text":"It\u0027s 3x minus 15."},{"Start":"07:38.535 ","End":"07:43.860","Text":"Then we put the remainder this bit over the divisor."},{"Start":"07:49.210 ","End":"07:54.900","Text":"2x^2 plus 92x plus 17."},{"Start":"07:54.900 ","End":"07:58.479","Text":"That\u0027s this over this."},{"Start":"07:59.060 ","End":"08:06.180","Text":"X^3 plus x^2 minus 7x plus 1,"},{"Start":"08:06.180 ","End":"08:08.005","Text":"and that\u0027s the answer."},{"Start":"08:08.005 ","End":"08:13.530","Text":"I shall highlight it and we are done with Part B, so that\u0027s it."}],"ID":5253},{"Watched":false,"Name":"Exercise 5","Duration":"4m 36s","ChapterTopicVideoID":5253,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5253.jpeg","UploadDate":"2016-03-07T08:41:33.8070000","DurationForVideoObject":"PT4M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"Yet another pair of long division of polynomials."},{"Start":"00:04.755 ","End":"00:08.745","Text":"Let\u0027s start with the first one, part a."},{"Start":"00:08.745 ","End":"00:16.382","Text":"As usual, we begin with writing the symbol for the long division,"},{"Start":"00:16.382 ","End":"00:20.775","Text":"then we write this part here and this part here."},{"Start":"00:20.775 ","End":"00:23.530","Text":"Just did that for you."},{"Start":"00:23.720 ","End":"00:28.320","Text":"We look at the leading coefficient"},{"Start":"00:28.320 ","End":"00:32.520","Text":"and ask how many times it goes into this leading coefficient."},{"Start":"00:32.520 ","End":"00:37.030","Text":"Clearly x^2 goes into x^3 x times."},{"Start":"00:37.120 ","End":"00:41.190","Text":"I multiply this x by the divisor"},{"Start":"00:41.190 ","End":"00:49.805","Text":"and I get x^3+3x^2-4x."},{"Start":"00:49.805 ","End":"00:53.650","Text":"Then we subtract and see what we get."},{"Start":"00:53.650 ","End":"00:55.674","Text":"This minus this is nothing,"},{"Start":"00:55.674 ","End":"00:57.490","Text":"how it should be,"},{"Start":"00:57.490 ","End":"01:04.443","Text":"x^2-3x^2 is -2x^2,"},{"Start":"01:04.443 ","End":"01:10.040","Text":"-10 -(-4) is -6x,"},{"Start":"01:10.040 ","End":"01:12.100","Text":"then drop down another term,"},{"Start":"01:12.100 ","End":"01:16.100","Text":"the last one in fact, and ask,"},{"Start":"01:16.100 ","End":"01:22.260","Text":"how many times does x^2 go into the leading term here."},{"Start":"01:22.260 ","End":"01:25.300","Text":"It\u0027s just -2."},{"Start":"01:25.730 ","End":"01:28.920","Text":"Put a -2 up here,"},{"Start":"01:28.920 ","End":"01:31.920","Text":"and then multiply -2 by this,"},{"Start":"01:31.920 ","End":"01:36.766","Text":"we go -2x^2-6x,"},{"Start":"01:36.766 ","End":"01:42.790","Text":"-2 times -4 is 8."},{"Start":"01:42.790 ","End":"01:45.370","Text":"How nice, there is no remainder,"},{"Start":"01:45.370 ","End":"01:48.346","Text":"I write a 0 there."},{"Start":"01:48.346 ","End":"01:50.133","Text":"Goes in evenly."},{"Start":"01:50.133 ","End":"01:56.090","Text":"So the answer to this is an even x-2."},{"Start":"01:56.280 ","End":"01:59.335","Text":"This part is called the quotient by the way."},{"Start":"01:59.335 ","End":"02:04.055","Text":"Anyway, that\u0027s the answer to part a and so on to part b."},{"Start":"02:04.055 ","End":"02:09.560","Text":"Who knows, maybe we\u0027ll get an even division here also."},{"Start":"02:09.970 ","End":"02:14.150","Text":"If time I already set the thing up,"},{"Start":"02:14.150 ","End":"02:16.205","Text":"we put the dividing sign,"},{"Start":"02:16.205 ","End":"02:19.490","Text":"and here we put the dividend,"},{"Start":"02:19.490 ","End":"02:20.570","Text":"which is this part,"},{"Start":"02:20.570 ","End":"02:21.860","Text":"and here the divisor,"},{"Start":"02:21.860 ","End":"02:23.480","Text":"which is this part."},{"Start":"02:23.480 ","End":"02:26.990","Text":"Now we look at leading terms and ask,"},{"Start":"02:26.990 ","End":"02:31.175","Text":"how many times is 2x^2 go into 2x^4?"},{"Start":"02:31.175 ","End":"02:35.120","Text":"The answer is x^2 times."},{"Start":"02:35.120 ","End":"02:37.520","Text":"Multiply x^2 by this,"},{"Start":"02:37.520 ","End":"02:45.750","Text":"we\u0027ve got 2x^4 minus 3x^3 plus x^2."},{"Start":"02:45.750 ","End":"02:47.550","Text":"Now a subtraction,"},{"Start":"02:47.550 ","End":"02:50.090","Text":"and see what we get."},{"Start":"02:50.090 ","End":"02:51.950","Text":"This minus this cancels,"},{"Start":"02:51.950 ","End":"02:57.850","Text":"-x^3 -(-3x^3) is"},{"Start":"02:57.850 ","End":"03:04.622","Text":"-1+3 is 2x^3,"},{"Start":"03:04.622 ","End":"03:13.730","Text":"-14-1-15x^2, and then drop the next one down, so +19x."},{"Start":"03:13.730 ","End":"03:19.530","Text":"This time I ask how many times does 2x^2 go into 2x^3?"},{"Start":"03:19.530 ","End":"03:23.262","Text":"The answer is an even x times,"},{"Start":"03:23.262 ","End":"03:24.360","Text":"x times this, 2x^3-3x^2,"},{"Start":"03:24.360 ","End":"03:32.655","Text":"and then +1 means +x, subtract."},{"Start":"03:32.655 ","End":"03:41.460","Text":"This is what we get, this cancels -15 -(-3) is -12x^2"},{"Start":"03:41.460 ","End":"03:50.800","Text":"and 19-1 is 18."},{"Start":"03:50.800 ","End":"03:53.140","Text":"Then drop the next term,"},{"Start":"03:53.140 ","End":"03:54.925","Text":"which is the last also."},{"Start":"03:54.925 ","End":"04:02.380","Text":"Now we are asked how many times does 2x^2 go into -12x^2?"},{"Start":"04:02.380 ","End":"04:04.560","Text":"The x^2 goes in evenly,"},{"Start":"04:04.560 ","End":"04:08.070","Text":"2 into -12 is -6 times,"},{"Start":"04:08.070 ","End":"04:11.588","Text":"so I multiply by this -12x^2,"},{"Start":"04:11.588 ","End":"04:16.260","Text":"-6 times -3 is +18x,"},{"Start":"04:16.260 ","End":"04:19.410","Text":"-6 times 1 is -6."},{"Start":"04:19.410 ","End":"04:23.870","Text":"Once again, it comes out evenly, there\u0027s no remainder."},{"Start":"04:23.870 ","End":"04:27.980","Text":"The answer is simply the quotient that\u0027s written above,"},{"Start":"04:27.980 ","End":"04:33.790","Text":"which is x^2+x-6,"},{"Start":"04:33.790 ","End":"04:36.560","Text":"and I\u0027ll highlight it."}],"ID":5254},{"Watched":false,"Name":"Exercise 6","Duration":"13m 27s","ChapterTopicVideoID":5254,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5254.jpeg","UploadDate":"2016-03-07T08:43:51.3430000","DurationForVideoObject":"PT13M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"This exercise, we have a pair of problems,"},{"Start":"00:02.550 ","End":"00:06.345","Text":"each of them is a long division of polynomials,"},{"Start":"00:06.345 ","End":"00:13.245","Text":"and they at first glance appear quite routine but we\u0027ll see not completely."},{"Start":"00:13.245 ","End":"00:16.215","Text":"Some points to look out for."},{"Start":"00:16.215 ","End":"00:22.320","Text":"For example, in the first one the denominator is not written as a polynomial,"},{"Start":"00:22.320 ","End":"00:23.880","Text":"we have to expand."},{"Start":"00:23.880 ","End":"00:29.880","Text":"Fair enough. The second thing to notice is that the exponent is not in decreasing order,"},{"Start":"00:29.880 ","End":"00:33.720","Text":"it\u0027s something that usually is given in that form but not always,"},{"Start":"00:33.720 ","End":"00:35.737","Text":"we have a 6, a 5, and then a 3 and a 4,"},{"Start":"00:35.737 ","End":"00:38.935","Text":"and a 1 and a 2 at the end."},{"Start":"00:38.935 ","End":"00:41.825","Text":"We\u0027re going to have to do some rearranging."},{"Start":"00:41.825 ","End":"00:48.615","Text":"As usual, we\u0027ll begin by writing the division sign."},{"Start":"00:48.615 ","End":"00:51.365","Text":"Inside we\u0027ll copy this but in the right order."},{"Start":"00:51.365 ","End":"00:55.960","Text":"So 2x^6, so far so good."},{"Start":"00:55.960 ","End":"00:58.560","Text":"Minus 13x^5."},{"Start":"00:58.560 ","End":"01:02.295","Text":"Then we need the x^4 term,"},{"Start":"01:02.295 ","End":"01:05.340","Text":"so plus 31x^4,"},{"Start":"01:05.340 ","End":"01:09.780","Text":"and then minus 31x^3."},{"Start":"01:09.780 ","End":"01:13.860","Text":"Then, let\u0027s see,"},{"Start":"01:13.860 ","End":"01:18.323","Text":"we need the plus 7x^2,"},{"Start":"01:18.323 ","End":"01:21.060","Text":"and then the plus 8x,"},{"Start":"01:21.060 ","End":"01:23.505","Text":"and then minus 4."},{"Start":"01:23.505 ","End":"01:29.210","Text":"That\u0027s 1 thing to watch out for make sure the exponents are in decreasing order."},{"Start":"01:29.210 ","End":"01:32.810","Text":"Next thing here is that we just have to expand this,"},{"Start":"01:32.810 ","End":"01:36.740","Text":"we can do this in our heads 2x times x is"},{"Start":"01:36.740 ","End":"01:43.733","Text":"2x^2 plus x minus 4x is minus 3x,"},{"Start":"01:43.733 ","End":"01:47.295","Text":"and 1 times minus 2 is minus 2."},{"Start":"01:47.295 ","End":"01:52.205","Text":"Now everything\u0027s in standard form and we can proceed."},{"Start":"01:52.205 ","End":"01:54.215","Text":"First thing is we ask,"},{"Start":"01:54.215 ","End":"01:58.700","Text":"how many times does this leading term go into this leading term?"},{"Start":"01:58.700 ","End":"02:00.259","Text":"The 2 goes into 2 once,"},{"Start":"02:00.259 ","End":"02:06.185","Text":"and x^2 into x^6 goes x^4 times,"},{"Start":"02:06.185 ","End":"02:12.170","Text":"x^4 times all this is 2x^6,"},{"Start":"02:12.170 ","End":"02:21.390","Text":"minus 3x^5, and then minus 2x^4."},{"Start":"02:21.390 ","End":"02:23.535","Text":"We do a subtraction here."},{"Start":"02:23.535 ","End":"02:25.385","Text":"Let\u0027s see what we get."},{"Start":"02:25.385 ","End":"02:27.890","Text":"This minus this, nothing,"},{"Start":"02:27.890 ","End":"02:33.570","Text":"this minus this is minus 10x^5,"},{"Start":"02:33.570 ","End":"02:40.715","Text":"and 31 minus minus 2 is 33x^4."},{"Start":"02:40.715 ","End":"02:45.125","Text":"Drop down the next term, 31x^3."},{"Start":"02:45.125 ","End":"02:50.920","Text":"Now we ask how many times does 2x^2 go into minus 10x^5."},{"Start":"02:50.920 ","End":"02:54.710","Text":"2 into minus 10 goes minus 5 times,"},{"Start":"02:54.710 ","End":"02:59.870","Text":"and x^2 into x^5 goes x^3 times."},{"Start":"02:59.870 ","End":"03:05.165","Text":"Multiply out and we\u0027ve got minus 10x^5,"},{"Start":"03:05.165 ","End":"03:15.690","Text":"plus 15x^4, and plus 10x^3."},{"Start":"03:15.690 ","End":"03:18.728","Text":"Again, we subtract,"},{"Start":"03:18.728 ","End":"03:21.360","Text":"and what we get is this minus, this is nothing,"},{"Start":"03:21.360 ","End":"03:27.510","Text":"33 minus 15 is 18x^4."},{"Start":"03:27.510 ","End":"03:31.140","Text":"Then this minus this minus 41x^3."},{"Start":"03:31.140 ","End":"03:36.030","Text":"Drop down the next 7x^2."},{"Start":"03:36.030 ","End":"03:42.150","Text":"2x^2 into 18x^4,"},{"Start":"03:42.150 ","End":"03:44.070","Text":"2 into 18 goes 9,"},{"Start":"03:44.070 ","End":"03:51.450","Text":"and that\u0027s x^2, so plus 9x^2."},{"Start":"03:51.450 ","End":"03:53.910","Text":"Then multiply out 9 times 2,"},{"Start":"03:53.910 ","End":"03:56.745","Text":"we get 18x^4,"},{"Start":"03:56.745 ","End":"04:03.315","Text":"minus 27x^3,"},{"Start":"04:03.315 ","End":"04:08.175","Text":"and then minus 18x^2."},{"Start":"04:08.175 ","End":"04:12.150","Text":"Again, subtract, this cancels."},{"Start":"04:12.150 ","End":"04:22.380","Text":"Minus 41 minus minus 27 is minus 14x^3."},{"Start":"04:22.380 ","End":"04:25.380","Text":"7 takeaway minus 18,"},{"Start":"04:25.380 ","End":"04:29.760","Text":"7 plus 18, 25x^2."},{"Start":"04:29.760 ","End":"04:32.880","Text":"We drop the 8x from here."},{"Start":"04:32.880 ","End":"04:43.104","Text":"2x^2 into minus 14x^3 is minus 7x, multiply out,"},{"Start":"04:43.104 ","End":"04:48.215","Text":"that minus 7 x times this, minus 14x^3,"},{"Start":"04:48.215 ","End":"04:56.370","Text":"plus 21x^2 plus 14x."},{"Start":"04:56.380 ","End":"04:59.675","Text":"Again subtract, this cancels,"},{"Start":"04:59.675 ","End":"05:04.165","Text":"25 minus 21 is 4x^2."},{"Start":"05:04.165 ","End":"05:08.760","Text":"This minus this, minus 6x,"},{"Start":"05:08.760 ","End":"05:11.775","Text":"and then minus 4."},{"Start":"05:11.775 ","End":"05:15.547","Text":"This goes into this,"},{"Start":"05:15.547 ","End":"05:17.120","Text":"2 into 4 goes twice,"},{"Start":"05:17.120 ","End":"05:20.890","Text":"x^2 is just x^2,"},{"Start":"05:21.170 ","End":"05:24.460","Text":"so plus 2 here."},{"Start":"05:24.460 ","End":"05:27.425","Text":"Then multiply out,"},{"Start":"05:27.425 ","End":"05:33.700","Text":"4x^2 minus 6x minus 4,"},{"Start":"05:33.700 ","End":"05:37.565","Text":"and it came out nice and even, no remainder."},{"Start":"05:37.565 ","End":"05:42.125","Text":"Which means that the quotient here is the answer to this thing,"},{"Start":"05:42.125 ","End":"05:50.925","Text":"is equal to x^4 minus 5x^3 plus 9x^2, I\u0027m copying from here,"},{"Start":"05:50.925 ","End":"05:54.675","Text":"minus 7x plus 2,"},{"Start":"05:54.675 ","End":"05:57.000","Text":"and it deserves to be highlighted,"},{"Start":"05:57.000 ","End":"06:02.440","Text":"that\u0027s the answer for part a, onto part b."},{"Start":"06:03.380 ","End":"06:08.060","Text":"In this one, in part b there\u0027s a few things to notice."},{"Start":"06:08.060 ","End":"06:13.445","Text":"The denominator will have to expand to get it as a regular polynomial,"},{"Start":"06:13.445 ","End":"06:15.580","Text":"and in the numerator,"},{"Start":"06:15.580 ","End":"06:19.900","Text":"not only are the exponents not in decreasing order."},{"Start":"06:19.900 ","End":"06:23.390","Text":"For example, I have x^3,"},{"Start":"06:23.390 ","End":"06:28.335","Text":"then x^2, so something is wrong here."},{"Start":"06:28.335 ","End":"06:30.120","Text":"But also there\u0027s missing exponents,"},{"Start":"06:30.120 ","End":"06:34.290","Text":"there is no x^5 or x^6,"},{"Start":"06:34.290 ","End":"06:40.898","Text":"so that\u0027s something else we have to watch out for."},{"Start":"06:40.898 ","End":"06:43.810","Text":"But these are just things that, not difficult,"},{"Start":"06:43.810 ","End":"06:46.730","Text":"you just have to be alert to constantly check and not to"},{"Start":"06:46.730 ","End":"06:50.765","Text":"assume that they\u0027re always given in decreasing order with no gaps."},{"Start":"06:50.765 ","End":"06:56.810","Text":"I\u0027ll write the division line like this,"},{"Start":"06:56.810 ","End":"06:59.250","Text":"and a line like this."},{"Start":"06:59.620 ","End":"07:04.740","Text":"Over here I\u0027ll write x minus 1 squared,"},{"Start":"07:04.740 ","End":"07:10.620","Text":"which is, if you computed x^2 minus 2x plus 1."},{"Start":"07:10.620 ","End":"07:14.555","Text":"Here I\u0027m going to copy this with a couple of changes."},{"Start":"07:14.555 ","End":"07:16.380","Text":"Let\u0027s see, 2x^7."},{"Start":"07:16.380 ","End":"07:19.560","Text":"Now there is no x^6,"},{"Start":"07:19.560 ","End":"07:21.860","Text":"you can either leave it blank or write zeros,"},{"Start":"07:21.860 ","End":"07:23.240","Text":"I\u0027m going to choose to write 0,"},{"Start":"07:23.240 ","End":"07:25.385","Text":"some people just leave a blank."},{"Start":"07:25.385 ","End":"07:29.260","Text":"I\u0027m going to write plus 0x^6."},{"Start":"07:29.260 ","End":"07:34.070","Text":"You choose, if you think it\u0027s neater with a blank, that\u0027s fine too."},{"Start":"07:34.070 ","End":"07:37.190","Text":"Even x^5 is missing."},{"Start":"07:37.190 ","End":"07:40.685","Text":"I put 0x^5,"},{"Start":"07:40.685 ","End":"07:46.015","Text":"x^4 is next, 25 of them."},{"Start":"07:46.015 ","End":"07:49.800","Text":"Then 20x^3."},{"Start":"07:50.710 ","End":"07:56.835","Text":"Then this one minus a 170x^2."},{"Start":"07:56.835 ","End":"08:01.380","Text":"Then the x plus 166x,"},{"Start":"08:01.380 ","End":"08:04.915","Text":"and then minus 43."},{"Start":"08:04.915 ","End":"08:07.670","Text":"Looks like we have a long one this time."},{"Start":"08:07.670 ","End":"08:17.250","Text":"Never mind."},{"Start":"08:17.250 ","End":"08:17.251","Text":"Oops. I notice there 2 instead of 20. Can we fix that? We\u0027re okay now."},{"Start":"08:17.251 ","End":"08:19.245","Text":"We look at the leading terms,"},{"Start":"08:19.245 ","End":"08:21.650","Text":"see how many times this goes into this."},{"Start":"08:21.650 ","End":"08:24.020","Text":"It will be 2, and it\u0027ll be x^5,"},{"Start":"08:24.020 ","End":"08:26.650","Text":"7 minus 2 is 5."},{"Start":"08:26.650 ","End":"08:30.030","Text":"I write 2x^5 here,"},{"Start":"08:30.030 ","End":"08:32.910","Text":"and then multiply by the divisor,"},{"Start":"08:32.910 ","End":"08:34.860","Text":"so I get 2x^7,"},{"Start":"08:34.860 ","End":"08:44.990","Text":"minus 4x^6 plus 2x^5,"},{"Start":"08:44.990 ","End":"08:48.625","Text":"and subtract."},{"Start":"08:48.625 ","End":"08:51.690","Text":"This minus this cancels,"},{"Start":"08:51.690 ","End":"08:55.875","Text":"0 minus minus 4 is 4x^6,"},{"Start":"08:55.875 ","End":"08:58.770","Text":"and 0 minus 2 is minus 2x^5,"},{"Start":"08:58.770 ","End":"09:03.580","Text":"and then we bring down the next one."},{"Start":"09:06.370 ","End":"09:11.765","Text":"This time the leading coefficient is 4x^6,"},{"Start":"09:11.765 ","End":"09:14.075","Text":"and x^2 goes into it,"},{"Start":"09:14.075 ","End":"09:19.340","Text":"4x^4 times, and multiplying gives us"},{"Start":"09:19.340 ","End":"09:27.975","Text":"4x^6 minus 8x^5 plus 4x^4."},{"Start":"09:27.975 ","End":"09:31.620","Text":"Subtract. This cancels,"},{"Start":"09:31.620 ","End":"09:39.450","Text":"minus 2 minus minus 8 is 6x^5, plus 21x^4."},{"Start":"09:39.450 ","End":"09:43.910","Text":"Bring another term down."},{"Start":"09:43.910 ","End":"09:47.675","Text":"This time we have 6x^5,"},{"Start":"09:47.675 ","End":"09:55.930","Text":"and x^2 goes into it 6x^3 times,"},{"Start":"09:55.930 ","End":"10:01.650","Text":"and then we\u0027ve got 6x^5 minus"},{"Start":"10:01.650 ","End":"10:07.492","Text":"12x^4 plus 6x^3,"},{"Start":"10:07.492 ","End":"10:09.025","Text":"from this times this."},{"Start":"10:09.025 ","End":"10:11.350","Text":"Again subtract."},{"Start":"10:11.350 ","End":"10:14.395","Text":"This first one always cancels,"},{"Start":"10:14.395 ","End":"10:18.670","Text":"21 less minus 12 is 33,"},{"Start":"10:18.670 ","End":"10:24.490","Text":"20 less 6 is 14."},{"Start":"10:24.490 ","End":"10:29.400","Text":"That\u0027s x^3 or x^3."},{"Start":"10:29.400 ","End":"10:34.350","Text":"Drop another term. Minus 170x^2."},{"Start":"10:34.350 ","End":"10:42.899","Text":"Now we ask how many times x^2 goes into 33x^4,"},{"Start":"10:42.899 ","End":"10:48.690","Text":"so it\u0027s 33x^2 times,"},{"Start":"10:48.690 ","End":"10:53.020","Text":"and then it gives us 33x^4"},{"Start":"10:53.020 ","End":"11:00.570","Text":"minus 66x^3 plus 33x^2."},{"Start":"11:00.570 ","End":"11:04.259","Text":"Subtract, put a minus sign here."},{"Start":"11:04.259 ","End":"11:05.875","Text":"This minus this cancels."},{"Start":"11:05.875 ","End":"11:11.965","Text":"14 minus minus 66 gives me 80x^3,"},{"Start":"11:11.965 ","End":"11:20.900","Text":"this minus this makes it minus 203x^2."},{"Start":"11:21.020 ","End":"11:27.960","Text":"We can drop the plus 166x from up there."},{"Start":"11:27.960 ","End":"11:30.825","Text":"Leading term, 80x^3,"},{"Start":"11:30.825 ","End":"11:36.030","Text":"x^2 goes into it 80x times,"},{"Start":"11:36.030 ","End":"11:46.290","Text":"so it\u0027s 80x^3 minus 160x^2,"},{"Start":"11:46.290 ","End":"11:47.850","Text":"that\u0027s from 80 times 2."},{"Start":"11:47.850 ","End":"11:54.525","Text":"Plus 80x."},{"Start":"11:54.525 ","End":"11:57.960","Text":"Let\u0027s see, subtracting."},{"Start":"11:57.960 ","End":"12:02.745","Text":"We shall get, this cancels,"},{"Start":"12:02.745 ","End":"12:06.960","Text":"minus this minus minus this was"},{"Start":"12:06.960 ","End":"12:17.010","Text":"still minus 43 down, minus 43x^2."},{"Start":"12:17.010 ","End":"12:23.760","Text":"This minus this plus 86x,"},{"Start":"12:23.760 ","End":"12:29.055","Text":"and then minus 43."},{"Start":"12:29.055 ","End":"12:36.765","Text":"x^2 goes into minus 43x^2 minus 43 times,"},{"Start":"12:36.765 ","End":"12:41.310","Text":"so minus 43x^2,"},{"Start":"12:41.310 ","End":"12:43.155","Text":"minus 43 minus 2,"},{"Start":"12:43.155 ","End":"12:50.085","Text":"plus 86x, and then minus 43."},{"Start":"12:50.085 ","End":"12:53.265","Text":"We subtract, it comes out evenly,"},{"Start":"12:53.265 ","End":"12:55.380","Text":"there is no remainder,"},{"Start":"12:55.380 ","End":"13:02.614","Text":"and that means that this quotient at the top here is the answer."},{"Start":"13:02.614 ","End":"13:06.559","Text":"I write this as,"},{"Start":"13:06.559 ","End":"13:08.590","Text":"just copy from here,"},{"Start":"13:08.590 ","End":"13:13.040","Text":"and I\u0027m going to literally copy-paste, why not?"},{"Start":"13:13.040 ","End":"13:17.875","Text":"This is our answer that I copied from here."},{"Start":"13:17.875 ","End":"13:27.030","Text":"I\u0027ll highlight it and say that we are done with the whole exercise."}],"ID":5255},{"Watched":false,"Name":"Exercise 7","Duration":"6m 7s","ChapterTopicVideoID":5255,"CourseChapterTopicPlaylistID":56152,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5255.jpeg","UploadDate":"2016-03-07T08:44:49.5270000","DurationForVideoObject":"PT6M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.590 ","End":"00:07.620","Text":"Here, we have another pair of calculations with long division of polynomials,"},{"Start":"00:07.620 ","End":"00:09.510","Text":"but with a difference."},{"Start":"00:09.510 ","End":"00:12.090","Text":"Each of them involves a parameter."},{"Start":"00:12.090 ","End":"00:14.715","Text":"We have here the parameter a,"},{"Start":"00:14.715 ","End":"00:17.295","Text":"and here the parameter b."},{"Start":"00:17.295 ","End":"00:23.040","Text":"It\u0027s not much more difficult so let\u0027s just see how we tackle"},{"Start":"00:23.040 ","End":"00:29.935","Text":"and see what difference it makes if there\u0027s a parameter beginning with Part A."},{"Start":"00:29.935 ","End":"00:32.540","Text":"The setup is the same."},{"Start":"00:32.540 ","End":"00:37.470","Text":"We write division."},{"Start":"00:38.020 ","End":"00:43.850","Text":"Here, we write the divisor, x minus a."},{"Start":"00:43.850 ","End":"00:47.120","Text":"Here we write the dividend, that\u0027s the numerator."},{"Start":"00:47.120 ","End":"00:55.110","Text":"But also note that there is a missing x"},{"Start":"00:55.110 ","End":"00:59.120","Text":"squared so we write it as x to"},{"Start":"00:59.120 ","End":"01:04.850","Text":"the fourth minus ax cubed and then either a blank as some people do,"},{"Start":"01:04.850 ","End":"01:10.070","Text":"but I prefer to write a 0 x^2."},{"Start":"01:10.070 ","End":"01:15.496","Text":"It really emphasizes that it\u0027s missing and you can see what\u0027s missing."},{"Start":"01:15.496 ","End":"01:21.545","Text":"It\u0027s the x^2+x-a."},{"Start":"01:21.545 ","End":"01:25.385","Text":"Then we look at the leading coefficients,"},{"Start":"01:25.385 ","End":"01:27.990","Text":"the x, and the x 4th."},{"Start":"01:27.990 ","End":"01:31.130","Text":"This goes into this,"},{"Start":"01:31.130 ","End":"01:36.650","Text":"x^3 times multiply the x cubed by x and"},{"Start":"01:36.650 ","End":"01:44.855","Text":"get x to the fourth and by whole thing so we get minus ax cubed."},{"Start":"01:44.855 ","End":"01:47.510","Text":"Then we subtract."},{"Start":"01:47.510 ","End":"01:51.320","Text":"This time there\u0027s nothing left here."},{"Start":"01:51.320 ","End":"01:54.810","Text":"We have to bring down more terms."},{"Start":"01:55.730 ","End":"02:01.490","Text":"The 0 \u0027s not going to be any good so we just bring down all the rest of it,"},{"Start":"02:01.490 ","End":"02:06.820","Text":"0x squared plus x minus a."},{"Start":"02:06.820 ","End":"02:11.700","Text":"The 0 is useless."},{"Start":"02:11.700 ","End":"02:13.680","Text":"I mean, we don\u0027t need that."},{"Start":"02:13.680 ","End":"02:16.625","Text":"Really we\u0027re focusing on the x minus a bit,"},{"Start":"02:16.625 ","End":"02:20.935","Text":"x into x goes once."},{"Start":"02:20.935 ","End":"02:26.874","Text":"We write the 1 over here."},{"Start":"02:26.874 ","End":"02:33.407","Text":"Because this is the leading term."},{"Start":"02:33.407 ","End":"02:35.265","Text":"This is ignored."},{"Start":"02:35.265 ","End":"02:37.305","Text":"That\u0027s how we organize it."},{"Start":"02:37.305 ","End":"02:45.649","Text":"Then 1(x-a)=x minus a and we get a remainder of 0."},{"Start":"02:45.649 ","End":"02:51.500","Text":"This comes out nicely as the quotient on the top."},{"Start":"02:51.500 ","End":"02:54.230","Text":"The answer to this exercise"},{"Start":"02:54.230 ","End":"03:05.230","Text":"is equal to x^3+1."},{"Start":"03:05.230 ","End":"03:07.700","Text":"Even though we\u0027re at the end,"},{"Start":"03:07.700 ","End":"03:16.850","Text":"I\u0027ll just create some more space and I\u0027ll also highlight the answer."},{"Start":"03:16.850 ","End":"03:21.150","Text":"We are done."},{"Start":"03:21.150 ","End":"03:23.790","Text":"On to Part B."},{"Start":"03:24.200 ","End":"03:26.985","Text":"Again, a parameter."},{"Start":"03:26.985 ","End":"03:35.070","Text":"Again, we set up the division and a little bit more space this time."},{"Start":"03:36.040 ","End":"03:41.569","Text":"Again, note, you need to watch out for the order because it should be in descending"},{"Start":"03:41.569 ","End":"03:51.450","Text":"order of powers so this 1 is actually x^2+x-2."},{"Start":"03:51.450 ","End":"03:55.170","Text":"Here we\u0027re okay, you have x^3, x^2,"},{"Start":"03:55.170 ","End":"03:58.860","Text":"x, and each constant. That\u0027s fine."},{"Start":"03:58.860 ","End":"04:00.510","Text":"That just goes as is,"},{"Start":"04:00.510 ","End":"04:04.080","Text":"x cubed plus 1 minus b x"},{"Start":"04:04.080 ","End":"04:13.150","Text":"squared minus 2 plus b x plus 2b,"},{"Start":"04:14.720 ","End":"04:19.110","Text":"x squared goes into x cubed."},{"Start":"04:19.110 ","End":"04:23.165","Text":"I\u0027ll just highlight them, x times."},{"Start":"04:23.165 ","End":"04:30.880","Text":"Multiply x by the divisor and we get x cubed plus x"},{"Start":"04:30.880 ","End":"04:40.220","Text":"squared minus 2x and then we subtract,"},{"Start":"04:40.230 ","End":"04:44.320","Text":"x cubed minus x cubed is nothing."},{"Start":"04:44.320 ","End":"04:53.455","Text":"As for the x squared, so1 minus b take away 1 is just minus b(x)^2."},{"Start":"04:53.455 ","End":"04:57.325","Text":"Now we want to subtract this minus this,"},{"Start":"04:57.325 ","End":"04:59.325","Text":"I\u0027ll just do it at the side."},{"Start":"04:59.325 ","End":"05:07.010","Text":"Minus 2 plus b minus 2. What do we get?"},{"Start":"05:07.010 ","End":"05:11.590","Text":"Minus 2 minus b plus 2,"},{"Start":"05:11.590 ","End":"05:17.935","Text":"which is just minus b so minus b(x),"},{"Start":"05:17.935 ","End":"05:22.300","Text":"and then drop the next term plus 2b."},{"Start":"05:23.040 ","End":"05:25.600","Text":"The leading term is this."},{"Start":"05:25.600 ","End":"05:34.905","Text":"How many times does x squared go into minus b(x)^2 minus b times and then multiply,"},{"Start":"05:34.905 ","End":"05:39.985","Text":"get minus b(x)^2 minus"},{"Start":"05:39.985 ","End":"05:45.900","Text":"b(x) minus b times minus 2 is plus 2(b)."},{"Start":"05:45.900 ","End":"05:47.955","Text":"Then if I subtract,"},{"Start":"05:47.955 ","End":"05:53.745","Text":"everything disappears, 0, no remainder."},{"Start":"05:53.745 ","End":"05:59.770","Text":"The answer to this question is just x minus b."},{"Start":"05:59.770 ","End":"06:02.065","Text":"What I got here."},{"Start":"06:02.065 ","End":"06:06.830","Text":"I\u0027ll just highlight that and that\u0027s the answer."}],"ID":5256}],"Thumbnail":null,"ID":56152},{"Name":"Roots (Zeros) of Polynomials","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Roots of Polynomials - Part 1","Duration":"20m 8s","ChapterTopicVideoID":6347,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6347.jpeg","UploadDate":"2020-09-30T14:18:29.2530000","DurationForVideoObject":"PT20M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"In this clip, I\u0027m going to be talking about roots of polynomials."},{"Start":"00:05.550 ","End":"00:10.470","Text":"In this instance, roots and zeros are exactly the same thing."},{"Start":"00:10.470 ","End":"00:13.230","Text":"They\u0027re just 2 different names for the same thing and"},{"Start":"00:13.230 ","End":"00:16.725","Text":"sometimes I\u0027ll use the word roots and sometimes I\u0027ll use the word zeros."},{"Start":"00:16.725 ","End":"00:19.185","Text":"Now, what is a root of a polynomial?"},{"Start":"00:19.185 ","End":"00:26.250","Text":"A polynomial, say p for polynomial in the variable x so we\u0027ll call it P(x),"},{"Start":"00:26.250 ","End":"00:27.510","Text":"which means a polynomial."},{"Start":"00:27.510 ","End":"00:29.010","Text":"I\u0027ll give an example."},{"Start":"00:29.010 ","End":"00:34.530","Text":"Suppose I have x^2 minus 3x plus 2."},{"Start":"00:34.530 ","End":"00:37.005","Text":"That\u0027s a degree 2 polynomial."},{"Start":"00:37.005 ","End":"00:46.710","Text":"Now, a root of p or a 0 of p. Like I said,"},{"Start":"00:46.710 ","End":"00:50.300","Text":"the root and zero are going to be completely synonymous."},{"Start":"00:50.300 ","End":"00:54.350","Text":"Root of p is a solution of"},{"Start":"00:54.350 ","End":"01:01.910","Text":"the equation where p=0 so P(x)=0."},{"Start":"01:01.910 ","End":"01:04.355","Text":"Let\u0027s say the polynomials in the variable x,"},{"Start":"01:04.355 ","End":"01:06.290","Text":"x is the most popular variable."},{"Start":"01:06.290 ","End":"01:07.490","Text":"But of course, in future,"},{"Start":"01:07.490 ","End":"01:12.875","Text":"we could be using other letters like t or z or any other letter."},{"Start":"01:12.875 ","End":"01:18.435","Text":"For example, if I wanted to find the root of this x^2 minus 3x plus 2,"},{"Start":"01:18.435 ","End":"01:28.670","Text":"then it would be a solution of P(x)=0 or x^2 minus 3x plus 2=0."},{"Start":"01:28.670 ","End":"01:37.680","Text":"Now, I could solve this using a quadratic equation and we could get the 2 solutions."},{"Start":"01:38.920 ","End":"01:41.105","Text":"That\u0027s not the point."},{"Start":"01:41.105 ","End":"01:43.265","Text":"The point is for quadratic equations,"},{"Start":"01:43.265 ","End":"01:45.784","Text":"degree 2 polynomials, we have a formula."},{"Start":"01:45.784 ","End":"01:51.215","Text":"But in general, we\u0027re going to be talking about polynomials of degree higher than 2,"},{"Start":"01:51.215 ","End":"01:52.745","Text":"3, and upwards."},{"Start":"01:52.745 ","End":"01:55.730","Text":"I just wanted to stay with degree 2 to see what we"},{"Start":"01:55.730 ","End":"02:00.155","Text":"can summarize here and then generalize."},{"Start":"02:00.155 ","End":"02:03.620","Text":"Now if I had this polynomial factorized,"},{"Start":"02:03.620 ","End":"02:08.240","Text":"suppose I told you that it was x minus 1 times"},{"Start":"02:08.240 ","End":"02:13.670","Text":"x minus 2 is the factorization of this polynomial."},{"Start":"02:13.670 ","End":"02:15.890","Text":"Then if I had it in this form,"},{"Start":"02:15.890 ","End":"02:20.410","Text":"then I could say that if a product of 2 things is zero,"},{"Start":"02:20.410 ","End":"02:23.025","Text":"then 1 of them must be zero."},{"Start":"02:23.025 ","End":"02:27.020","Text":"Actually, there\u0027s a name for this property of the numbers,"},{"Start":"02:27.020 ","End":"02:30.710","Text":"and it\u0027s called the zero factor property,"},{"Start":"02:30.710 ","End":"02:36.334","Text":"which says that if you multiply 2 or more numbers and get zero,"},{"Start":"02:36.334 ","End":"02:39.770","Text":"then at least 1 of them has to be zero."},{"Start":"02:39.770 ","End":"02:48.979","Text":"In this case, we get that either x minus 1 = 0 or x minus 2 = 0,"},{"Start":"02:48.979 ","End":"02:55.645","Text":"which means that x = 1 or 2."},{"Start":"02:55.645 ","End":"03:00.485","Text":"These are the 2 roots or zeros of this polynomial."},{"Start":"03:00.485 ","End":"03:06.220","Text":"You can check if you substitute x=1, what is p(1)?"},{"Start":"03:06.220 ","End":"03:10.410","Text":"It\u0027s 1^2 minus 3 times 1 plus 2,"},{"Start":"03:10.410 ","End":"03:12.720","Text":"which is 1 minus 3 plus 2 = 0,"},{"Start":"03:12.720 ","End":"03:21.570","Text":"and p(2)= 2^2 minus 3 times 2 plus 2,"},{"Start":"03:21.570 ","End":"03:27.675","Text":"which is 4 minus 6 plus 2 = 0 so indeed these 2,"},{"Start":"03:27.675 ","End":"03:31.400","Text":"they are zeros of the polynomial,"},{"Start":"03:31.400 ","End":"03:36.770","Text":"they make the equation like p(x)=0 to be true."},{"Start":"03:36.770 ","End":"03:39.875","Text":"Or if you substitute them in the polynomial, they give you zero."},{"Start":"03:39.875 ","End":"03:42.065","Text":"It\u0027s the same thing."},{"Start":"03:42.065 ","End":"03:45.435","Text":"Yes, we have 2 roots."},{"Start":"03:45.435 ","End":"03:48.700","Text":"Now, as I say, factorization is going to be"},{"Start":"03:48.700 ","End":"03:53.290","Text":"our main tool in finding zeros of polynomials."},{"Start":"03:53.290 ","End":"03:55.270","Text":"In the case of quadratics,"},{"Start":"03:55.270 ","End":"03:58.525","Text":"we also have something called the quadratic formula."},{"Start":"03:58.525 ","End":"04:01.840","Text":"Let me just show you that we would get the same thing if we used the formula."},{"Start":"04:01.840 ","End":"04:05.710","Text":"We would say x = minus b,"},{"Start":"04:05.710 ","End":"04:11.140","Text":"which is plus 3 plus or minus the square root of b squared,"},{"Start":"04:11.140 ","End":"04:15.310","Text":"which is 9 minus 4 times 1 times 2,"},{"Start":"04:15.310 ","End":"04:17.410","Text":"4 times 1 times 2,"},{"Start":"04:17.410 ","End":"04:22.125","Text":"all over 2a, which is 2."},{"Start":"04:22.125 ","End":"04:28.290","Text":"Since the square root of 9 minus 8 is the square root of 1, which is 1."},{"Start":"04:28.290 ","End":"04:36.795","Text":"We get either 3 plus 1/2 or 3 minus 1/2."},{"Start":"04:36.795 ","End":"04:38.310","Text":"This 1 is 4/2,"},{"Start":"04:38.310 ","End":"04:40.500","Text":"which is 2, and this 1 is 2/2,"},{"Start":"04:40.500 ","End":"04:45.610","Text":"which is 1 so we get the same 2 roots."},{"Start":"04:45.610 ","End":"04:49.220","Text":"1 and 2 are both roots,"},{"Start":"04:49.220 ","End":"04:51.005","Text":"same as the 2 and the 1,"},{"Start":"04:51.005 ","End":"04:53.900","Text":"they are not in the same order here."},{"Start":"04:53.900 ","End":"04:58.235","Text":"Let\u0027s take another example and we\u0027ll stay with degree 2."},{"Start":"04:58.235 ","End":"05:07.625","Text":"Suppose I had Q(x) = x^2 minus 2x plus 1 and I ask,"},{"Start":"05:07.625 ","End":"05:11.750","Text":"what are its roots or zeros?"},{"Start":"05:11.750 ","End":"05:19.185","Text":"To find the roots of"},{"Start":"05:19.185 ","End":"05:28.615","Text":"Q(x) so I take Q(x)=0 so I do x^2 minus 2x plus 1=0."},{"Start":"05:28.615 ","End":"05:30.760","Text":"Of course, because it\u0027s a quadratic,"},{"Start":"05:30.760 ","End":"05:32.395","Text":"I can use the formula,"},{"Start":"05:32.395 ","End":"05:34.930","Text":"but I don\u0027t want to this time because in general,"},{"Start":"05:34.930 ","End":"05:38.665","Text":"we\u0027re going to use factorization as the main method for roots."},{"Start":"05:38.665 ","End":"05:45.715","Text":"But let me factorize it for you and later we\u0027ll discuss how to factorize."},{"Start":"05:45.715 ","End":"05:52.690","Text":"Let me just give you the answer and say that x^2 minus 2x plus 1 = x minus 1^2."},{"Start":"05:52.690 ","End":"05:56.905","Text":"Of course, you can multiply it out and check =0."},{"Start":"05:56.905 ","End":"06:06.180","Text":"In this case, we only have 1 possible solution it seems because if x minus 1^2 = 0,"},{"Start":"06:06.180 ","End":"06:08.310","Text":"then x minus 1 = 0,"},{"Start":"06:08.310 ","End":"06:11.680","Text":"so we have that x=1."},{"Start":"06:11.880 ","End":"06:17.260","Text":"It seems like we only have 1 root but there is a different way of looking at it."},{"Start":"06:17.260 ","End":"06:22.435","Text":"I could look at it like I did here and say instead of x minus 1^2,"},{"Start":"06:22.435 ","End":"06:27.415","Text":"I could say x minus 1 times x minus 1=0."},{"Start":"06:27.415 ","End":"06:33.270","Text":"Then I can say either x minus 1 = 0 or x minus 1 = 0."},{"Start":"06:33.270 ","End":"06:37.505","Text":"I could say that x = 1 or 1."},{"Start":"06:37.505 ","End":"06:40.030","Text":"I could say I have 2 roots,"},{"Start":"06:40.030 ","End":"06:42.940","Text":"they just happen to both be the same."},{"Start":"06:42.940 ","End":"06:46.690","Text":"When we have a square factor like this,"},{"Start":"06:46.690 ","End":"06:49.780","Text":"then we call this a double root,"},{"Start":"06:49.780 ","End":"06:55.530","Text":"so I actually would either say that x=1 or 1 like this,"},{"Start":"06:55.530 ","End":"06:59.810","Text":"or I could say x=1 is the only different solution,"},{"Start":"06:59.810 ","End":"07:03.390","Text":"but I could say that it\u0027s a double root."},{"Start":"07:03.730 ","End":"07:11.930","Text":"Similarly, when we have other examples of degree 3,"},{"Start":"07:11.930 ","End":"07:14.660","Text":"we can have something called a triple root,"},{"Start":"07:14.660 ","End":"07:17.180","Text":"where we have x minus 1^3=0,"},{"Start":"07:17.180 ","End":"07:20.069","Text":"let\u0027s say, then it would be a triple root,"},{"Start":"07:20.069 ","End":"07:26.285","Text":"and in general, I want to introduce a new word called multiplicity."},{"Start":"07:26.285 ","End":"07:32.085","Text":"x=1 is a root of multiplicity 2."},{"Start":"07:32.085 ","End":"07:35.910","Text":"I\u0027ll use a new word,"},{"Start":"07:35.910 ","End":"07:45.325","Text":"multiplicity 2."},{"Start":"07:45.325 ","End":"07:49.070","Text":"It\u0027s a matter of how you look at it."},{"Start":"07:49.070 ","End":"07:53.855","Text":"We prefer to look at it mathematicians as a double root,"},{"Start":"07:53.855 ","End":"07:55.892","Text":"a root of multiplicity 2,"},{"Start":"07:55.892 ","End":"07:58.520","Text":"and to say that we actually have 2 solutions,"},{"Start":"07:58.520 ","End":"08:02.260","Text":"1 and 1, they just both happen to be the same."},{"Start":"08:02.260 ","End":"08:05.750","Text":"Now I want to take a third example,"},{"Start":"08:05.750 ","End":"08:08.795","Text":"I\u0027m still staying with degree 2 or quadratic."},{"Start":"08:08.795 ","End":"08:10.730","Text":"I\u0027ve used letters P and Q."},{"Start":"08:10.730 ","End":"08:20.000","Text":"Next letter is R. Let\u0027s take the example x^2 plus 4."},{"Start":"08:20.000 ","End":"08:22.220","Text":"I want to know,"},{"Start":"08:22.220 ","End":"08:25.380","Text":"what are the roots"},{"Start":"08:27.920 ","End":"08:36.530","Text":"of R(x) if the variable is x."},{"Start":"08:36.530 ","End":"08:39.695","Text":"The question mark. I say,"},{"Start":"08:39.695 ","End":"08:43.100","Text":"what happens when R(x)=0?"},{"Start":"08:43.100 ","End":"08:47.250","Text":"This gives me that x^2 plus 4=0."},{"Start":"08:48.610 ","End":"08:52.960","Text":"Here it can\u0027t be factorized."},{"Start":"08:52.960 ","End":"08:56.315","Text":"In fact, even if you try the quadratic equation method,"},{"Start":"08:56.315 ","End":"09:01.634","Text":"there are no solutions so there are no roots."},{"Start":"09:01.634 ","End":"09:06.105","Text":"R(x) has no roots."},{"Start":"09:06.105 ","End":"09:11.035","Text":"If the equation polynomial =0 has no solutions,"},{"Start":"09:11.035 ","End":"09:17.500","Text":"then the polynomial has no roots or no zeros."},{"Start":"09:17.500 ","End":"09:22.630","Text":"Not even sure if zeros is spelled with an e or without an e. I think it\u0027s without an e,"},{"Start":"09:22.630 ","End":"09:24.680","Text":"but I\u0027m not sure."},{"Start":"09:25.200 ","End":"09:31.300","Text":"What this comes down to is that a degree 2 can have 2 solutions,"},{"Start":"09:31.300 ","End":"09:33.820","Text":"or 1 solution,"},{"Start":"09:33.820 ","End":"09:36.565","Text":"which is a double solution or no solutions."},{"Start":"09:36.565 ","End":"09:40.680","Text":"In general, what I want to say is, not the same P,"},{"Start":"09:40.680 ","End":"09:46.940","Text":"is that if a polynomial has degree n,"},{"Start":"09:47.110 ","End":"09:50.195","Text":"which in our case was 2,"},{"Start":"09:50.195 ","End":"09:54.860","Text":"then I can\u0027t say it has n roots,"},{"Start":"09:54.860 ","End":"10:03.255","Text":"but I can say it has at most n roots or zeros."},{"Start":"10:03.255 ","End":"10:05.695","Text":"If I want to rephrase this,"},{"Start":"10:05.695 ","End":"10:10.835","Text":"I can rephrase the second half by saying that the equation"},{"Start":"10:10.835 ","End":"10:18.065","Text":"P(x)=0 has at most n solutions."},{"Start":"10:18.065 ","End":"10:20.345","Text":"That\u0027s just the re-phrase."},{"Start":"10:20.345 ","End":"10:28.625","Text":"Because a root of a polynomial is a solution of the polynomial equals 0 equation."},{"Start":"10:28.625 ","End":"10:32.240","Text":"I think it\u0027s important enough that I\u0027ll highlight this,"},{"Start":"10:32.240 ","End":"10:36.325","Text":"that a polynomial of degree n has at most n roots."},{"Start":"10:36.325 ","End":"10:39.410","Text":"We saw that this is so for degree 2,"},{"Start":"10:39.410 ","End":"10:46.760","Text":"but what is interesting is for degree 3 and higher so I\u0027m going to start a new subtopic,"},{"Start":"10:46.760 ","End":"10:54.720","Text":"polynomials of degree bigger than 2."},{"Start":"10:54.720 ","End":"10:59.409","Text":"Let\u0027s say 3 or higher."},{"Start":"10:59.409 ","End":"11:02.165","Text":"I\u0027ll just emphasize this."},{"Start":"11:02.165 ","End":"11:05.435","Text":"Let me start with an example of degree 3."},{"Start":"11:05.435 ","End":"11:13.205","Text":"Let\u0027s take P(x)=x^3 minus 3x^2 plus 2x."},{"Start":"11:13.205 ","End":"11:19.355","Text":"I want to know what are its roots or zeros?"},{"Start":"11:19.355 ","End":"11:26.494","Text":"As I said, I\u0027ll be using the word roots and zeros interchangeably throughout."},{"Start":"11:26.494 ","End":"11:30.310","Text":"The roots of this are the same as the solutions of"},{"Start":"11:30.310 ","End":"11:37.795","Text":"the equation x^3 minus 3x^2 plus 2x equals 0."},{"Start":"11:37.795 ","End":"11:42.040","Text":"The main tool we have will be factorization."},{"Start":"11:42.040 ","End":"11:44.890","Text":"We can\u0027t always factorize the polynomial,"},{"Start":"11:44.890 ","End":"11:48.490","Text":"but I\u0027m going to show you later some techniques of how to do it."},{"Start":"11:48.490 ","End":"11:54.670","Text":"In this case we can factorize it because we see there\u0027s a missing constant term."},{"Start":"11:54.670 ","End":"11:56.440","Text":"I can take x out."},{"Start":"11:56.440 ","End":"12:06.835","Text":"I can say that this gives me that x(x^2-3x+2)=0."},{"Start":"12:06.835 ","End":"12:10.255","Text":"Isn\u0027t this just the example we had above,"},{"Start":"12:10.255 ","End":"12:11.845","Text":"where we saw that this"},{"Start":"12:11.845 ","End":"12:12.430","Text":"is equal"},{"Start":"12:12.430 ","End":"12:22.165","Text":"to x(x-1)(x-2)=0."},{"Start":"12:22.165 ","End":"12:27.205","Text":"Now again, we\u0027re going to use this 0 factor property."},{"Start":"12:27.205 ","End":"12:31.510","Text":"It works on any number of factors that if I have a product of things is 0,"},{"Start":"12:31.510 ","End":"12:34.150","Text":"then one of them at least has to be 0."},{"Start":"12:34.150 ","End":"12:42.640","Text":"We get that either x=0 or x-1=0 or x-2=0."},{"Start":"12:42.640 ","End":"12:50.545","Text":"This gives us that x=0,1 if x-1=0,"},{"Start":"12:50.545 ","End":"12:55.480","Text":"then x=1, or x-2=0, so x=2."},{"Start":"12:55.480 ","End":"12:57.595","Text":"So I can now answer the question."},{"Start":"12:57.595 ","End":"13:05.845","Text":"The roots of this polynomial are 0,1 and 2 or x= 0,1,2."},{"Start":"13:05.845 ","End":"13:11.785","Text":"And we found three roots and we couldn\u0027t expect anymore because this has degree 3."},{"Start":"13:11.785 ","End":"13:16.030","Text":"Now I want to point out something which may seem obvious,"},{"Start":"13:16.030 ","End":"13:21.940","Text":"but it isn\u0027t and it\u0027s going to help us in solving degree 3 or higher."},{"Start":"13:21.940 ","End":"13:23.515","Text":"It\u0027s going to help us to factorize."},{"Start":"13:23.515 ","End":"13:27.040","Text":"Notice I\u0027ll just take the 2, for example,"},{"Start":"13:27.040 ","End":"13:36.505","Text":"that 2 is a root of the polynomial and also x-2 is a factor of the polynomial."},{"Start":"13:36.505 ","End":"13:41.230","Text":"I mean this bit here is just p(x) factorized."},{"Start":"13:41.230 ","End":"13:42.985","Text":"And so in general,"},{"Start":"13:42.985 ","End":"13:46.570","Text":"what we can say is that if x-a,"},{"Start":"13:46.570 ","End":"13:49.330","Text":"where a is 2, for example,"},{"Start":"13:49.330 ","End":"13:58.120","Text":"here is a factor of a polynomial p, p(x)."},{"Start":"13:58.120 ","End":"14:05.170","Text":"Then we know that x equals a is a root or"},{"Start":"14:05.170 ","End":"14:12.570","Text":"0 of p. Now the thing I want to stress is it works the other way round as well,"},{"Start":"14:12.570 ","End":"14:15.450","Text":"that I can put the arrow both ways."},{"Start":"14:15.450 ","End":"14:19.405","Text":"That if I find a root by chance,"},{"Start":"14:19.405 ","End":"14:21.220","Text":"by luck, by technique,"},{"Start":"14:21.220 ","End":"14:25.840","Text":"whatever, if I find a root of the polynomial,"},{"Start":"14:25.840 ","End":"14:28.150","Text":"then I also have a factor."},{"Start":"14:28.150 ","End":"14:32.200","Text":"And this is going to help us to solve polynomials."},{"Start":"14:32.200 ","End":"14:35.410","Text":"As we shall see. This is important,"},{"Start":"14:35.410 ","End":"14:37.910","Text":"so I\u0027m going to highlight it."},{"Start":"14:38.820 ","End":"14:42.730","Text":"Now let\u0027s take an example of how this is useful."},{"Start":"14:42.730 ","End":"14:50.965","Text":"Let\u0027s take an example I don\u0027t want to use p I\u0027ll use q polynomial also of degree 3."},{"Start":"14:50.965 ","End":"15:00.160","Text":"Let\u0027s take x^3-x^2-4x+4."},{"Start":"15:00.160 ","End":"15:05.840","Text":"And I want to know what are its roots."},{"Start":"15:06.540 ","End":"15:11.530","Text":"Suppose that you could somehow get just one root"},{"Start":"15:11.530 ","End":"15:16.825","Text":"either by guesswork or by other techniques which I\u0027ll show you later."},{"Start":"15:16.825 ","End":"15:18.100","Text":"Imagine even try guessing,"},{"Start":"15:18.100 ","End":"15:25.390","Text":"let\u0027s say is x equals 0 or root while you substitute 0-0-0+4. No, it\u0027s not a root."},{"Start":"15:25.390 ","End":"15:28.400","Text":"You try may be x equals1, 1-1-4+4."},{"Start":"15:29.820 ","End":"15:32.770","Text":"Bingo, yes, it\u0027s 0."},{"Start":"15:32.770 ","End":"15:35.755","Text":"So if you can guess one of the roots,"},{"Start":"15:35.755 ","End":"15:37.150","Text":"x equals 1,"},{"Start":"15:37.150 ","End":"15:38.830","Text":"let\u0027s just write it as a guess,"},{"Start":"15:38.830 ","End":"15:42.250","Text":"but later we\u0027ll get some techniques."},{"Start":"15:42.250 ","End":"15:47.710","Text":"Then one root can help us find other roots."},{"Start":"15:47.710 ","End":"15:54.880","Text":"What I wrote here highlighted is that if x equals one is a root,"},{"Start":"15:54.880 ","End":"15:58.810","Text":"then x-1 is going to be a factor."},{"Start":"15:58.810 ","End":"16:02.050","Text":"So I can now say that q(x)."},{"Start":"16:02.050 ","End":"16:07.630","Text":"Which is x^3 minus x^2 minus 4x plus"},{"Start":"16:07.630 ","End":"16:14.320","Text":"4 is equal to x-1 times something."},{"Start":"16:14.320 ","End":"16:16.690","Text":"I don\u0027t know what that something is."},{"Start":"16:16.690 ","End":"16:24.265","Text":"But I\u0027m assuming that you\u0027ve studied something called long division of polynomials."},{"Start":"16:24.265 ","End":"16:26.320","Text":"If you\u0027re a bit rusty,"},{"Start":"16:26.320 ","End":"16:28.450","Text":"I\u0027ll remind you of how to do this."},{"Start":"16:28.450 ","End":"16:33.280","Text":"What I want to do is to get to this unknown by dividing this,"},{"Start":"16:33.280 ","End":"16:37.360","Text":"divided by this, I\u0027m going to do a long division of polynomials."},{"Start":"16:37.360 ","End":"16:43.645","Text":"Now, we\u0027re going to do x-1 into"},{"Start":"16:43.645 ","End":"16:48.985","Text":"x^3 minus"},{"Start":"16:48.985 ","End":"16:55.135","Text":"x^2 minus 4x plus 4."},{"Start":"16:55.135 ","End":"16:58.390","Text":"I\u0027m going to continue on the next page."},{"Start":"16:58.590 ","End":"17:02.605","Text":"Here we are about to do the long division."},{"Start":"17:02.605 ","End":"17:06.174","Text":"What we do is we look at the leading term,"},{"Start":"17:06.174 ","End":"17:08.260","Text":"which is the x,"},{"Start":"17:08.260 ","End":"17:12.070","Text":"and we ask how many times it goes into the leading term here."},{"Start":"17:12.070 ","End":"17:16.645","Text":"And the answer is x^2, of course."},{"Start":"17:16.645 ","End":"17:26.335","Text":"Then we do x^2 times x-1 is x^3 minus x^2."},{"Start":"17:26.335 ","End":"17:29.050","Text":"That goes in evenly."},{"Start":"17:29.050 ","End":"17:33.640","Text":"So we have to drop the next 2 terms down."},{"Start":"17:33.640 ","End":"17:36.865","Text":"We have -4x plus 4."},{"Start":"17:36.865 ","End":"17:43.855","Text":"And then we ask how many times does x go into leading term -4x?"},{"Start":"17:43.855 ","End":"17:47.373","Text":"And the answer is -4 times,"},{"Start":"17:47.373 ","End":"17:50.710","Text":"-4 times x-1 is -4x,"},{"Start":"17:50.710 ","End":"17:53.410","Text":"-4 times -1 is plus 4."},{"Start":"17:53.410 ","End":"17:56.710","Text":"Again, we subtract, I should have mentioned this is a subtraction."},{"Start":"17:56.710 ","End":"17:58.705","Text":"This is a subtraction."},{"Start":"17:58.705 ","End":"18:00.685","Text":"Looked like nothing left,"},{"Start":"18:00.685 ","End":"18:02.545","Text":"so it goes in evenly."},{"Start":"18:02.545 ","End":"18:08.305","Text":"So now we can say that this question mark is x^2 minus 4."},{"Start":"18:08.305 ","End":"18:12.700","Text":"Now we can say that q(x) factorizes"},{"Start":"18:12.700 ","End":"18:19.975","Text":"as x-1 times x^2 minus 4."},{"Start":"18:19.975 ","End":"18:26.005","Text":"Already helps us because now the other factor is only of degree 2 quadratic."},{"Start":"18:26.005 ","End":"18:30.340","Text":"And we know how to factorize quadratics. There are many techniques."},{"Start":"18:30.340 ","End":"18:35.020","Text":"I\u0027ll just write the answer using the difference of squares formula,"},{"Start":"18:35.020 ","End":"18:40.345","Text":"for example, is (x-2)(x+2)."},{"Start":"18:40.345 ","End":"18:43.795","Text":"And once I have q factorized,"},{"Start":"18:43.795 ","End":"18:45.970","Text":"then I know what the roots are,"},{"Start":"18:45.970 ","End":"18:49.510","Text":"because what I wrote above is that if x-a is a factor,"},{"Start":"18:49.510 ","End":"18:51.265","Text":"then a is a solution."},{"Start":"18:51.265 ","End":"18:58.015","Text":"So I can say that the roots of q are,"},{"Start":"18:58.015 ","End":"19:05.125","Text":"that x is either equal to 1 or 2 or -2."},{"Start":"19:05.125 ","End":"19:09.130","Text":"Three roots, and there can\u0027t be anymore because this is degree 3,"},{"Start":"19:09.130 ","End":"19:11.785","Text":"it has at most 3 roots."},{"Start":"19:11.785 ","End":"19:14.230","Text":"Just to emphasize the steps,"},{"Start":"19:14.230 ","End":"19:20.035","Text":"since this is one of our first examples of a bigger than 2 degree polynomial,"},{"Start":"19:20.035 ","End":"19:22.450","Text":"we get the polynomial."},{"Start":"19:22.450 ","End":"19:28.330","Text":"We somehow get one of the roots."},{"Start":"19:28.330 ","End":"19:31.090","Text":"And we\u0027ll discuss this more later,"},{"Start":"19:31.090 ","End":"19:33.385","Text":"how we can do it other than guesswork."},{"Start":"19:33.385 ","End":"19:34.870","Text":"Once we have that,"},{"Start":"19:34.870 ","End":"19:38.270","Text":"then we do a long division."},{"Start":"19:38.730 ","End":"19:44.185","Text":"That long division helps us to do the factorization."},{"Start":"19:44.185 ","End":"19:46.900","Text":"Once we have the factorization,"},{"Start":"19:46.900 ","End":"19:50.695","Text":"then we have the roots."},{"Start":"19:50.695 ","End":"19:57.415","Text":"Let\u0027s take a break now and continue in the next clip more formally,"},{"Start":"19:57.415 ","End":"20:00.535","Text":"how to guess a root."},{"Start":"20:00.535 ","End":"20:04.345","Text":"We\u0027ll also return to the subject of multiplicity of roots,"},{"Start":"20:04.345 ","End":"20:08.180","Text":"but that will be in the next clip."}],"ID":6350},{"Watched":false,"Name":"Roots of Polynomials - Part 2","Duration":"10m 47s","ChapterTopicVideoID":6348,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6348.jpeg","UploadDate":"2016-06-22T08:57:34.7230000","DurationForVideoObject":"PT10M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.715","Text":"Here we are after the break."},{"Start":"00:02.715 ","End":"00:06.165","Text":"Before the break I said I\u0027d be talking about two things,"},{"Start":"00:06.165 ","End":"00:10.650","Text":"multiplicity and also how to guess a root."},{"Start":"00:10.650 ","End":"00:13.455","Text":"Let me just continue on the topic of multiplicity."},{"Start":"00:13.455 ","End":"00:16.350","Text":"I\u0027d like to give a few more examples of degree 3,"},{"Start":"00:16.350 ","End":"00:19.380","Text":"and they\u0027re pretty typical of all higher degrees."},{"Start":"00:19.380 ","End":"00:22.845","Text":"We had this example."},{"Start":"00:22.845 ","End":"00:25.785","Text":"Let us take another example,"},{"Start":"00:25.785 ","End":"00:30.090","Text":"which is x minus 1."},{"Start":"00:30.090 ","End":"00:33.770","Text":"Suppose we had in the previous example after the long division,"},{"Start":"00:33.770 ","End":"00:37.590","Text":"maybe we got x^2 plus 4."},{"Start":"00:38.420 ","End":"00:41.240","Text":"I\u0027m giving it, you have to factorization."},{"Start":"00:41.240 ","End":"00:42.785","Text":"Then we would say, okay,"},{"Start":"00:42.785 ","End":"00:47.270","Text":"we have one root, is that x=1."},{"Start":"00:47.270 ","End":"00:56.040","Text":"We say that either x=1 or x^2 plus 4=0."},{"Start":"00:56.040 ","End":"00:59.756","Text":"Either this is 0, which is x is 1, or this is 0."},{"Start":"00:59.756 ","End":"01:02.765","Text":"But this equals 0 has no solutions."},{"Start":"01:02.765 ","End":"01:10.640","Text":"This one is impossible because we can\u0027t have x^2 equals minus 4."},{"Start":"01:10.640 ","End":"01:14.430","Text":"We only have one root,"},{"Start":"01:14.830 ","End":"01:18.940","Text":"and that root is x=1."},{"Start":"01:18.940 ","End":"01:23.355","Text":"We don\u0027t always get three roots for a cubic,"},{"Start":"01:23.355 ","End":"01:26.710","Text":"sometimes we get just one root."},{"Start":"01:27.080 ","End":"01:29.460","Text":"This is an example."},{"Start":"01:29.460 ","End":"01:31.955","Text":"I\u0027m going to give a few more examples."},{"Start":"01:31.955 ","End":"01:36.920","Text":"Perhaps I\u0027ll label them instead of using different letters, p1 and then p2."},{"Start":"01:36.920 ","End":"01:38.615","Text":"Second polynomial."},{"Start":"01:38.615 ","End":"01:41.030","Text":"I\u0027m going to give them factorized because"},{"Start":"01:41.030 ","End":"01:43.610","Text":"there\u0027s no point in wasting time in factorizing."},{"Start":"01:43.610 ","End":"01:47.065","Text":"Suppose I had after factorization,"},{"Start":"01:47.065 ","End":"01:54.010","Text":"x minus 1^2 times x plus 3."},{"Start":"01:54.010 ","End":"01:57.260","Text":"I could expand this and give you what it was before,"},{"Start":"01:57.260 ","End":"01:59.645","Text":"but let\u0027s just take it from here."},{"Start":"01:59.645 ","End":"02:03.860","Text":"In this case, we would say, there\u0027s 2 solutions."},{"Start":"02:03.860 ","End":"02:06.869","Text":"You could say that the roots,"},{"Start":"02:08.050 ","End":"02:11.550","Text":"either x minus 1^2 is 0."},{"Start":"02:11.550 ","End":"02:16.860","Text":"In other words, we get x minus 1 is 0, so x=1."},{"Start":"02:16.860 ","End":"02:22.085","Text":"We could also say that x plus 3 is 0,"},{"Start":"02:22.085 ","End":"02:25.900","Text":"in which case x is equal to minus 3."},{"Start":"02:25.900 ","End":"02:30.310","Text":"Now, you\u0027ve left a gap here deliberately because we could\u0027ve said that this is"},{"Start":"02:30.310 ","End":"02:36.670","Text":"x minus 1 times x minus 1 times x plus 3."},{"Start":"02:36.670 ","End":"02:41.500","Text":"When you have a factor appearing more than once or squared,"},{"Start":"02:41.500 ","End":"02:44.640","Text":"then we say it has a multiplicity of 2."},{"Start":"02:44.640 ","End":"02:47.670","Text":"Sometimes you even write 1,"},{"Start":"02:47.670 ","End":"02:49.215","Text":"1 and minus 3."},{"Start":"02:49.215 ","End":"02:50.880","Text":"You say the three roots,"},{"Start":"02:50.880 ","End":"02:53.045","Text":"1, 1 and minus 3."},{"Start":"02:53.045 ","End":"02:55.735","Text":"Two of them happen to be the same."},{"Start":"02:55.735 ","End":"02:57.670","Text":"It all depends on your point of view."},{"Start":"02:57.670 ","End":"03:01.825","Text":"You either say it\u0027s one root and it has multiplicity 2,"},{"Start":"03:01.825 ","End":"03:05.675","Text":"or you say there are well,"},{"Start":"03:05.675 ","End":"03:09.620","Text":"altogether three, but these two are two different ones."},{"Start":"03:09.810 ","End":"03:12.980","Text":"I phrased that badly."},{"Start":"03:14.040 ","End":"03:16.775","Text":"Take two on the last bit."},{"Start":"03:16.775 ","End":"03:21.185","Text":"You could say there are just two different routes and they are 1 and minus 3."},{"Start":"03:21.185 ","End":"03:24.245","Text":"Or you could say there are three roots, 1,"},{"Start":"03:24.245 ","End":"03:25.670","Text":"1 and minus 3,"},{"Start":"03:25.670 ","End":"03:28.800","Text":"just that two of them happen to be the same."},{"Start":"03:29.030 ","End":"03:36.140","Text":"The root 1 has multiplicity 2 and root minus 3 has multiplicity 1."},{"Start":"03:36.140 ","End":"03:38.900","Text":"Now we could even have another example."},{"Start":"03:38.900 ","End":"03:45.510","Text":"Suppose we had x minus 1^3."},{"Start":"03:45.790 ","End":"03:49.520","Text":"I don\u0027t know why I\u0027m sticking with the x minus 1."},{"Start":"03:49.520 ","End":"03:51.454","Text":"That\u0027s convenient."},{"Start":"03:51.454 ","End":"03:53.540","Text":"In this case, you could,"},{"Start":"03:53.540 ","End":"03:56.360","Text":"just for emphasis, write it as x minus 1,"},{"Start":"03:56.360 ","End":"03:57.790","Text":"x minus 1,"},{"Start":"03:57.790 ","End":"03:59.370","Text":"x minus 1,"},{"Start":"03:59.370 ","End":"04:01.425","Text":"which is reminiscent of this."},{"Start":"04:01.425 ","End":"04:04.065","Text":"Here we got that the roots were 1,"},{"Start":"04:04.065 ","End":"04:06.150","Text":"2 and minus 2."},{"Start":"04:06.150 ","End":"04:12.246","Text":"Here we can say that the roots are 1, 1 and 1."},{"Start":"04:12.246 ","End":"04:16.135","Text":"You could say we have three roots and they\u0027re all 1,"},{"Start":"04:16.135 ","End":"04:22.401","Text":"or you could say we have the only root is x=1,"},{"Start":"04:22.401 ","End":"04:28.575","Text":"but it has multiplicity 3,"},{"Start":"04:28.575 ","End":"04:37.135","Text":"just like here we could say that this is 1 with multiplicity 2."},{"Start":"04:37.135 ","End":"04:40.730","Text":"You might wonder why we need this concept of multiplicity."},{"Start":"04:40.730 ","End":"04:46.440","Text":"It turns out to be important for higher math if you continue."},{"Start":"04:46.440 ","End":"04:48.710","Text":"You could just say naively,"},{"Start":"04:48.710 ","End":"04:50.240","Text":"if I want to know the solutions,"},{"Start":"04:50.240 ","End":"04:53.000","Text":"what numbers make this 0,"},{"Start":"04:53.000 ","End":"04:55.720","Text":"then yeah, just 1 or minus 3."},{"Start":"04:55.720 ","End":"05:01.295","Text":"Here, you could say that the only thing that makes this polynomial 0 is x=1."},{"Start":"05:01.295 ","End":"05:05.240","Text":"But it turns out that the concepts of multiplicity is important for other topics,"},{"Start":"05:05.240 ","End":"05:08.070","Text":"so I\u0027m sticking it in here."},{"Start":"05:08.350 ","End":"05:13.075","Text":"Let\u0027s just take one more example."},{"Start":"05:13.075 ","End":"05:18.888","Text":"Let\u0027s take p4 of x is going to be similar to p2,"},{"Start":"05:18.888 ","End":"05:24.775","Text":"only this time it\u0027s x minus 1 times x plus 3^2."},{"Start":"05:24.775 ","End":"05:30.500","Text":"This time, the roots are going to be, if you look at it,"},{"Start":"05:30.500 ","End":"05:32.525","Text":"1 appears one time,"},{"Start":"05:32.525 ","End":"05:37.005","Text":"but minus 3 appears twice."},{"Start":"05:37.005 ","End":"05:39.815","Text":"Both of these polynomials,"},{"Start":"05:39.815 ","End":"05:44.635","Text":"the only numbers that make them 0 are 1 and minus 3."},{"Start":"05:44.635 ","End":"05:50.030","Text":"But here one has a multiplicity of 2 and minus 3 has the multiplicity of 1."},{"Start":"05:50.030 ","End":"05:51.830","Text":"Here, it\u0027s the other way around,"},{"Start":"05:51.830 ","End":"05:55.700","Text":"that this one has a multiplicity of 2."},{"Start":"05:55.700 ","End":"06:02.710","Text":"I\u0027d like to continue the topic of multiplicity with another couple of examples,"},{"Start":"06:02.710 ","End":"06:06.280","Text":"this time with degree higher than 3."},{"Start":"06:06.280 ","End":"06:10.640","Text":"Here\u0027s an example of degree 5."},{"Start":"06:10.640 ","End":"06:13.610","Text":"It has another thing that\u0027s different."},{"Start":"06:13.610 ","End":"06:14.995","Text":"Up till now for some reason,"},{"Start":"06:14.995 ","End":"06:18.760","Text":"we\u0027ve always had the leading coefficient as 1."},{"Start":"06:18.760 ","End":"06:21.770","Text":"But here, the leading coefficient is 5."},{"Start":"06:21.770 ","End":"06:26.855","Text":"What I don\u0027t expect you to do at this stage is to factorize."},{"Start":"06:26.855 ","End":"06:29.360","Text":"I\u0027ll give you the factorization."},{"Start":"06:29.360 ","End":"06:33.335","Text":"Here it is by magic factorized."},{"Start":"06:33.335 ","End":"06:41.110","Text":"What I expect you to know how to write the roots given the factorization."},{"Start":"06:41.110 ","End":"06:42.776","Text":"There\u0027s two ways of doing it,"},{"Start":"06:42.776 ","End":"06:44.050","Text":"just like I said before,"},{"Start":"06:44.050 ","End":"06:46.450","Text":"with or without the multiplicity."},{"Start":"06:46.450 ","End":"06:50.740","Text":"Let me just first note that the constant makes no difference at all."},{"Start":"06:50.740 ","End":"06:52.225","Text":"I just ignore it."},{"Start":"06:52.225 ","End":"06:54.640","Text":"Because if I was solving an equation,"},{"Start":"06:54.640 ","End":"06:57.790","Text":"p(x)=0, I could divide both sides by 5."},{"Start":"06:57.790 ","End":"07:01.675","Text":"A leading constant has no bearing at all on the roots."},{"Start":"07:01.675 ","End":"07:06.865","Text":"Now I see I have either x plus 1 is 0 or x minus 2 is 0."},{"Start":"07:06.865 ","End":"07:10.655","Text":"I could say that x is either equal to minus 1."},{"Start":"07:10.655 ","End":"07:15.525","Text":"I could write minus 1 twice because this is x plus 1 and x plus 1."},{"Start":"07:15.525 ","End":"07:18.085","Text":"Here I see x=2."},{"Start":"07:18.085 ","End":"07:19.933","Text":"I could write that three times,"},{"Start":"07:19.933 ","End":"07:22.595","Text":"and I can say yes, we have five roots."},{"Start":"07:22.595 ","End":"07:23.780","Text":"There are minus 1,"},{"Start":"07:23.780 ","End":"07:26.525","Text":"minus 1, 2, 2 and 2."},{"Start":"07:26.525 ","End":"07:29.729","Text":"That\u0027s one possibility."},{"Start":"07:29.729 ","End":"07:33.425","Text":"Or I could say, it\u0027s minus 1."},{"Start":"07:33.425 ","End":"07:38.890","Text":"Then add that the multiplicity is equal to 2."},{"Start":"07:38.890 ","End":"07:41.965","Text":"Then say I have another root 2,"},{"Start":"07:41.965 ","End":"07:47.405","Text":"but it has a multiplicity of 3,"},{"Start":"07:47.405 ","End":"07:49.775","Text":"whichever way you prefer."},{"Start":"07:49.775 ","End":"07:52.290","Text":"Another example."},{"Start":"07:52.310 ","End":"07:57.300","Text":"Here we are a degree 8 polynomial."},{"Start":"07:57.300 ","End":"07:59.705","Text":"I\u0027m going to do the factoring for you,"},{"Start":"07:59.705 ","End":"08:01.655","Text":"although you could start it yourself,"},{"Start":"08:01.655 ","End":"08:05.870","Text":"you could see that we\u0027re only from x to the fourth and higher."},{"Start":"08:05.870 ","End":"08:10.190","Text":"We could by ourselves say it\u0027s x to the fourth times something."},{"Start":"08:10.190 ","End":"08:12.280","Text":"But I\u0027ll complete the work for you."},{"Start":"08:12.280 ","End":"08:16.495","Text":"It happens to be x minus 3^3 times x plus 5."},{"Start":"08:16.495 ","End":"08:20.900","Text":"Now notice that x is the same as x minus 0."},{"Start":"08:20.900 ","End":"08:23.690","Text":"When I see x to the fourth,"},{"Start":"08:23.690 ","End":"08:27.830","Text":"then it means that 0 is a root."},{"Start":"08:27.830 ","End":"08:29.960","Text":"But it\u0027s a root four times."},{"Start":"08:29.960 ","End":"08:31.610","Text":"If I do it the long way,"},{"Start":"08:31.610 ","End":"08:35.385","Text":"I can say that my roots are 0,"},{"Start":"08:35.385 ","End":"08:38.260","Text":"0, 0, 0."},{"Start":"08:38.260 ","End":"08:40.145","Text":"Let me just write the word roots."},{"Start":"08:40.145 ","End":"08:44.550","Text":"You know what, I should have written it here also, roots."},{"Start":"08:44.570 ","End":"08:46.800","Text":"Then x minus 3,"},{"Start":"08:46.800 ","End":"08:48.350","Text":"if it\u0027s 0, x is 3."},{"Start":"08:48.350 ","End":"08:49.940","Text":"But since we have a power of 3,"},{"Start":"08:49.940 ","End":"08:56.750","Text":"then we take this three times and this one gives us a root of minus 5, but only once."},{"Start":"08:56.750 ","End":"09:00.300","Text":"We could say that we have eight roots,"},{"Start":"09:00.560 ","End":"09:03.385","Text":"0, 0, 0, 0, 3, 3, 3 and minus 5."},{"Start":"09:03.385 ","End":"09:09.265","Text":"Or as before, we could write it as 0 with a multiplicity,"},{"Start":"09:09.265 ","End":"09:14.005","Text":"3 with a multiplicity, and minus 5."},{"Start":"09:14.005 ","End":"09:18.240","Text":"Here the multiplicity is 4,"},{"Start":"09:18.240 ","End":"09:21.190","Text":"here the multiplicity is 3,"},{"Start":"09:21.190 ","End":"09:25.830","Text":"and here the multiplicity is 1."},{"Start":"09:26.650 ","End":"09:29.630","Text":"Just one more thing I wanted to note."},{"Start":"09:29.630 ","End":"09:33.190","Text":"Just by coincidence, we had a degree 8"},{"Start":"09:33.190 ","End":"09:38.655","Text":"polynomial and we had eight roots depending on how you count it."},{"Start":"09:38.655 ","End":"09:42.720","Text":"Here we had a degree 5 and we had five roots."},{"Start":"09:42.720 ","End":"09:49.030","Text":"Here we have degree 3 with three roots,"},{"Start":"09:49.030 ","End":"09:56.945","Text":"but as we saw, it\u0027s possible for a degree n polynomial to have less than n roots."},{"Start":"09:56.945 ","End":"09:59.960","Text":"I remember backup here, we had,"},{"Start":"09:59.960 ","End":"10:04.215","Text":"there we are, a cubic but only one root."},{"Start":"10:04.215 ","End":"10:10.385","Text":"We said before, if the number of roots is at most the degree of the polynomial,"},{"Start":"10:10.385 ","End":"10:11.880","Text":"and in most cases,"},{"Start":"10:11.880 ","End":"10:13.180","Text":"we\u0027ve encountered it is,"},{"Start":"10:13.180 ","End":"10:15.845","Text":"but that\u0027s not necessarily the case."},{"Start":"10:15.845 ","End":"10:18.365","Text":"Because if you count multiplicities,"},{"Start":"10:18.365 ","End":"10:23.615","Text":"the number of roots really is when you add the multiplicities as eight roots,"},{"Start":"10:23.615 ","End":"10:25.715","Text":"4 plus 3 plus 1 is 8."},{"Start":"10:25.715 ","End":"10:29.420","Text":"Although some people would still argue with that there are only three roots."},{"Start":"10:29.420 ","End":"10:32.240","Text":"It really just depends how you look at it."},{"Start":"10:32.240 ","End":"10:35.375","Text":"We\u0027re done with the topic of multiplicity."},{"Start":"10:35.375 ","End":"10:38.475","Text":"I\u0027m going to take a break here."},{"Start":"10:38.475 ","End":"10:41.510","Text":"When we continue, we\u0027ll talk about what I mentioned"},{"Start":"10:41.510 ","End":"10:46.890","Text":"earlier of how to find or guess a root for a polynomial."}],"ID":6351},{"Watched":false,"Name":"Roots of Polynomials - Part 3","Duration":"9m 14s","ChapterTopicVideoID":6349,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6349.jpeg","UploadDate":"2020-09-30T14:22:48.9700000","DurationForVideoObject":"PT9M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.700","Text":"Again, continuing the topic of roots or zeros of polynomials."},{"Start":"00:05.700 ","End":"00:11.325","Text":"So I\u0027ll write the word continued and as I said in the previous clip,"},{"Start":"00:11.325 ","End":"00:15.165","Text":"what we\u0027re going to do next is learn ways how to find"},{"Start":"00:15.165 ","End":"00:21.119","Text":"even just 1 root of a polynomial and if we can find or guess 1 root of a polynomial,"},{"Start":"00:21.119 ","End":"00:24.225","Text":"it will help us to find all the rest."},{"Start":"00:24.225 ","End":"00:27.690","Text":"I wrote down what we\u0027re going to do is learn how to find"},{"Start":"00:27.690 ","End":"00:31.380","Text":"just partially guesswork root of a polynomial."},{"Start":"00:31.380 ","End":"00:34.350","Text":"We\u0027re going to work with polynomials of degree"},{"Start":"00:34.350 ","End":"00:38.125","Text":"3 and higher because we already know how to do quadratics."},{"Start":"00:38.125 ","End":"00:42.720","Text":"Let me take an example which we already had before."},{"Start":"00:42.720 ","End":"00:53.630","Text":"X^3 minus x^2 minus 4x plus 4."},{"Start":"00:53.630 ","End":"00:58.270","Text":"You might recall that what we did there was we just started guessing."},{"Start":"00:58.270 ","End":"01:00.250","Text":"We said let\u0027s try a whole number."},{"Start":"01:00.250 ","End":"01:01.840","Text":"Let\u0027s try 0,"},{"Start":"01:01.840 ","End":"01:03.355","Text":"1, 2, etc."},{"Start":"01:03.355 ","End":"01:10.515","Text":"We try plugging in 0 and we found that 0 minus 0 minus 0 plus 4 is not 0."},{"Start":"01:10.515 ","End":"01:16.170","Text":"Then we tried 1 and we got 1 minus 1 minus 4 plus 4 and yeah, we got lucky."},{"Start":"01:16.170 ","End":"01:24.750","Text":"We found that x equals 1 is a root or 0 of this polynomial."},{"Start":"01:24.750 ","End":"01:28.010","Text":"But I don\u0027t want to rely on this method."},{"Start":"01:28.010 ","End":"01:29.390","Text":"It\u0027s too haphazard."},{"Start":"01:29.390 ","End":"01:36.320","Text":"What I want to talk about is something called the integer root theorem."},{"Start":"01:36.320 ","End":"01:43.010","Text":"It\u0027s going to help us to find an integer root of a polynomial under certain conditions."},{"Start":"01:43.010 ","End":"01:45.650","Text":"Let me just write what the conditions are."},{"Start":"01:45.650 ","End":"01:54.390","Text":"First condition is that the polynomial p(x) must have integer coefficients."},{"Start":"01:54.890 ","End":"02:00.335","Text":"The other thing we need is that the leading coefficient should be 1."},{"Start":"02:00.335 ","End":"02:04.355","Text":"I could write 1 here though it\u0027s not necessary."},{"Start":"02:04.355 ","End":"02:12.795","Text":"So P(x) has a leading coefficient of 1,"},{"Start":"02:12.795 ","End":"02:15.255","Text":"which it does here."},{"Start":"02:15.255 ","End":"02:18.815","Text":"The conclusion if these 2 are true,"},{"Start":"02:18.815 ","End":"02:24.110","Text":"is that any integer root of"},{"Start":"02:24.110 ","End":"02:31.285","Text":"p is a factor of the constant term."},{"Start":"02:31.285 ","End":"02:35.150","Text":"These 2 are the ones that are given."},{"Start":"02:35.150 ","End":"02:37.250","Text":"If I have these 2,"},{"Start":"02:37.250 ","End":"02:39.950","Text":"then they imply this."},{"Start":"02:39.950 ","End":"02:45.085","Text":"The constant term is just the term without any x\u0027s in it."},{"Start":"02:45.085 ","End":"02:49.040","Text":"Just this one here in our case,"},{"Start":"02:49.040 ","End":"02:52.145","Text":"that\u0027s the constant term of a polynomial."},{"Start":"02:52.145 ","End":"02:59.680","Text":"Factor, just means divides into a whole number of times like 3 is a factor of 6,"},{"Start":"02:59.680 ","End":"03:03.495","Text":"and also minus 3 is a factor of 6."},{"Start":"03:03.495 ","End":"03:08.345","Text":"Note that this theorem only helps us to find integer roots."},{"Start":"03:08.345 ","End":"03:10.835","Text":"Back to our example."},{"Start":"03:10.835 ","End":"03:15.170","Text":"Let\u0027s forget that we knew that x=1 is"},{"Start":"03:15.170 ","End":"03:21.469","Text":"a solution and let\u0027s look for any possible integer roots."},{"Start":"03:21.469 ","End":"03:27.575","Text":"So according to the theorem these can only be factors of 4,"},{"Start":"03:27.575 ","End":"03:33.540","Text":"and the factors of 4 could be plus or minus 1 plus or minus 2,"},{"Start":"03:33.540 ","End":"03:37.530","Text":"that goes into 4 and it could be plus or minus 4."},{"Start":"03:37.530 ","End":"03:39.855","Text":"There\u0027s 6 possibilities to check."},{"Start":"03:39.855 ","End":"03:41.250","Text":"1, minus 1, 2,"},{"Start":"03:41.250 ","End":"03:43.305","Text":"minus 2, 4, minus 4."},{"Start":"03:43.305 ","End":"03:48.530","Text":"Now this exercise we\u0027ve done before earlier in an earlier clip,"},{"Start":"03:48.530 ","End":"03:54.170","Text":"and I know that x equals 1 is a root."},{"Start":"03:54.170 ","End":"03:55.820","Text":"So if you tried 1 first,"},{"Start":"03:55.820 ","End":"03:57.905","Text":"you\u0027d get lucky first time."},{"Start":"03:57.905 ","End":"03:59.390","Text":"Let\u0027s just check again."},{"Start":"03:59.390 ","End":"04:02.225","Text":"1^3 is 1^2 is 1."},{"Start":"04:02.225 ","End":"04:05.945","Text":"We get 1 minus 1 minus 4 plus 4 is 0."},{"Start":"04:05.945 ","End":"04:11.420","Text":"We\u0027re okay. Now as soon as I find a root, I stop."},{"Start":"04:11.420 ","End":"04:13.280","Text":"I\u0027ll say more about this later."},{"Start":"04:13.280 ","End":"04:16.475","Text":"The goal is to find all the roots of the polynomial."},{"Start":"04:16.475 ","End":"04:21.740","Text":"I copy pasted a line that we did earlier in an earlier clip that says in"},{"Start":"04:21.740 ","End":"04:27.755","Text":"general that if x equals a is a root of a polynomial P,"},{"Start":"04:27.755 ","End":"04:30.125","Text":"in our case x equals 1,"},{"Start":"04:30.125 ","End":"04:32.375","Text":"then x minus a,"},{"Start":"04:32.375 ","End":"04:38.130","Text":"x minus 1 would be a factor of p. What this means is"},{"Start":"04:38.130 ","End":"04:44.790","Text":"that I can write p(x) as x minus 1 times something else,"},{"Start":"04:44.790 ","End":"04:47.560","Text":"at the moment I\u0027ll call it q(x)."},{"Start":"04:47.560 ","End":"04:51.110","Text":"That\u0027s what it means to be a factor of."},{"Start":"04:51.110 ","End":"04:54.335","Text":"The way we find q(x) is by long division."},{"Start":"04:54.335 ","End":"04:56.854","Text":"We already did the long division before."},{"Start":"04:56.854 ","End":"05:05.005","Text":"I remember, we took this and did a long division of this by x minus 1."},{"Start":"05:05.005 ","End":"05:15.955","Text":"The answer we got was that q(x) was in fact x^2 minus 4."},{"Start":"05:15.955 ","End":"05:19.130","Text":"I\u0027d like to point out the essence of what\u0027s going on here."},{"Start":"05:19.130 ","End":"05:23.285","Text":"We started off with a polynomial in this case of degree 3,"},{"Start":"05:23.285 ","End":"05:25.295","Text":"which was this 1."},{"Start":"05:25.295 ","End":"05:32.469","Text":"Then we found 1 root by whatever method."},{"Start":"05:32.620 ","End":"05:35.195","Text":"Once we have a root,"},{"Start":"05:35.195 ","End":"05:38.870","Text":"we then did a division and then we were able to factorize"},{"Start":"05:38.870 ","End":"05:42.500","Text":"this polynomial as x minus 1 times something else,"},{"Start":"05:42.500 ","End":"05:49.190","Text":"which is this and it turns out that the theorem that if you do this process,"},{"Start":"05:49.190 ","End":"05:56.180","Text":"the remaining roots besides x=1 are all going to be roots of this quotient,"},{"Start":"05:56.180 ","End":"05:59.659","Text":"this bit that you get when you do the long division."},{"Start":"05:59.659 ","End":"06:06.380","Text":"So what happens is we\u0027ve reduced our case from a degree 3 to a degree 2 polynomial."},{"Start":"06:06.380 ","End":"06:08.660","Text":"In general, if we started,"},{"Start":"06:08.660 ","End":"06:10.160","Text":"say, with a degree 7,"},{"Start":"06:10.160 ","End":"06:13.040","Text":"and here we\u0027d have a degree 6 and then we could keep repeating"},{"Start":"06:13.040 ","End":"06:17.660","Text":"the process until we get to degree 2 which we know how to solve."},{"Start":"06:17.660 ","End":"06:21.095","Text":"That\u0027s the essence of the method,"},{"Start":"06:21.095 ","End":"06:29.420","Text":"is you guess or use this method or another method to find 1 root, then you divide."},{"Start":"06:29.420 ","End":"06:30.920","Text":"If you\u0027ve have a root x equals a,"},{"Start":"06:30.920 ","End":"06:35.555","Text":"you divide by x minus a and you get a polynomial of a degree less,"},{"Start":"06:35.555 ","End":"06:37.880","Text":"and it makes it just easier to solve them."},{"Start":"06:37.880 ","End":"06:40.160","Text":"We have this root that we had here,"},{"Start":"06:40.160 ","End":"06:45.630","Text":"plus the ones that we are about to find from this part here."},{"Start":"06:45.740 ","End":"06:48.950","Text":"This bit is degree 2 and we know how to solve it."},{"Start":"06:48.950 ","End":"06:58.660","Text":"In fact, we did solve it before and I\u0027ll just say that the solutions are x=2 or minus 2."},{"Start":"06:58.660 ","End":"07:02.579","Text":"So those are the remaining roots."},{"Start":"07:03.040 ","End":"07:06.635","Text":"We found all 3 roots."},{"Start":"07:06.635 ","End":"07:12.845","Text":"The roots of p are 1,"},{"Start":"07:12.845 ","End":"07:20.555","Text":"2 and minus 2 and that\u0027s all of them because it\u0027s a degree 3."},{"Start":"07:20.555 ","End":"07:24.905","Text":"So we found 1 and that helped us to find the rest."},{"Start":"07:24.905 ","End":"07:29.765","Text":"I\u0027d like to point out that there is no guarantee that any of these"},{"Start":"07:29.765 ","End":"07:35.869","Text":"will work as a root and this method has failed."},{"Start":"07:35.869 ","End":"07:40.670","Text":"In case you\u0027re wondering why is it that I stop after I find the first root?"},{"Start":"07:40.670 ","End":"07:43.100","Text":"Why don\u0027t I continue?"},{"Start":"07:43.100 ","End":"07:44.750","Text":"Let me say this."},{"Start":"07:44.750 ","End":"07:46.970","Text":"Often to make substitutions,"},{"Start":"07:46.970 ","End":"07:54.215","Text":"it\u0027s computationally complicated and it really only helps if you\u0027re lucky."},{"Start":"07:54.215 ","End":"07:58.490","Text":"This turns out to be a lucky case what I call,"},{"Start":"07:58.490 ","End":"08:00.035","Text":"if you do continue,"},{"Start":"08:00.035 ","End":"08:05.470","Text":"you\u0027ll get that x equals 1 works Minus 1 doesn\u0027t work."},{"Start":"08:05.470 ","End":"08:13.980","Text":"2 works, minus 2 works 4 doesn\u0027t work and minus 4 doesn\u0027t work."},{"Start":"08:13.980 ","End":"08:21.780","Text":"In this case we got 3 different integer roots and it\u0027s a degree 3 polynomial."},{"Start":"08:21.780 ","End":"08:23.800","Text":"So we know there can\u0027t be anymore."},{"Start":"08:23.800 ","End":"08:27.355","Text":"In this case, we\u0027re lucky and we could stop here and say,"},{"Start":"08:27.355 ","End":"08:29.170","Text":"okay, we found all 3 roots."},{"Start":"08:29.170 ","End":"08:31.870","Text":"We don\u0027t need to do any long divisions."},{"Start":"08:31.870 ","End":"08:38.005","Text":"But often you\u0027re not lucky and it is complicated to make all these computations often,"},{"Start":"08:38.005 ","End":"08:43.090","Text":"there\u0027s many of them and all you\u0027ve saved is the long division."},{"Start":"08:43.090 ","End":"08:48.850","Text":"So if you hate long division and if you feel lucky, then go ahead."},{"Start":"08:48.850 ","End":"08:52.560","Text":"But my method is to just look for 1."},{"Start":"08:52.560 ","End":"08:54.500","Text":"As soon as you found the first 1,"},{"Start":"08:54.500 ","End":"08:56.400","Text":"you do the long division,"},{"Start":"08:56.400 ","End":"09:02.375","Text":"you cut the degree down by 1 and continue until you get down to a degree 2 or less."},{"Start":"09:02.375 ","End":"09:04.655","Text":"That\u0027s all I\u0027ll say about that."},{"Start":"09:04.655 ","End":"09:09.005","Text":"This is the end of this clip and coming up,"},{"Start":"09:09.005 ","End":"09:15.600","Text":"we\u0027ll do some more examples and I\u0027ll also show you a generalization of this theorem."}],"ID":6352},{"Watched":false,"Name":"Roots of Polynomials - Part 4a","Duration":"6m 40s","ChapterTopicVideoID":6350,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6350.jpeg","UploadDate":"2020-09-30T13:34:05.4670000","DurationForVideoObject":"PT6M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"Continuing with the roots and zeros of polynomials,"},{"Start":"00:03.720 ","End":"00:09.195","Text":"I want to give another example that\u0027s going to use the Integer Root Theorem."},{"Start":"00:09.195 ","End":"00:11.070","Text":"I\u0027ll use the same letter P,"},{"Start":"00:11.070 ","End":"00:14.715","Text":"but I\u0027ll use a different color so we don\u0027t get confused."},{"Start":"00:14.715 ","End":"00:21.180","Text":"This time I\u0027ll take p(x) =x^4 + x ^3"},{"Start":"00:21.180 ","End":"00:30.910","Text":"- 3x^2 - 5x - 2."},{"Start":"00:30.910 ","End":"00:35.930","Text":"We want to find all the 0\u0027s or roots of this"},{"Start":"00:35.930 ","End":"00:43.510","Text":"P. We look for integer roots first so let\u0027s see."},{"Start":"00:43.510 ","End":"00:46.055","Text":"Possible integer roots."},{"Start":"00:46.055 ","End":"00:53.025","Text":"They have to be factors of this constant term, -2."},{"Start":"00:53.025 ","End":"00:58.445","Text":"The only possibilities are plus or minus 1,"},{"Start":"00:58.445 ","End":"01:02.930","Text":"plus or minus 2 for possibilities."},{"Start":"01:02.930 ","End":"01:05.810","Text":"We just start substituting in whatever order,"},{"Start":"01:05.810 ","End":"01:07.520","Text":"let\u0027s say we try 1 first."},{"Start":"01:07.520 ","End":"01:13.400","Text":"Well, 1+1-3-5-2 obviously not 0."},{"Start":"01:13.400 ","End":"01:22.235","Text":"I happen to know because I tried them all that -1 and 2 are the only ones that work."},{"Start":"01:22.235 ","End":"01:26.280","Text":"But let\u0027s just say we tried the -1 next."},{"Start":"01:26.280 ","End":"01:27.600","Text":"Although you might have tried the 2."},{"Start":"01:27.600 ","End":"01:30.300","Text":"If we try the -1, and it works."},{"Start":"01:30.300 ","End":"01:35.550","Text":"Because if we substitute -1, we get 1."},{"Start":"01:35.550 ","End":"01:43.305","Text":"Then -1-3+5-2, and it comes out 0."},{"Start":"01:43.305 ","End":"01:53.055","Text":"We have x=-1 is 1 of the roots."},{"Start":"01:53.055 ","End":"01:56.040","Text":"We can find up to 4."},{"Start":"01:56.040 ","End":"02:00.670","Text":"Then what we do is because x-1 is a root,"},{"Start":"02:00.670 ","End":"02:04.435","Text":"we now do a long division and we do it over here."},{"Start":"02:04.435 ","End":"02:14.540","Text":"We take the original x^4+x^3-3x^2-5x-2,"},{"Start":"02:14.700 ","End":"02:19.690","Text":"and divide it by x minus the root."},{"Start":"02:19.690 ","End":"02:22.465","Text":"It\u0027s x - - 1."},{"Start":"02:22.465 ","End":"02:27.350","Text":"It\u0027s x+1, note."},{"Start":"02:31.610 ","End":"02:34.085","Text":"Look at the leading coefficient."},{"Start":"02:34.085 ","End":"02:36.845","Text":"How many times does this go into this?"},{"Start":"02:36.845 ","End":"02:41.430","Text":"x^3 times. Write x^3 here, then multiply."},{"Start":"02:41.430 ","End":"02:48.540","Text":"We get x^4+x^3 then"},{"Start":"02:48.540 ","End":"02:54.785","Text":"we subtract and a bit more space here."},{"Start":"02:54.785 ","End":"02:59.376","Text":"This leaves nothing so we lower,"},{"Start":"02:59.376 ","End":"03:04.970","Text":"we drop 2 more terms, -3x^2-5x."},{"Start":"03:04.970 ","End":"03:07.260","Text":"Then compare the leading terms,"},{"Start":"03:07.260 ","End":"03:14.040","Text":"x into -3x^2 goes -3x times,"},{"Start":"03:14.040 ","End":"03:22.370","Text":"-3x times x +1 is -3x^2-3x."},{"Start":"03:22.370 ","End":"03:26.224","Text":"Again, subtract this minus, this is nothing."},{"Start":"03:26.224 ","End":"03:29.840","Text":"This minus this is -2x."},{"Start":"03:29.840 ","End":"03:32.435","Text":"Drop the -2."},{"Start":"03:32.435 ","End":"03:34.955","Text":"Look at the leading term."},{"Start":"03:34.955 ","End":"03:39.215","Text":"X goes into -2x, -2 times."},{"Start":"03:39.215 ","End":"03:43.505","Text":"We have -2x and then -2."},{"Start":"03:43.505 ","End":"03:48.305","Text":"Then it goes into it evenly. Nothing left."},{"Start":"03:48.305 ","End":"03:55.260","Text":"This long division tells us that we can factor P(x)"},{"Start":"03:55.260 ","End":"04:03.440","Text":"as (x +1) (x^3-3x-2)."},{"Start":"04:03.440 ","End":"04:09.770","Text":"Now we continue looking for up to 3 more roots just from this part here."},{"Start":"04:09.770 ","End":"04:15.275","Text":"We have the same constant term as left times so we\u0027re still looking at 4 possibilities."},{"Start":"04:15.275 ","End":"04:17.450","Text":"Let\u0027s see, we try again."},{"Start":"04:17.450 ","End":"04:20.705","Text":"There is no point trying x=1."},{"Start":"04:20.705 ","End":"04:23.775","Text":"If you failed here,"},{"Start":"04:23.775 ","End":"04:26.560","Text":"it\u0027s not going to work here."},{"Start":"04:26.570 ","End":"04:33.360","Text":"Again, we could try -1 and indeed -1 works again."},{"Start":"04:33.360 ","End":"04:34.620","Text":"If put in -1,"},{"Start":"04:34.620 ","End":"04:38.820","Text":"you get -1+3-2, and that\u0027s 0."},{"Start":"04:38.820 ","End":"04:45.670","Text":"Once again, I\u0027ll make a note that x=-1 is a root again,"},{"Start":"04:45.670 ","End":"04:47.375","Text":"so it\u0027s already a double root,"},{"Start":"04:47.375 ","End":"04:48.799","Text":"but it could be a triple."},{"Start":"04:48.799 ","End":"04:52.970","Text":"Anyway held multiplicity, at least 2 already and then we\u0027re"},{"Start":"04:52.970 ","End":"04:57.715","Text":"going to do a long division of this by x+1,"},{"Start":"04:57.715 ","End":"05:01.295","Text":"but I\u0027m going to save time and do it for you."},{"Start":"05:01.295 ","End":"05:05.185","Text":"Now, this x+1 is just what I\u0027m copying."},{"Start":"05:05.185 ","End":"05:09.860","Text":"x+1 because I had x=minus 1 again."},{"Start":"05:09.860 ","End":"05:13.345","Text":"Divide this into this."},{"Start":"05:13.345 ","End":"05:15.815","Text":"As I said I\u0027m just saving time."},{"Start":"05:15.815 ","End":"05:25.230","Text":"It comes out to x^2-x-2 from the long division."},{"Start":"05:25.230 ","End":"05:27.860","Text":"Now we\u0027re down to a quadratic."},{"Start":"05:27.860 ","End":"05:30.965","Text":"Once we\u0027re down to a quadratic, I mean,"},{"Start":"05:30.965 ","End":"05:33.980","Text":"you could keep using the integer root theorem,"},{"Start":"05:33.980 ","End":"05:35.120","Text":"but no need to."},{"Start":"05:35.120 ","End":"05:37.650","Text":"We know how to solve quadratics."},{"Start":"05:37.790 ","End":"05:40.070","Text":"Again, to save time,"},{"Start":"05:40.070 ","End":"05:42.710","Text":"I\u0027ll just tell you that the roots of this quadratic"},{"Start":"05:42.710 ","End":"05:52.050","Text":"are x=-1 and x=2."},{"Start":"05:52.050 ","End":"05:59.945","Text":"All together, what we have for our roots are that x can equal as 4 roots,"},{"Start":"05:59.945 ","End":"06:01.820","Text":"but some of them are the same."},{"Start":"06:01.820 ","End":"06:04.805","Text":"-1, -1,"},{"Start":"06:04.805 ","End":"06:08.430","Text":"-1, and 2."},{"Start":"06:08.890 ","End":"06:11.615","Text":"I\u0027ll highlight the answer."},{"Start":"06:11.615 ","End":"06:17.900","Text":"Although you could also say that the roots,"},{"Start":"06:17.900 ","End":"06:22.685","Text":"the different roots are just -1 and 2."},{"Start":"06:22.685 ","End":"06:26.350","Text":"You could then note that the multiplicities,"},{"Start":"06:26.350 ","End":"06:30.350","Text":"this has multiplicity 3 and this has multiplicity 1."},{"Start":"06:30.350 ","End":"06:33.590","Text":"It\u0027s just a different way of bookkeeping."},{"Start":"06:33.590 ","End":"06:37.430","Text":"Anyway, there we are."},{"Start":"06:37.430 ","End":"06:41.249","Text":"I think it\u0027s time for a break."}],"ID":6353},{"Watched":false,"Name":"Roots of Polynomials - Part 4b","Duration":"9m 5s","ChapterTopicVideoID":6351,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6351.jpeg","UploadDate":"2020-09-30T13:42:23.9330000","DurationForVideoObject":"PT9M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.140 ","End":"00:02.970","Text":"Here we are after the break."},{"Start":"00:02.970 ","End":"00:05.260","Text":"Just to remind you where we were,"},{"Start":"00:05.260 ","End":"00:09.100","Text":"I\u0027ll scroll back to the top of the page."},{"Start":"00:09.100 ","End":"00:13.839","Text":"We were doing the integer root theorem."},{"Start":"00:13.839 ","End":"00:16.420","Text":"Note that in the integer root theorem,"},{"Start":"00:16.420 ","End":"00:21.835","Text":"we require that the polynomial has a leading coefficient of 1."},{"Start":"00:21.835 ","End":"00:24.310","Text":"That was true for our example."},{"Start":"00:24.310 ","End":"00:26.575","Text":"I\u0027m going to change the example,"},{"Start":"00:26.575 ","End":"00:31.900","Text":"this time I\u0027ll take p(x) equals not 1,"},{"Start":"00:31.900 ","End":"00:35.580","Text":"2 I\u0027m going to make a a 4th degree,"},{"Start":"00:35.580 ","End":"00:44.245","Text":"x^4 plus x^3 plus 3x^2 plus 3x minus 9."},{"Start":"00:44.245 ","End":"00:47.510","Text":"Now the integer root theorem is no good to me anymore."},{"Start":"00:47.510 ","End":"00:53.149","Text":"I\u0027m going to instead talk about the rational root theorem."},{"Start":"00:53.149 ","End":"00:57.410","Text":"Remember a rational number is a fraction, like a/b."},{"Start":"00:57.410 ","End":"01:02.110","Text":"I need to make some changes here."},{"Start":"01:02.110 ","End":"01:07.400","Text":"I\u0027m not going to assume any more that the polynomial has a leading coefficient of 1."},{"Start":"01:07.400 ","End":"01:10.820","Text":"I\u0027ll just put this out of the way,"},{"Start":"01:10.820 ","End":"01:13.180","Text":"I can use it for recycling."},{"Start":"01:13.180 ","End":"01:18.235","Text":"This time all assuming is that P is a polynomial with integer coefficients."},{"Start":"01:18.235 ","End":"01:24.545","Text":"Then I conclude that instead of talking about integer,"},{"Start":"01:24.545 ","End":"01:34.050","Text":"I\u0027ll talk about rational roots of P. An irrational root of P is of the form,"},{"Start":"01:34.050 ","End":"01:43.985","Text":"it\u0027s a fraction, we call it a/b and now I\u0027m going to give some conditions on a and b."},{"Start":"01:43.985 ","End":"01:48.170","Text":"We\u0027ll say a something and then b something,"},{"Start":"01:48.170 ","End":"01:53.335","Text":"a, is a factor of the constant term."},{"Start":"01:53.335 ","End":"01:57.050","Text":"In our example, this is the constant term."},{"Start":"01:57.050 ","End":"01:58.820","Text":"This also has a name,"},{"Start":"01:58.820 ","End":"02:01.925","Text":"this is the leading coefficient,"},{"Start":"02:01.925 ","End":"02:04.700","Text":"the coefficient of the highest order term,"},{"Start":"02:04.700 ","End":"02:09.080","Text":"and this is going to be useful for b. P is also a factor,"},{"Start":"02:09.080 ","End":"02:14.960","Text":"but this time of the leading coefficient."},{"Start":"02:15.080 ","End":"02:17.670","Text":"Getting back to our example,"},{"Start":"02:17.670 ","End":"02:21.385","Text":"let\u0027s see what are the possibilities for a and for b."},{"Start":"02:21.385 ","End":"02:23.955","Text":"The factors of 9,"},{"Start":"02:23.955 ","End":"02:26.200","Text":"are 1,3 and 9."},{"Start":"02:26.200 ","End":"02:34.200","Text":"Let\u0027s just ignore the plus or minus for the moment and b has to be a factor of 2."},{"Start":"02:34.200 ","End":"02:39.370","Text":"So b could be 1 or 2 again, plus or minus."},{"Start":"02:39.890 ","End":"02:43.130","Text":"Our root, call it x,"},{"Start":"02:43.130 ","End":"02:48.575","Text":"has possibility of being a/b and we\u0027ll put in the plus or minus."},{"Start":"02:48.575 ","End":"02:52.770","Text":"It could be 1/1"},{"Start":"02:54.830 ","End":"03:02.730","Text":"it could be plus or minus 3/1 plus or minus 9/1."},{"Start":"03:02.730 ","End":"03:05.840","Text":"That\u0027s just taking b equals 1 and we have more."},{"Start":"03:05.840 ","End":"03:09.155","Text":"We have plus or minus 1/2,"},{"Start":"03:09.155 ","End":"03:18.375","Text":"plus or minus 3/2 and plus or minus 9/2."},{"Start":"03:18.375 ","End":"03:23.070","Text":"Let\u0027s see the 6 times plus or minus 12 possibilities."},{"Start":"03:23.070 ","End":"03:32.695","Text":"Let me just rewrite the ones that are over one as integers 1, 3 and 9."},{"Start":"03:32.695 ","End":"03:38.440","Text":"Now all these substitutions can be quite tedious computationally."},{"Start":"03:38.440 ","End":"03:44.155","Text":"I\u0027d recommend you start with the integers first and then go to the fractions."},{"Start":"03:44.155 ","End":"03:47.770","Text":"I cooked it so we get lucky first time x equals"},{"Start":"03:47.770 ","End":"03:53.605","Text":"1 will work as the root if you plug it in,"},{"Start":"03:53.605 ","End":"04:00.445","Text":"but you can see 2 plus 1 plus 3 plus 3 is 9 minus 9 is 0."},{"Start":"04:00.445 ","End":"04:03.460","Text":"What we do next is long division."},{"Start":"04:03.460 ","End":"04:05.815","Text":"We take this polynomial,"},{"Start":"04:05.815 ","End":"04:08.225","Text":"I just copied this over here."},{"Start":"04:08.225 ","End":"04:14.105","Text":"Now we need to divide it by x minus 1,"},{"Start":"04:14.105 ","End":"04:21.620","Text":"x into 2x^4 goes 2x^3 times,"},{"Start":"04:21.620 ","End":"04:31.665","Text":"multiplying we get 2x^3 minus x^4 here, of course."},{"Start":"04:31.665 ","End":"04:37.760","Text":"Then subtracting, I get 3x^3,"},{"Start":"04:37.760 ","End":"04:40.290","Text":"bring down the 3x^2,"},{"Start":"04:40.290 ","End":"04:42.555","Text":"make some more space."},{"Start":"04:42.555 ","End":"04:47.605","Text":"Then x into 3x^3 goes"},{"Start":"04:47.605 ","End":"04:54.970","Text":"3x^2 times multiplying 3x^3 minus 3x^2."},{"Start":"04:55.490 ","End":"05:02.490","Text":"Subtracting, it\u0027s 6x^2 plus 3x,"},{"Start":"05:02.490 ","End":"05:05.880","Text":"x goes into 6x^2,"},{"Start":"05:05.880 ","End":"05:12.520","Text":"6x times, so it\u0027s 6x^2 minus 6x."},{"Start":"05:12.830 ","End":"05:15.990","Text":"Again, subtracting."},{"Start":"05:15.990 ","End":"05:20.130","Text":"This is 9x minus 9,"},{"Start":"05:20.130 ","End":"05:22.995","Text":"x goes into 9x,"},{"Start":"05:22.995 ","End":"05:30.225","Text":"9 times and we get 9x minus 9 so it goes in evenly."},{"Start":"05:30.225 ","End":"05:33.500","Text":"I\u0027m going to copy this quotient over here."},{"Start":"05:33.500 ","End":"05:36.190","Text":"Now we\u0027re down to a cubic polynomial,"},{"Start":"05:36.190 ","End":"05:42.780","Text":"but we have the same leading coefficient and the constant term."},{"Start":"05:42.780 ","End":"05:48.870","Text":"We can still use the same possible values here."},{"Start":"05:48.870 ","End":"05:52.405","Text":"Once again, because it\u0027s tedious to substitute each time,"},{"Start":"05:52.405 ","End":"06:00.895","Text":"I\u0027m going to tell you that the one that works is actually x equals minus 3 over 2."},{"Start":"06:00.895 ","End":"06:04.060","Text":"You would\u0027ve tried all these 6,7,8,9,"},{"Start":"06:04.060 ","End":"06:07.040","Text":"the 10th one, 10th time, lucky."},{"Start":"06:07.040 ","End":"06:08.610","Text":"I\u0027ll write that down,"},{"Start":"06:08.610 ","End":"06:11.415","Text":"x equals minus 3/2."},{"Start":"06:11.415 ","End":"06:15.920","Text":"This time we have to do another long division."},{"Start":"06:15.920 ","End":"06:18.275","Text":"I\u0027m going to copy this over here."},{"Start":"06:18.275 ","End":"06:22.160","Text":"Put a dividing sign and remember to change the sign."},{"Start":"06:22.160 ","End":"06:26.300","Text":"It\u0027s x plus 3/2 because it\u0027s x minus the root."},{"Start":"06:26.300 ","End":"06:32.595","Text":"X into 2x^3 goes 2x^2 times."},{"Start":"06:32.595 ","End":"06:38.280","Text":"Then this multiplied by this is 2x^3 plus,"},{"Start":"06:38.280 ","End":"06:42.550","Text":"2 times 3/2 is 3, so it\u0027s 3x^2."},{"Start":"06:42.980 ","End":"06:45.825","Text":"This subtraction, nothing left,"},{"Start":"06:45.825 ","End":"06:48.045","Text":"we bring down 2 more terms."},{"Start":"06:48.045 ","End":"06:49.875","Text":"6x plus 9,"},{"Start":"06:49.875 ","End":"06:55.905","Text":"x into minus 6x goes minus 6 times,"},{"Start":"06:55.905 ","End":"07:01.080","Text":"so we get multiplying out minus 6x minus,"},{"Start":"07:01.080 ","End":"07:04.930","Text":"6 times 3/2 is,"},{"Start":"07:07.120 ","End":"07:14.930","Text":"I just noticed this as a plus 6 so has to be a plus 6 I better fix it over"},{"Start":"07:14.930 ","End":"07:24.165","Text":"here as well so there\u0027s no minus here and it\u0027s 6x plus 9 and that goes in evenly."},{"Start":"07:24.165 ","End":"07:26.820","Text":"This is the quotient."},{"Start":"07:26.820 ","End":"07:28.875","Text":"I\u0027ll copy it over here."},{"Start":"07:28.875 ","End":"07:32.925","Text":"We\u0027re now down to a quadratic 2x^2 plus 6,"},{"Start":"07:32.925 ","End":"07:36.150","Text":"but this is an irreducible quadratic,"},{"Start":"07:36.150 ","End":"07:37.965","Text":"it has no roots."},{"Start":"07:37.965 ","End":"07:40.590","Text":"If you tried solving it,"},{"Start":"07:40.590 ","End":"07:44.150","Text":"if you let it equal 0, you get x^2 is minus 3."},{"Start":"07:44.150 ","End":"07:45.805","Text":"You can see this is irreducible,"},{"Start":"07:45.805 ","End":"07:49.030","Text":"I\u0027ll just write that reducible."},{"Start":"07:49.030 ","End":"07:53.190","Text":"P(x) only has 2 roots,"},{"Start":"07:53.190 ","End":"07:57.105","Text":"that is 1 and minus 3/2."},{"Start":"07:57.105 ","End":"07:59.410","Text":"But actually I didn\u0027t ask a question,"},{"Start":"07:59.410 ","End":"08:00.760","Text":"I didn\u0027t say find the roots."},{"Start":"08:00.760 ","End":"08:03.880","Text":"The question could have been factorize P(x)."},{"Start":"08:03.880 ","End":"08:06.595","Text":"Let\u0027s say I asked you to factorize P(x),"},{"Start":"08:06.595 ","End":"08:09.505","Text":"then I would be able to say that P(x) is,"},{"Start":"08:09.505 ","End":"08:10.690","Text":"first of all from the 1,"},{"Start":"08:10.690 ","End":"08:13.220","Text":"I get x minus 1."},{"Start":"08:13.220 ","End":"08:15.780","Text":"Then from the minus 3 over 2,"},{"Start":"08:15.780 ","End":"08:18.540","Text":"I get x plus 3 over 2."},{"Start":"08:18.540 ","End":"08:20.265","Text":"Then I get the last bit,"},{"Start":"08:20.265 ","End":"08:24.040","Text":"the 2x^2 plus 6."},{"Start":"08:24.160 ","End":"08:27.110","Text":"That would be just fine as an answer,"},{"Start":"08:27.110 ","End":"08:30.155","Text":"but there is some slight improvement I could make."},{"Start":"08:30.155 ","End":"08:32.990","Text":"If I took the 2 out here,"},{"Start":"08:32.990 ","End":"08:35.120","Text":"and that\u0027s really part of the factorization,"},{"Start":"08:35.120 ","End":"08:37.925","Text":"I\u0027d get 2 times x^2 plus 3."},{"Start":"08:37.925 ","End":"08:40.010","Text":"If I put the 2 in here,"},{"Start":"08:40.010 ","End":"08:45.070","Text":"then I can write this thing as x minus 1 and then 2x plus"},{"Start":"08:45.070 ","End":"08:50.315","Text":"3 could just multiply everything by 2 times x^2 plus 3."},{"Start":"08:50.315 ","End":"08:54.205","Text":"Then I\u0027ve got a nice factorization without any fractions."},{"Start":"08:54.205 ","End":"08:57.620","Text":"There\u0027s plenty more examples of the rational root theorem in"},{"Start":"08:57.620 ","End":"09:02.705","Text":"the example exercises following the tutorial."},{"Start":"09:02.705 ","End":"09:05.550","Text":"We are done here."}],"ID":6354},{"Watched":false,"Name":"Complex Roots of Polynomials","Duration":"13m 34s","ChapterTopicVideoID":6338,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6338.jpeg","UploadDate":"2020-09-30T13:57:54.3730000","DurationForVideoObject":"PT13M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.570","Text":"In this clip, I\u0027m going to be talking again about roots or zeros of polynomials."},{"Start":"00:06.570 ","End":"00:10.425","Text":"I say again, because there already was a tutorial,"},{"Start":"00:10.425 ","End":"00:16.755","Text":"but this time I\u0027m going to assume that you\u0027ve studied complex numbers."},{"Start":"00:16.755 ","End":"00:23.564","Text":"The things we said earlier become a bit different if you allow complex roots also."},{"Start":"00:23.564 ","End":"00:25.755","Text":"When I say polynomial,"},{"Start":"00:25.755 ","End":"00:30.130","Text":"we\u0027re going to assume that they have integer coefficients."},{"Start":"00:30.130 ","End":"00:32.210","Text":"In the previous tutorial,"},{"Start":"00:32.210 ","End":"00:34.760","Text":"we started off by discussing,"},{"Start":"00:34.760 ","End":"00:37.685","Text":"first of all, quadratic polynomials,"},{"Start":"00:37.685 ","End":"00:41.460","Text":"and we actually had three examples there."},{"Start":"00:41.620 ","End":"00:50.210","Text":"Our first example was polynomial a^2 minus 3x plus 2."},{"Start":"00:50.210 ","End":"00:53.970","Text":"Then we had two more examples."},{"Start":"00:54.430 ","End":"01:00.335","Text":"The example we used here was x^2 minus 2x plus 1,"},{"Start":"01:00.335 ","End":"01:04.350","Text":"and here we had x^2 plus 4."},{"Start":"01:04.550 ","End":"01:10.129","Text":"What we concluded there was we solve the quadratic equation,"},{"Start":"01:10.129 ","End":"01:13.285","Text":"x^2 minus 3x plus 2 equals 0."},{"Start":"01:13.285 ","End":"01:15.960","Text":"We got two solutions,"},{"Start":"01:15.960 ","End":"01:18.120","Text":"which was 2,"},{"Start":"01:18.120 ","End":"01:21.690","Text":"it\u0027s 4 plus 1 over 2 and 2 over 2 is 1,"},{"Start":"01:21.690 ","End":"01:24.915","Text":"and we had two separate roots."},{"Start":"01:24.915 ","End":"01:34.665","Text":"The roots where x =1 or x=2, two different roots."},{"Start":"01:34.665 ","End":"01:37.384","Text":"In the second case,"},{"Start":"01:37.384 ","End":"01:41.825","Text":"we tried the formula also,"},{"Start":"01:41.825 ","End":"01:44.434","Text":"which left us with the same,"},{"Start":"01:44.434 ","End":"01:47.540","Text":"whether you take the plus or the minus 1 and 1."},{"Start":"01:47.540 ","End":"01:52.190","Text":"Rather than saying that we only have one root, which is 1,"},{"Start":"01:52.190 ","End":"01:59.400","Text":"we say that we have two roots, x=1 or 1."},{"Start":"01:59.400 ","End":"02:01.665","Text":"We count the 1 twice."},{"Start":"02:01.665 ","End":"02:06.500","Text":"This is a different bookkeeping and we allow repeated roots."},{"Start":"02:06.500 ","End":"02:09.235","Text":"So in this case, we also have two roots."},{"Start":"02:09.235 ","End":"02:13.380","Text":"We allow for repeated roots,"},{"Start":"02:13.380 ","End":"02:16.455","Text":"and we also introduced the term multiplicity,"},{"Start":"02:16.455 ","End":"02:23.545","Text":"so we said that x=1 has multiplicity of 2,"},{"Start":"02:23.545 ","End":"02:26.045","Text":"or we sometimes call this a double root."},{"Start":"02:26.045 ","End":"02:28.850","Text":"Now, where we are going to greatly differ from"},{"Start":"02:28.850 ","End":"02:32.825","Text":"the case of real numbers only and we allow complex,"},{"Start":"02:32.825 ","End":"02:36.050","Text":"it was in the third case where we found there was no solution."},{"Start":"02:36.050 ","End":"02:40.760","Text":"If you set x^2 plus 4 equals 0 and don\u0027t use the formula,"},{"Start":"02:40.760 ","End":"02:43.715","Text":"we get x^2 equals minus 4,"},{"Start":"02:43.715 ","End":"02:48.120","Text":"and then we couldn\u0027t solve it with real numbers,"},{"Start":"02:48.120 ","End":"02:49.985","Text":"but then when you know complex numbers,"},{"Start":"02:49.985 ","End":"02:57.480","Text":"you\u0027ll see that this is plus or minus square root of 4 times i,"},{"Start":"02:57.480 ","End":"02:59.940","Text":"which is plus or minus 2i."},{"Start":"02:59.940 ","End":"03:07.090","Text":"The roots, in this case are 2i and minus 2i."},{"Start":"03:07.850 ","End":"03:14.285","Text":"What I can say then is that if you allow for complex roots,"},{"Start":"03:14.285 ","End":"03:17.420","Text":"then we have three possibilities,"},{"Start":"03:17.420 ","End":"03:21.880","Text":"either two different roots and then they are always real numbers,"},{"Start":"03:21.880 ","End":"03:27.785","Text":"or we have a repeated or double root or root with multiplicity two,"},{"Start":"03:27.785 ","End":"03:29.720","Text":"in this case, it\u0027s also real,"},{"Start":"03:29.720 ","End":"03:32.935","Text":"or we have two complex numbers."},{"Start":"03:32.935 ","End":"03:34.430","Text":"But in all cases,"},{"Start":"03:34.430 ","End":"03:38.940","Text":"we have two roots if you allow for multiplicity."},{"Start":"03:40.150 ","End":"03:42.350","Text":"I\u0027ll just make a note of that,"},{"Start":"03:42.350 ","End":"03:44.090","Text":"in the case of complex numbers,"},{"Start":"03:44.090 ","End":"03:49.550","Text":"we have always two roots,"},{"Start":"03:49.550 ","End":"03:53.269","Text":"allowing for complex and allowing for repeated."},{"Start":"03:53.269 ","End":"03:55.610","Text":"Let me do another example."},{"Start":"03:55.610 ","End":"03:57.830","Text":"Let\u0027s see P, Q, R,"},{"Start":"03:57.830 ","End":"03:59.425","Text":"let\u0027s call it S(x) will be"},{"Start":"03:59.425 ","End":"04:07.922","Text":"x^2 minus"},{"Start":"04:07.922 ","End":"04:11.395","Text":"2x plus 5."},{"Start":"04:11.395 ","End":"04:13.500","Text":"S looks too much like 5,"},{"Start":"04:13.500 ","End":"04:17.710","Text":"I\u0027ll change that to, how about T?"},{"Start":"04:17.980 ","End":"04:21.260","Text":"Let\u0027s see what we get in this case."},{"Start":"04:21.260 ","End":"04:25.490","Text":"What are the roots, the roots are the solutions of the equation where this equals 0,"},{"Start":"04:25.490 ","End":"04:31.219","Text":"and we get that x can be 2 plus"},{"Start":"04:31.219 ","End":"04:37.290","Text":"or minus the square root of b^2 is 4 minus 4 times a is 1,"},{"Start":"04:37.290 ","End":"04:43.830","Text":"I don\u0027t write it, times c over 2a is 2."},{"Start":"04:43.830 ","End":"04:49.250","Text":"Let\u0027s see, 4 minus 20 is minus 16."},{"Start":"04:49.250 ","End":"04:53.375","Text":"Here, we have the square root of minus 16,"},{"Start":"04:53.375 ","End":"05:01.825","Text":"2 plus or minus 4i over 2,"},{"Start":"05:01.825 ","End":"05:09.745","Text":"which comes out to be 1 plus or minus 2i."},{"Start":"05:09.745 ","End":"05:15.380","Text":"In other words, the roots for this one,"},{"Start":"05:15.380 ","End":"05:23.200","Text":"there\u0027s two of them, 1 plus 2i and 1 minus 2i."},{"Start":"05:23.770 ","End":"05:30.440","Text":"What I want you to notice is that whenever you have two complex roots,"},{"Start":"05:30.440 ","End":"05:33.230","Text":"they\u0027re always conjugates of each other."},{"Start":"05:33.230 ","End":"05:35.836","Text":"This could be useful later on."},{"Start":"05:35.836 ","End":"05:39.320","Text":"I\u0027d like to highlight the roots,"},{"Start":"05:39.320 ","End":"05:43.740","Text":"and in fact, I meant to highlight them all along."},{"Start":"05:43.740 ","End":"05:45.555","Text":"Here we had 2i and minus 2i,"},{"Start":"05:45.555 ","End":"05:49.475","Text":"notice that these are also complex conjugates of each other."},{"Start":"05:49.475 ","End":"05:52.385","Text":"This was the case where we had two the same,"},{"Start":"05:52.385 ","End":"05:55.405","Text":"this is the case where we had two different ones."},{"Start":"05:55.405 ","End":"05:59.960","Text":"That\u0027s four examples, and that\u0027s quadratic."},{"Start":"05:59.960 ","End":"06:02.510","Text":"Now, I\u0027m going to generalize this."},{"Start":"06:02.510 ","End":"06:05.910","Text":"A quadratic is degree 2,"},{"Start":"06:05.910 ","End":"06:11.880","Text":"so let\u0027s generalize to degree n where n is anything."},{"Start":"06:11.920 ","End":"06:17.845","Text":"In this case, we always"},{"Start":"06:17.845 ","End":"06:25.400","Text":"get n roots and as before allowing for repeated roots,"},{"Start":"06:25.400 ","End":"06:30.715","Text":"in other words, roots with multiplicity higher than 1 and complex roots."},{"Start":"06:30.715 ","End":"06:35.210","Text":"I just copied an example that we had from the previous clip."},{"Start":"06:35.210 ","End":"06:37.100","Text":"Before we had complex numbers,"},{"Start":"06:37.100 ","End":"06:44.880","Text":"we had this and we managed to factorize it and we got that"},{"Start":"06:44.880 ","End":"06:55.455","Text":"the roots were 1 and from here minus 3/2."},{"Start":"06:55.455 ","End":"07:00.800","Text":"This was irreducible and those were the only two roots that we got."},{"Start":"07:00.800 ","End":"07:04.399","Text":"But now that we have complex numbers,"},{"Start":"07:04.399 ","End":"07:10.025","Text":"we can get two additional roots by setting this equal to 0,"},{"Start":"07:10.025 ","End":"07:14.150","Text":"and we get x^2 plus 3 is 0,"},{"Start":"07:14.150 ","End":"07:19.550","Text":"so we get that x and also b,"},{"Start":"07:19.550 ","End":"07:23.740","Text":"plus or minus root 3i,"},{"Start":"07:23.740 ","End":"07:26.685","Text":"because x squared is minus 3."},{"Start":"07:26.685 ","End":"07:31.065","Text":"I\u0027ll write this as two separate roots,"},{"Start":"07:31.065 ","End":"07:34.080","Text":"root 3i and minus root 3i,"},{"Start":"07:34.080 ","End":"07:35.670","Text":"and we count them 1,"},{"Start":"07:35.670 ","End":"07:37.335","Text":"2, 3, 4,"},{"Start":"07:37.335 ","End":"07:40.125","Text":"and we have a degree for polynomial,"},{"Start":"07:40.125 ","End":"07:44.410","Text":"and so we get a full n roots."},{"Start":"07:44.440 ","End":"07:50.375","Text":"I\u0027ll just highlight them and then we\u0027ll go and do another exercise."},{"Start":"07:50.375 ","End":"07:53.370","Text":"I\u0027m going to disguise it."},{"Start":"07:53.740 ","End":"08:01.515","Text":"I\u0027ll just ask you to solve the equation x^3 equals 8."},{"Start":"08:01.515 ","End":"08:04.304","Text":"It doesn\u0027t look like a roots problem,"},{"Start":"08:04.304 ","End":"08:07.190","Text":"but if I rephrase it as,"},{"Start":"08:07.190 ","End":"08:15.455","Text":"find the roots of the polynomial x cubed minus 8,"},{"Start":"08:15.455 ","End":"08:18.695","Text":"now it is a roots problem and it\u0027s the same thing"},{"Start":"08:18.695 ","End":"08:24.810","Text":"because to find the roots of x^3 minus 8 means to solve x^3 minus 8 equals 0,"},{"Start":"08:24.810 ","End":"08:26.660","Text":"which is the same as x^8 equals 8."},{"Start":"08:26.660 ","End":"08:29.720","Text":"I just thought it would be more interesting to phrase it this way."},{"Start":"08:29.720 ","End":"08:33.260","Text":"Normally, if I say to you what number cubed is 8,"},{"Start":"08:33.260 ","End":"08:35.110","Text":"you\u0027ll say it\u0027s only 2,"},{"Start":"08:35.110 ","End":"08:39.380","Text":"and you\u0027d be right if we\u0027re only considering real numbers."},{"Start":"08:39.380 ","End":"08:41.195","Text":"But according to this theorem,"},{"Start":"08:41.195 ","End":"08:43.870","Text":"this is a degree three polynomial,"},{"Start":"08:43.870 ","End":"08:45.560","Text":"we have to have three roots."},{"Start":"08:45.560 ","End":"08:48.620","Text":"In complex numbers, we should find three solutions,"},{"Start":"08:48.620 ","End":"08:53.245","Text":"three numbers whose cube is 8."},{"Start":"08:53.245 ","End":"08:55.665","Text":"Well, let\u0027s go ahead and do it."},{"Start":"08:55.665 ","End":"08:58.770","Text":"I\u0027ll just give it a name, I\u0027ll call it p(x),"},{"Start":"08:58.770 ","End":"09:01.830","Text":"which is x^3 minus 8,"},{"Start":"09:01.830 ","End":"09:10.970","Text":"and the usual technique is to somehow find one root of this."},{"Start":"09:10.970 ","End":"09:18.710","Text":"Now, you can use the rational factor theorem or the integer factor theorem."},{"Start":"09:18.710 ","End":"09:26.490","Text":"But the easiest thing is just to look at this and we know one solution of this is x=2."},{"Start":"09:26.560 ","End":"09:30.110","Text":"We know that x=2 is one root,"},{"Start":"09:30.110 ","End":"09:31.550","Text":"and once you have one root,"},{"Start":"09:31.550 ","End":"09:36.140","Text":"then we can do a long division and take our x^3 minus 8,"},{"Start":"09:36.140 ","End":"09:37.220","Text":"but we leave spaces."},{"Start":"09:37.220 ","End":"09:43.390","Text":"In fact, let me write it as 0x^2 plus 0x minus 8,"},{"Start":"09:43.390 ","End":"09:47.640","Text":"and then put a division sign."},{"Start":"09:47.640 ","End":"09:51.410","Text":"We know it\u0027s going to be divisible by x minus 2."},{"Start":"09:51.410 ","End":"09:53.355","Text":"It\u0027s x minus the root."},{"Start":"09:53.355 ","End":"09:58.070","Text":"Let\u0027s see, x goes into x^3,"},{"Start":"09:58.740 ","End":"10:09.015","Text":"x^2 times multiply x^2 by x minus 2 and we have x^3 minus 2x^2."},{"Start":"10:09.015 ","End":"10:11.085","Text":"Now, a subtraction,"},{"Start":"10:11.085 ","End":"10:12.975","Text":"this minus this is nothing,"},{"Start":"10:12.975 ","End":"10:15.750","Text":"this minus this is 2x^2,"},{"Start":"10:15.750 ","End":"10:19.065","Text":"x into 2x^2,"},{"Start":"10:19.065 ","End":"10:24.165","Text":"goes 2x times,"},{"Start":"10:24.165 ","End":"10:27.915","Text":"drop down the next term,"},{"Start":"10:27.915 ","End":"10:31.860","Text":"then multiply 2x by x minus 2,"},{"Start":"10:31.860 ","End":"10:37.980","Text":"so it\u0027s 2x^2 minus 4x subtracting,"},{"Start":"10:37.980 ","End":"10:41.625","Text":"we get 4x, bring down the minus 8,"},{"Start":"10:41.625 ","End":"10:43.380","Text":"x into 4x,"},{"Start":"10:43.380 ","End":"10:45.690","Text":"goes 4 times,"},{"Start":"10:45.690 ","End":"10:53.670","Text":"here we get 4x minus 8 and so it goes in evenly."},{"Start":"10:53.670 ","End":"10:56.130","Text":"We didn\u0027t expect any remainder,"},{"Start":"10:56.130 ","End":"10:57.880","Text":"this is the quotient."},{"Start":"10:57.880 ","End":"11:02.000","Text":"Now, we have to find the roots of this."},{"Start":"11:02.000 ","End":"11:04.794","Text":"To find the roots of this, this is a quadratic,"},{"Start":"11:04.794 ","End":"11:06.970","Text":"we just solve the equation,"},{"Start":"11:06.970 ","End":"11:11.885","Text":"x^2 plus 2x plus 4 equals 0."},{"Start":"11:11.885 ","End":"11:14.690","Text":"Let\u0027s use the quadratic formula on this,"},{"Start":"11:14.690 ","End":"11:20.360","Text":"so we get that x is equal to minus b plus or minus"},{"Start":"11:20.360 ","End":"11:26.220","Text":"the square root of b^2 minus 4 times a,"},{"Start":"11:26.220 ","End":"11:29.820","Text":"which is 1 times c,"},{"Start":"11:29.820 ","End":"11:34.650","Text":"and all this over 2a which is 2."},{"Start":"11:34.650 ","End":"11:41.970","Text":"Now, 4 minus 16 is minus 12,"},{"Start":"11:41.970 ","End":"11:49.245","Text":"so we get minus 2 plus or minus the square root of minus 12 over 2."},{"Start":"11:49.245 ","End":"11:52.385","Text":"It\u0027s possible to simplify this a bit,"},{"Start":"11:52.385 ","End":"12:03.220","Text":"because minus 12 is 3 times 4 times minus 1."},{"Start":"12:03.710 ","End":"12:08.985","Text":"What we get is minus 2 plus or minus,"},{"Start":"12:08.985 ","End":"12:10.530","Text":"now the square root of minus 12,"},{"Start":"12:10.530 ","End":"12:12.220","Text":"we take the square root of each bit separately."},{"Start":"12:12.220 ","End":"12:14.470","Text":"I\u0027ll take first of all, the square root of 4,"},{"Start":"12:14.470 ","End":"12:15.955","Text":"which is 2,"},{"Start":"12:15.955 ","End":"12:18.200","Text":"then the square root of 3,"},{"Start":"12:18.200 ","End":"12:21.270","Text":"and then square root of minus 1 is i,"},{"Start":"12:21.270 ","End":"12:25.020","Text":"all this over 2."},{"Start":"12:25.020 ","End":"12:27.570","Text":"Now, the 2\u0027s cancel,"},{"Start":"12:27.570 ","End":"12:35.885","Text":"so I can get that this is minus 1 plus or minus the square root of 3i."},{"Start":"12:35.885 ","End":"12:42.485","Text":"So I really do get three roots."},{"Start":"12:42.485 ","End":"12:50.040","Text":"The roots are 2 minus 1 plus square root of 3i,"},{"Start":"12:50.040 ","End":"12:54.825","Text":"and minus 1 minus square root of 3i."},{"Start":"12:54.825 ","End":"12:57.225","Text":"Polynomial of degree 3,"},{"Start":"12:57.225 ","End":"12:59.130","Text":"and three roots 1,"},{"Start":"12:59.130 ","End":"13:01.785","Text":"2, and 3,"},{"Start":"13:01.785 ","End":"13:03.980","Text":"and that\u0027s about it."},{"Start":"13:03.980 ","End":"13:10.490","Text":"But I\u0027d like to just leave you with an extra assignment, optional,"},{"Start":"13:10.490 ","End":"13:20.330","Text":"as to verify that these really are solutions or at least verify one other one besides 2."},{"Start":"13:20.330 ","End":"13:22.070","Text":"I know that 2^3 is 8."},{"Start":"13:22.070 ","End":"13:25.970","Text":"Take this here, and as a complex number,"},{"Start":"13:25.970 ","End":"13:28.730","Text":"raise it to the power of 3 and see if you get 8."},{"Start":"13:28.730 ","End":"13:31.144","Text":"That\u0027s an optional extra assignment."},{"Start":"13:31.144 ","End":"13:33.780","Text":"Anyway, I\u0027m done."}],"ID":6355},{"Watched":false,"Name":"Roots of Polynomials Using Derivatives","Duration":"15m 11s","ChapterTopicVideoID":6339,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6339.jpeg","UploadDate":"2020-09-30T14:09:18.3470000","DurationForVideoObject":"PT15M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this clip, we discussed an additional technique for finding"},{"Start":"00:04.290 ","End":"00:08.505","Text":"the roots of polynomials using the derivative."},{"Start":"00:08.505 ","End":"00:11.310","Text":"Assuming that you\u0027ve studied derivatives,"},{"Start":"00:11.310 ","End":"00:15.460","Text":"if not, then you can skip this clip."},{"Start":"00:15.680 ","End":"00:19.830","Text":"Let me start right away with an example and we\u0027ll illustrate"},{"Start":"00:19.830 ","End":"00:24.090","Text":"the technique during the course of the example."},{"Start":"00:24.090 ","End":"00:27.030","Text":"I\u0027m going to take a fifth-degree polynomial."},{"Start":"00:27.030 ","End":"00:29.925","Text":"The one I\u0027m writing here."},{"Start":"00:29.925 ","End":"00:32.820","Text":"The x_5 plus 3x_4 plus 2x cubed,"},{"Start":"00:32.820 ","End":"00:37.785","Text":"minus 2x^2 minus 3x minus 1."},{"Start":"00:37.785 ","End":"00:41.230","Text":"I\u0027d like to find all its roots."},{"Start":"00:41.230 ","End":"00:45.785","Text":"Polynomial with integer coefficients and leading coefficient 1."},{"Start":"00:45.785 ","End":"00:49.430","Text":"Most natural thing to start with is the integer root theorem."},{"Start":"00:49.430 ","End":"00:53.570","Text":"I look at the constant term which is minus 1,"},{"Start":"00:53.570 ","End":"01:00.575","Text":"and I know that any possible routes divisors,"},{"Start":"01:00.575 ","End":"01:03.185","Text":"factors of minus 1,"},{"Start":"01:03.185 ","End":"01:05.120","Text":"and that\u0027s only 2 possibilities,"},{"Start":"01:05.120 ","End":"01:07.770","Text":"plus or minus 1."},{"Start":"01:08.720 ","End":"01:12.270","Text":"So far everything is as usual."},{"Start":"01:12.270 ","End":"01:14.460","Text":"We try in succession."},{"Start":"01:14.460 ","End":"01:18.030","Text":"Let\u0027s try x equals 1 and plug it in."},{"Start":"01:18.030 ","End":"01:21.000","Text":"1 plus 3 plus 2 minus 2 minus 3 minus 1."},{"Start":"01:21.000 ","End":"01:22.845","Text":"Yes, it\u0027s 0."},{"Start":"01:22.845 ","End":"01:27.585","Text":"We have that x equals 1 is a root."},{"Start":"01:27.585 ","End":"01:32.130","Text":"Now, this is where I diverge."},{"Start":"01:32.130 ","End":"01:36.260","Text":"Normally, what I would do is just find a single root,"},{"Start":"01:36.260 ","End":"01:40.430","Text":"and then divide this polynomial by x minus 1,"},{"Start":"01:40.430 ","End":"01:42.605","Text":"get a degree 4 polynomial,"},{"Start":"01:42.605 ","End":"01:46.225","Text":"and keep going until I get a degree 2."},{"Start":"01:46.225 ","End":"01:50.150","Text":"An alternative approach which could use any way,"},{"Start":"01:50.150 ","End":"01:53.225","Text":"is to try all the possible roots first."},{"Start":"01:53.225 ","End":"01:58.410","Text":"In this case, we would also try plugging in minus 1."},{"Start":"01:58.790 ","End":"02:01.200","Text":"If we plug in minus 1,"},{"Start":"02:01.200 ","End":"02:06.205","Text":"let\u0027s see the first 3 terms give us minus 1 plus 3,"},{"Start":"02:06.205 ","End":"02:08.910","Text":"minus 2, that\u0027s 0."},{"Start":"02:08.910 ","End":"02:14.625","Text":"The last 3 terms give us minus 2 plus 3 minus 1 also 0."},{"Start":"02:14.625 ","End":"02:19.990","Text":"So x equals minus 1 is also going to work."},{"Start":"02:20.410 ","End":"02:23.135","Text":"The question is now what?"},{"Start":"02:23.135 ","End":"02:29.975","Text":"I\u0027ve only got 2 roots on the could be up to 5 roots because it\u0027s a degree 5."},{"Start":"02:29.975 ","End":"02:33.830","Text":"I still end up having to do some polynomial division."},{"Start":"02:33.830 ","End":"02:38.765","Text":"I could divide by x minus 1 and then I could divide by x plus 1."},{"Start":"02:38.765 ","End":"02:40.520","Text":"Or I can straight away,"},{"Start":"02:40.520 ","End":"02:44.580","Text":"multiply x minus 1 times x plus 1 and divide by that."},{"Start":"02:44.580 ","End":"02:48.380","Text":"Anyway, I get down to a degree 3 and then I\u0027d have to continue."},{"Start":"02:48.380 ","End":"02:51.665","Text":"But using the method I\u0027m going to show you,"},{"Start":"02:51.665 ","End":"02:56.105","Text":"we might get lucky and not have to do any polynomial divisions at all."},{"Start":"02:56.105 ","End":"02:57.800","Text":"The idea is this,"},{"Start":"02:57.800 ","End":"03:00.215","Text":"we compute the derivative polynomial."},{"Start":"03:00.215 ","End":"03:04.715","Text":"In this case, it\u0027s 5x^4,"},{"Start":"03:04.715 ","End":"03:10.550","Text":"plus 12x cubed plus"},{"Start":"03:10.550 ","End":"03:17.290","Text":"6x^2 squared minus 4x minus 3."},{"Start":"03:17.290 ","End":"03:22.535","Text":"Then we try to substitute each of these roots into the derivative."},{"Start":"03:22.535 ","End":"03:25.940","Text":"If a root is also a root of the derivative,"},{"Start":"03:25.940 ","End":"03:27.640","Text":"then it\u0027s a double root."},{"Start":"03:27.640 ","End":"03:31.395","Text":"Let\u0027s see. We plug in x equals 1."},{"Start":"03:31.395 ","End":"03:35.760","Text":"Well, I see there\u0027s a lot of pluses 5 and 12,"},{"Start":"03:35.760 ","End":"03:41.295","Text":"and 6 is going be a lot more than 4 and 3."},{"Start":"03:41.295 ","End":"03:44.310","Text":"So,1 does not work here,"},{"Start":"03:44.310 ","End":"03:47.895","Text":"but minus 1, let\u0027s see."},{"Start":"03:47.895 ","End":"03:52.770","Text":"5 minus 12 plus 6."},{"Start":"03:52.770 ","End":"03:55.770","Text":"That\u0027s already minus 1."},{"Start":"03:55.770 ","End":"03:58.620","Text":"Then plus 4 minus 3x,"},{"Start":"03:58.620 ","End":"04:06.930","Text":"yes it\u0027s 0. x=minus 1 is also a root of this."},{"Start":"04:06.930 ","End":"04:12.275","Text":"That means that minus 1 is a double root of the original polynomial."},{"Start":"04:12.275 ","End":"04:13.940","Text":"But that\u0027s not all."},{"Start":"04:13.940 ","End":"04:17.690","Text":"As long as it\u0027s still keeps being a root of the derivative,"},{"Start":"04:17.690 ","End":"04:19.100","Text":"we can keep deriving."},{"Start":"04:19.100 ","End":"04:22.205","Text":"We can try p double prime of x."},{"Start":"04:22.205 ","End":"04:27.120","Text":"This time we get 20x cubed"},{"Start":"04:27.120 ","End":"04:35.655","Text":"plus 36x^2 plus 12 x minus 4."},{"Start":"04:35.655 ","End":"04:39.160","Text":"Now we try plugging in minus 1."},{"Start":"04:39.160 ","End":"04:45.680","Text":"Let\u0027s see, the only positive will be 36 and the negatives will be minus 20,"},{"Start":"04:45.680 ","End":"04:48.850","Text":"minus 12, minus 4, that\u0027s minus 36."},{"Start":"04:48.850 ","End":"04:55.220","Text":"Yeah, that says 0 were 2. x equals minus 1 is"},{"Start":"04:55.220 ","End":"05:03.350","Text":"also a root of the second derivative that makes it triple root."},{"Start":"05:03.350 ","End":"05:06.570","Text":"Maybe it\u0027s a quadruple root."},{"Start":"05:06.790 ","End":"05:13.320","Text":"Let\u0027s try P triple prime of x."},{"Start":"05:13.320 ","End":"05:23.310","Text":"Then we\u0027ve got the 60x^2 plus 72x plus 12,"},{"Start":"05:23.310 ","End":"05:26.010","Text":"plug in minus 1,"},{"Start":"05:26.010 ","End":"05:28.610","Text":"and we have on the negatives,"},{"Start":"05:28.610 ","End":"05:31.790","Text":"we have minus 72 positive 60 and 12."},{"Start":"05:31.790 ","End":"05:34.400","Text":"Also 72, so yes,"},{"Start":"05:34.400 ","End":"05:38.060","Text":"x equals minus 1 is a root."},{"Start":"05:38.060 ","End":"05:41.735","Text":"Now the question is, is there any point in continuing further?"},{"Start":"05:41.735 ","End":"05:45.320","Text":"I say no. Because if we count"},{"Start":"05:45.320 ","End":"05:53.020","Text":"1,2,3,4 minus 1 is a quadruple root and x equals 1 is just a root."},{"Start":"05:53.020 ","End":"05:56.490","Text":"We already have 5 roots."},{"Start":"05:56.490 ","End":"06:04.470","Text":"The roots of the original p. We have minus 1,"},{"Start":"06:04.470 ","End":"06:08.685","Text":"minus 1, minus 1 minus 1, and 1."},{"Start":"06:08.685 ","End":"06:13.275","Text":"We have minus 1 as a root of multiplicity 4,"},{"Start":"06:13.275 ","End":"06:15.735","Text":"and 1 is a root of multiplicity 1,"},{"Start":"06:15.735 ","End":"06:19.765","Text":"and that makes 5 altogether and it\u0027s a degree 5 polynomial."},{"Start":"06:19.765 ","End":"06:23.355","Text":"We know that these are all the roots."},{"Start":"06:23.355 ","End":"06:26.890","Text":"If the question was to factorize this,"},{"Start":"06:26.890 ","End":"06:31.355","Text":"then we could also answer that and say that p of x,"},{"Start":"06:31.355 ","End":"06:33.855","Text":"because of the minus 1,"},{"Start":"06:33.855 ","End":"06:43.700","Text":"the factor is x plus 1 and the 4 times the x=1 gives us an x minus 1."},{"Start":"06:43.700 ","End":"06:47.080","Text":"We have the roots and we have the factorization."},{"Start":"06:47.080 ","End":"06:51.370","Text":"That saved us doing any polynomial divisions,"},{"Start":"06:51.370 ","End":"06:53.755","Text":"and it came out a bit easier."},{"Start":"06:53.755 ","End":"06:56.800","Text":"Certainly we can do without the derivatives,"},{"Start":"06:56.800 ","End":"07:02.465","Text":"but on occasion when we\u0027re lucky it helps and really shortens the work."},{"Start":"07:02.465 ","End":"07:06.275","Text":"I\u0027m going to do another example."},{"Start":"07:06.275 ","End":"07:11.345","Text":"This time, I\u0027ll take a cubic example."},{"Start":"07:11.345 ","End":"07:20.895","Text":"x cubed minus 6x^2 squared plus 12x minus 8."},{"Start":"07:20.895 ","End":"07:27.625","Text":"I would like to factorize this and find the roots,"},{"Start":"07:27.625 ","End":"07:30.286","Text":"whatever I can find out."},{"Start":"07:30.286 ","End":"07:32.730","Text":"We\u0027re going to use"},{"Start":"07:32.730 ","End":"07:36.510","Text":"the rational root theorem or more"},{"Start":"07:36.510 ","End":"07:41.100","Text":"specifically the integer root theorem when we have a leading coefficient of 1."},{"Start":"07:41.100 ","End":"07:48.330","Text":"So we know that the possible roots are divisors of the constant term."},{"Start":"07:48.330 ","End":"07:52.215","Text":"The possible roots are factors of 8"},{"Start":"07:52.215 ","End":"07:55.875","Text":"and actually it\u0027s quite a few of them could be plus or minus 1,"},{"Start":"07:55.875 ","End":"07:57.705","Text":"plus or minus 2,"},{"Start":"07:57.705 ","End":"08:00.180","Text":"plus or minus 4,"},{"Start":"08:00.180 ","End":"08:02.445","Text":"plus or minus 8."},{"Start":"08:02.445 ","End":"08:05.289","Text":"There\u0027s quite a lot to check."},{"Start":"08:05.570 ","End":"08:08.820","Text":"I try x=1, doesn\u0027t work."},{"Start":"08:08.820 ","End":"08:12.540","Text":"Try x =-1 we won\u0027t do all the calculations doesn\u0027t work."},{"Start":"08:12.540 ","End":"08:15.450","Text":"Try x=2 substitute it,"},{"Start":"08:15.450 ","End":"08:17.055","Text":"you find that it does work."},{"Start":"08:17.055 ","End":"08:19.605","Text":"So x=2 is a root."},{"Start":"08:19.605 ","End":"08:21.690","Text":"Now if I was using the old method,"},{"Start":"08:21.690 ","End":"08:27.855","Text":"I would just go right ahead and divide by x minus 2 and then get a quadratic."},{"Start":"08:27.855 ","End":"08:30.915","Text":"But we\u0027re using derivatives,"},{"Start":"08:30.915 ","End":"08:33.600","Text":"which sometimes helps and sometimes doesn\u0027t."},{"Start":"08:33.600 ","End":"08:36.510","Text":"In this case, I go through all the rest of them and"},{"Start":"08:36.510 ","End":"08:40.260","Text":"find that all the rest of them minus 2,"},{"Start":"08:40.260 ","End":"08:42.985","Text":"4, minus 4, 8 don\u0027t work."},{"Start":"08:42.985 ","End":"08:53.365","Text":"I now try the derivative method so I do p\u0027 the derivative of p 3x^2 minus 12x"},{"Start":"08:53.365 ","End":"08:59.610","Text":"plus 12 and I try x=2 in here and I\u0027ve"},{"Start":"08:59.610 ","End":"09:07.005","Text":"got 3*4 is 12 minus 24 plus 12 yes, it is 0."},{"Start":"09:07.005 ","End":"09:17.220","Text":"So x=2 is a root of this also so I\u0027m still lucky I\u0027m on a streak so I try p\"(x),"},{"Start":"09:17.220 ","End":"09:22.755","Text":"which is 6x minus 12."},{"Start":"09:22.755 ","End":"09:24.285","Text":"I plug in x=2,"},{"Start":"09:24.285 ","End":"09:27.465","Text":"6 times 2 minus 12 yes, it\u0027s 0."},{"Start":"09:27.465 ","End":"09:32.055","Text":"So x=2 is again a root of the derivative."},{"Start":"09:32.055 ","End":"09:36.750","Text":"Now it\u0027s 3 times already and we\u0027re a degree 3 polynomial."},{"Start":"09:36.750 ","End":"09:40.380","Text":"There\u0027s no point in continuing so I know that the roots of"},{"Start":"09:40.380 ","End":"09:47.505","Text":"this polynomial counting duplicates so multiple roots are 2,"},{"Start":"09:47.505 ","End":"09:49.980","Text":"2 and 2,"},{"Start":"09:49.980 ","End":"09:54.465","Text":"2 with multiplicity 3 and if you want the factorization,"},{"Start":"09:54.465 ","End":"10:04.150","Text":"then p(x)= x minus 2 from the 2 but 3 times so it\u0027s cubed."},{"Start":"10:06.350 ","End":"10:13.300","Text":"This is how the derivative can help to find the multiplicity of a root."},{"Start":"10:13.790 ","End":"10:16.965","Text":"This is the last example,"},{"Start":"10:16.965 ","End":"10:22.215","Text":"let\u0027s take p(x) is again a fifth degree,"},{"Start":"10:22.215 ","End":"10:31.830","Text":"x^5 plus x^4 missing x^3 and x^2 terms minus x minus 1."},{"Start":"10:31.830 ","End":"10:40.360","Text":"Let\u0027s see I want to find its roots and maybe factorize it."},{"Start":"10:40.370 ","End":"10:45.209","Text":"Again, we\u0027ll use the integer root theorem."},{"Start":"10:45.209 ","End":"10:52.620","Text":"Just like before, the possible roots are factors of the constant term minus 1 so"},{"Start":"10:52.620 ","End":"11:00.060","Text":"only plus or minus 1 could be possible roots for the polynomial."},{"Start":"11:00.060 ","End":"11:03.885","Text":"So we can try each one."},{"Start":"11:03.885 ","End":"11:10.260","Text":"If I try x=1, I get 1 plus 1 minus 1 minus 1 is 0 so yeah,"},{"Start":"11:10.260 ","End":"11:16.260","Text":"very good x=1 works."},{"Start":"11:16.260 ","End":"11:17.910","Text":"If I wasn\u0027t using derivatives,"},{"Start":"11:17.910 ","End":"11:20.670","Text":"I might just go ahead and divide by x minus 1,"},{"Start":"11:20.670 ","End":"11:23.710","Text":"get a fourth degree and so on."},{"Start":"11:23.900 ","End":"11:31.410","Text":"But it\u0027s also easy to try the other one so let\u0027s try minus 1."},{"Start":"11:31.410 ","End":"11:41.100","Text":"If I do minus 1 then I\u0027ve got here minus 1 plus 1 plus 1 minus 1 also 0."},{"Start":"11:41.100 ","End":"11:45.870","Text":"So x=-1 also works."},{"Start":"11:45.870 ","End":"11:54.000","Text":"If I didn\u0027t have the derivatives technique then at this point I\u0027d have to divide,"},{"Start":"11:54.000 ","End":"11:58.890","Text":"I didn\u0027t make too long divisions or I\u0027d multiply x minus 1 times x plus 1 and do"},{"Start":"11:58.890 ","End":"12:04.905","Text":"one long division and be left with a cubic and then continue."},{"Start":"12:04.905 ","End":"12:10.490","Text":"But we\u0027re going to see if we can find the multiplicities using derivative."},{"Start":"12:10.490 ","End":"12:15.130","Text":"So the derivative p\u0027(x) is"},{"Start":"12:15.130 ","End":"12:21.415","Text":"5x^4 plus 4x^3 minus 1."},{"Start":"12:21.415 ","End":"12:24.540","Text":"Let\u0027s try each of these in here."},{"Start":"12:24.540 ","End":"12:27.465","Text":"If I try x=1,"},{"Start":"12:27.465 ","End":"12:32.880","Text":"I get 5 plus 4 minus 1 and no way is that 0."},{"Start":"12:32.880 ","End":"12:34.975","Text":"I try minus 1,"},{"Start":"12:34.975 ","End":"12:36.575","Text":"I\u0027ve got plus 5,"},{"Start":"12:36.575 ","End":"12:39.860","Text":"minus 4 minus 1 yes, that\u0027s good."},{"Start":"12:39.860 ","End":"12:44.809","Text":"So x=-1 is a derivative"},{"Start":"12:44.809 ","End":"12:50.565","Text":"here too so I\u0027ve got already 1 as a single root,"},{"Start":"12:50.565 ","End":"12:52.485","Text":"minus 1 as a double root."},{"Start":"12:52.485 ","End":"13:01.155","Text":"But let\u0027s keep going and see what happens in p\"(x) may be minus 1 will be good there too."},{"Start":"13:01.155 ","End":"13:10.665","Text":"Well, this is 20x^3 plus 12x^2 and if I substitute minus 1,"},{"Start":"13:10.665 ","End":"13:13.050","Text":"I get minus 20 plus 12,"},{"Start":"13:13.050 ","End":"13:15.585","Text":"not 0 so that\u0027s it."},{"Start":"13:15.585 ","End":"13:20.760","Text":"I\u0027ve run out of possibilities, we\u0027re stuck."},{"Start":"13:20.760 ","End":"13:24.990","Text":"I\u0027ll just give you an outline of how to continue what we"},{"Start":"13:24.990 ","End":"13:28.890","Text":"would normally do would be to write this"},{"Start":"13:28.890 ","End":"13:37.140","Text":"p(x) as we know it divides by x minus 1 and we know that minus 1 is a double root,"},{"Start":"13:37.140 ","End":"13:39.495","Text":"so it divides by x plus 1,"},{"Start":"13:39.495 ","End":"13:43.365","Text":"that\u0027s x minus minus 1 squared."},{"Start":"13:43.365 ","End":"13:48.825","Text":"What I would do would be to divide p(x) by this polynomial,"},{"Start":"13:48.825 ","End":"13:56.460","Text":"multiply it out and then divided into p(x) and get some quadratic."},{"Start":"13:56.460 ","End":"13:58.590","Text":"How do I know it\u0027s a quadratic?"},{"Start":"13:58.590 ","End":"14:02.145","Text":"Because altogether here I have a degree 3,"},{"Start":"14:02.145 ","End":"14:07.470","Text":"degree 1, degree 2 and I\u0027m missing another 2 to get 5."},{"Start":"14:07.470 ","End":"14:10.155","Text":"So with quadratic by long division"},{"Start":"14:10.155 ","End":"14:13.110","Text":"and then we\u0027d solve the quadratic which may or may not have"},{"Start":"14:13.110 ","End":"14:17.190","Text":"roots and we would\u0027ve got to the same quadratic if"},{"Start":"14:17.190 ","End":"14:22.035","Text":"we just gone one step at a time we\u0027d first divide by x minus 1,"},{"Start":"14:22.035 ","End":"14:23.460","Text":"we get a fourth degree."},{"Start":"14:23.460 ","End":"14:26.160","Text":"Then we divide by x plus 1 and get"},{"Start":"14:26.160 ","End":"14:31.290","Text":"a cubic and then divide again by x plus 1 and we\u0027d get that quadratic."},{"Start":"14:31.290 ","End":"14:37.170","Text":"As a matter of fact, this quadratic happens to be x^2 plus 1,"},{"Start":"14:37.170 ","End":"14:46.335","Text":"I did the calculation at the side somewhere and this turns out has no roots."},{"Start":"14:46.335 ","End":"14:48.270","Text":"We would\u0027ve got to that either way,"},{"Start":"14:48.270 ","End":"14:49.500","Text":"using derivatives or not,"},{"Start":"14:49.500 ","End":"14:52.500","Text":"so not always using derivatives is going to solve"},{"Start":"14:52.500 ","End":"14:56.490","Text":"all problems for us but it\u0027s a worthwhile technique to know,"},{"Start":"14:56.490 ","End":"15:02.025","Text":"it can sometimes save steps and save some long divisions."},{"Start":"15:02.025 ","End":"15:10.800","Text":"Okay, so that\u0027s all for this technique and we\u0027re done."}],"ID":6356},{"Watched":false,"Name":"Exercise 1","Duration":"18m 27s","ChapterTopicVideoID":6340,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6340.jpeg","UploadDate":"2016-06-22T08:41:18.1570000","DurationForVideoObject":"PT18M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.565","Text":"This exercise is really 7 equations."},{"Start":"00:05.565 ","End":"00:09.120","Text":"In each equation we have some polynomial in"},{"Start":"00:09.120 ","End":"00:13.350","Text":"x=0 and the polynomials of degree higher than 2,"},{"Start":"00:13.350 ","End":"00:14.820","Text":"factor 3 or 4,"},{"Start":"00:14.820 ","End":"00:18.239","Text":"and the coefficients are all whole numbers."},{"Start":"00:18.239 ","End":"00:24.855","Text":"There is a technique that we use and we\u0027ll start with Part A and demonstrate it."},{"Start":"00:24.855 ","End":"00:28.005","Text":"The idea is to,"},{"Start":"00:28.005 ","End":"00:31.710","Text":"I\u0027ll call it guess a solution."},{"Start":"00:31.710 ","End":"00:34.540","Text":"It\u0027s not exactly a guess,"},{"Start":"00:34.540 ","End":"00:37.010","Text":"there\u0027s a theorem that if you have"},{"Start":"00:37.010 ","End":"00:40.370","Text":"a whole number solution and we\u0027ll look for whole number solutions,"},{"Start":"00:40.370 ","End":"00:44.030","Text":"then that whole number must divide the free coefficient,"},{"Start":"00:44.030 ","End":"00:49.080","Text":"that\u0027s the constant at the end without the x."},{"Start":"00:49.080 ","End":"00:51.285","Text":"That\u0027s the free coefficient."},{"Start":"00:51.285 ","End":"00:56.885","Text":"If a whole number divides into minus 2 evenly,"},{"Start":"00:56.885 ","End":"00:59.675","Text":"the only possibilities that it could be,"},{"Start":"00:59.675 ","End":"01:01.190","Text":"is plus or minus 1."},{"Start":"01:01.190 ","End":"01:02.480","Text":"Let\u0027s write it as 1,"},{"Start":"01:02.480 ","End":"01:05.960","Text":"minus 1, or 2, or minus 2."},{"Start":"01:05.960 ","End":"01:10.340","Text":"These are the only numbers that go a whole number of times into minus 2."},{"Start":"01:10.340 ","End":"01:12.320","Text":"We just try them separately,"},{"Start":"01:12.320 ","End":"01:14.735","Text":"one at a time and see which ones divide."},{"Start":"01:14.735 ","End":"01:16.885","Text":"I put in x=1,"},{"Start":"01:16.885 ","End":"01:21.404","Text":"I get 1 minus 2 plus 1 minus 2, that\u0027s not 0."},{"Start":"01:21.404 ","End":"01:28.610","Text":"In short, I tried them all and the only one that works is x=2."},{"Start":"01:28.610 ","End":"01:33.980","Text":"You can see that 8 minus 2 times 4 is 0 when 2 minus 2 is 0,"},{"Start":"01:33.980 ","End":"01:36.310","Text":"so we do get 0."},{"Start":"01:36.310 ","End":"01:38.640","Text":"This is the only one,"},{"Start":"01:38.640 ","End":"01:48.650","Text":"and so we know that because 2 is a solution or root of the equation,"},{"Start":"01:48.650 ","End":"01:54.515","Text":"then x minus 2 will divide this polynomial."},{"Start":"01:54.515 ","End":"01:56.495","Text":"What we do is the long division."},{"Start":"01:56.495 ","End":"01:58.280","Text":"I\u0027ll do this at the side,"},{"Start":"01:58.280 ","End":"02:01.090","Text":"take a dividing line."},{"Start":"02:01.090 ","End":"02:08.090","Text":"Here we put the x cubed minus 2x squared plus x minus 2 from here."},{"Start":"02:08.090 ","End":"02:11.180","Text":"Here I put the x minus 2,"},{"Start":"02:11.180 ","End":"02:13.700","Text":"and I hope you remember your polynomial division,"},{"Start":"02:13.700 ","End":"02:17.465","Text":"but if not, we\u0027ll refresh your memory here."},{"Start":"02:17.465 ","End":"02:20.720","Text":"We say x into x cubed,"},{"Start":"02:20.720 ","End":"02:23.300","Text":"that goes x squared times,"},{"Start":"02:23.300 ","End":"02:25.625","Text":"we multiply x squared by this,"},{"Start":"02:25.625 ","End":"02:28.840","Text":"which is x cubed minus 2x squared."},{"Start":"02:28.840 ","End":"02:33.890","Text":"Then we subtract, and the whole thing cancels."},{"Start":"02:33.890 ","End":"02:38.279","Text":"We dropped the next pair of terms,"},{"Start":"02:38.510 ","End":"02:40.740","Text":"x minus 2,"},{"Start":"02:40.740 ","End":"02:42.075","Text":"then we say,"},{"Start":"02:42.075 ","End":"02:43.890","Text":"x goes into x,"},{"Start":"02:43.890 ","End":"02:46.960","Text":"once, so I write plus 1."},{"Start":"02:46.960 ","End":"02:51.860","Text":"Usually we write it over above the leading coefficient in each case."},{"Start":"02:51.860 ","End":"02:53.570","Text":"Sometimes I highlight them."},{"Start":"02:53.570 ","End":"02:54.770","Text":"In the first instance,"},{"Start":"02:54.770 ","End":"02:56.795","Text":"I had x goes into x cubed,"},{"Start":"02:56.795 ","End":"03:00.830","Text":"the next instance I asked how many times does x goes into x?"},{"Start":"03:00.830 ","End":"03:05.245","Text":"If I multiply plus 1 by x minus 2,"},{"Start":"03:05.245 ","End":"03:08.485","Text":"then I get x minus 2,"},{"Start":"03:08.485 ","End":"03:11.630","Text":"then I subtract, I get nothing left."},{"Start":"03:11.630 ","End":"03:12.770","Text":"It goes in evenly,"},{"Start":"03:12.770 ","End":"03:15.330","Text":"I can say a remainder of 0."},{"Start":"03:15.590 ","End":"03:19.310","Text":"Now that we know that this is the quotient,"},{"Start":"03:19.310 ","End":"03:29.430","Text":"it means that this polynomial can be factored as x minus 2 times x squared plus 1,"},{"Start":"03:30.350 ","End":"03:34.150","Text":"and it still equals 0."},{"Start":"03:35.620 ","End":"03:39.800","Text":"Now, we know that this means that x=2,"},{"Start":"03:39.800 ","End":"03:40.999","Text":"which we knew already,"},{"Start":"03:40.999 ","End":"03:44.435","Text":"this is either 0 or this is 0 when the product is 0."},{"Start":"03:44.435 ","End":"03:47.865","Text":"This gives us x=2, what\u0027s new."},{"Start":"03:47.865 ","End":"03:50.690","Text":"This one gives us the equation,"},{"Start":"03:50.690 ","End":"03:53.235","Text":"x squared plus 1 equals 0,"},{"Start":"03:53.235 ","End":"03:55.995","Text":"but that has no solution."},{"Start":"03:55.995 ","End":"04:02.900","Text":"The only solution to this equation is x=2."},{"Start":"04:02.900 ","End":"04:07.490","Text":"Before I move on, let me make an important note that"},{"Start":"04:07.490 ","End":"04:11.660","Text":"if you have studied what are known as complex numbers,"},{"Start":"04:11.660 ","End":"04:15.334","Text":"and if you\u0027re not sure what I mean by complex numbers,"},{"Start":"04:15.334 ","End":"04:19.055","Text":"then if the following looks like gibberish to you,"},{"Start":"04:19.055 ","End":"04:21.290","Text":"that the square root of minus 1 equals i,"},{"Start":"04:21.290 ","End":"04:23.225","Text":"then you probably haven\u0027t studied them."},{"Start":"04:23.225 ","End":"04:25.040","Text":"Forget what I\u0027m about to say,"},{"Start":"04:25.040 ","End":"04:27.260","Text":"but if you have studied complex numbers,"},{"Start":"04:27.260 ","End":"04:29.990","Text":"I\u0027m going to return to these exercises,"},{"Start":"04:29.990 ","End":"04:31.370","Text":"and at the end,"},{"Start":"04:31.370 ","End":"04:37.235","Text":"we\u0027ll modify the solution for those who have studied complex numbers."},{"Start":"04:37.235 ","End":"04:43.295","Text":"I\u0027m going to erase this remark and move on to the next one."},{"Start":"04:43.295 ","End":"04:46.550","Text":"We\u0027ll speed it up a bit now."},{"Start":"04:46.550 ","End":"04:47.870","Text":"After Part A,"},{"Start":"04:47.870 ","End":"04:51.380","Text":"you\u0027ve learned that the technique is to guess"},{"Start":"04:51.380 ","End":"04:55.790","Text":"a whole number solution which divides minus 6."},{"Start":"04:55.790 ","End":"05:00.455","Text":"The only possibilities are plus or minus 1,"},{"Start":"05:00.455 ","End":"05:02.285","Text":"plus or minus 2,"},{"Start":"05:02.285 ","End":"05:04.825","Text":"plus or minus 3,"},{"Start":"05:04.825 ","End":"05:08.400","Text":"and plus or minus 6."},{"Start":"05:08.400 ","End":"05:11.210","Text":"That\u0027s actually 8 possibilities to check."},{"Start":"05:11.210 ","End":"05:16.670","Text":"In each case we have to substitute in the left-hand side and see if we get 0."},{"Start":"05:16.670 ","End":"05:20.321","Text":"In principle, we have to try all of them,"},{"Start":"05:20.321 ","End":"05:22.970","Text":"but as soon as we find 3 roots,"},{"Start":"05:22.970 ","End":"05:27.800","Text":"we can stop because a degree 3 polynomial can have at most 3 roots."},{"Start":"05:27.800 ","End":"05:30.080","Text":"Now, if I try plugging in x=1,"},{"Start":"05:30.080 ","End":"05:33.425","Text":"I get 1 plus 2 minus 5 minus 6, that\u0027s not 0."},{"Start":"05:33.425 ","End":"05:35.095","Text":"I put minus 1,"},{"Start":"05:35.095 ","End":"05:40.670","Text":"then I get minus 1 plus 2 plus 5 minus 6,"},{"Start":"05:40.670 ","End":"05:42.705","Text":"7 minus 7 is 0,"},{"Start":"05:42.705 ","End":"05:44.290","Text":"and minus 1 works."},{"Start":"05:44.290 ","End":"05:48.260","Text":"Let me just give you the answers because the substitution is tedious."},{"Start":"05:48.260 ","End":"05:50.060","Text":"Minus 1 works,"},{"Start":"05:50.060 ","End":"05:54.020","Text":"and 2 works,"},{"Start":"05:54.020 ","End":"05:57.455","Text":"and minus 3 works."},{"Start":"05:57.455 ","End":"06:03.920","Text":"These are the 3 roots or the solutions to the equation."},{"Start":"06:03.920 ","End":"06:07.205","Text":"Moving on to Part C as before,"},{"Start":"06:07.205 ","End":"06:12.375","Text":"what we have to do is guess 1 or more roots,"},{"Start":"06:12.375 ","End":"06:14.175","Text":"and we\u0027re going to look for whole numbers."},{"Start":"06:14.175 ","End":"06:16.100","Text":"Like I said, if it\u0027s a whole number root,"},{"Start":"06:16.100 ","End":"06:20.240","Text":"it has to divide a whole number of times into 6,"},{"Start":"06:20.240 ","End":"06:23.405","Text":"so the possibilities are plus or minus 1,"},{"Start":"06:23.405 ","End":"06:25.405","Text":"plus or minus 2,"},{"Start":"06:25.405 ","End":"06:27.450","Text":"plus or minus 3,"},{"Start":"06:27.450 ","End":"06:29.080","Text":"plus or minus 6."},{"Start":"06:29.080 ","End":"06:31.820","Text":"This is very similar to the previous exercise,"},{"Start":"06:31.820 ","End":"06:34.520","Text":"just the signs are different."},{"Start":"06:34.520 ","End":"06:40.860","Text":"This time, the ones that work are 1,"},{"Start":"06:40.860 ","End":"06:43.500","Text":"minus 2, and 3,"},{"Start":"06:43.500 ","End":"06:45.660","Text":"and because there are 3 of them,"},{"Start":"06:45.660 ","End":"06:53.325","Text":"and there are most 3 roots to a polynomial of degree 3,"},{"Start":"06:53.325 ","End":"06:55.550","Text":"this is actually the answer."},{"Start":"06:55.550 ","End":"06:58.460","Text":"We can stop here, but I just like to point out that this"},{"Start":"06:58.460 ","End":"07:01.490","Text":"also tells us how to factorize this polynomial,"},{"Start":"07:01.490 ","End":"07:06.710","Text":"and this polynomial actually factors as x minus 1."},{"Start":"07:06.710 ","End":"07:09.230","Text":"Subtract the x minus each route,"},{"Start":"07:09.230 ","End":"07:13.760","Text":"x plus 2 and x minus 3."},{"Start":"07:13.760 ","End":"07:18.145","Text":"This is not what was asked, but I thought it\u0027s worth mentioning anyway."},{"Start":"07:18.145 ","End":"07:20.330","Text":"Onto Part D,"},{"Start":"07:20.330 ","End":"07:24.650","Text":"and this time we have a degree 4 polynomial,"},{"Start":"07:24.650 ","End":"07:27.560","Text":"so we can have at most 4 roots."},{"Start":"07:27.560 ","End":"07:33.380","Text":"Again, we have 6 here in the free term,"},{"Start":"07:33.380 ","End":"07:36.350","Text":"the coefficient of x to the 0,"},{"Start":"07:36.350 ","End":"07:38.945","Text":"or the constant term, or whatever you want to call it."},{"Start":"07:38.945 ","End":"07:41.990","Text":"That means there are 8 possibilities to try,"},{"Start":"07:41.990 ","End":"07:43.650","Text":"the usual plus or minus 1,"},{"Start":"07:43.650 ","End":"07:45.020","Text":"plus or minus 2,"},{"Start":"07:45.020 ","End":"07:46.795","Text":"plus or minus 3,"},{"Start":"07:46.795 ","End":"07:49.650","Text":"plus or minus 6."},{"Start":"07:49.650 ","End":"07:53.175","Text":"I\u0027m not going to compute each one,"},{"Start":"07:53.175 ","End":"07:55.290","Text":"I\u0027ll try one of them, see if we get lucky."},{"Start":"07:55.290 ","End":"07:57.020","Text":"The 1 is easiest to substitute."},{"Start":"07:57.020 ","End":"08:03.275","Text":"If I substitute 1, I get 1 minus 1 minus 7 plus 1 plus 6."},{"Start":"08:03.275 ","End":"08:06.890","Text":"Actually, we did get lucky, because 1 works."},{"Start":"08:06.890 ","End":"08:08.300","Text":"1 minus 1 is 0,"},{"Start":"08:08.300 ","End":"08:11.180","Text":"and minus 7 plus 1 plus 6 is also 0,"},{"Start":"08:11.180 ","End":"08:14.657","Text":"so 1 is one of the possibilities."},{"Start":"08:14.657 ","End":"08:16.747","Text":"I\u0027ll tell you,"},{"Start":"08:16.747 ","End":"08:21.085","Text":"the other ones are: Minus 1,"},{"Start":"08:21.085 ","End":"08:23.985","Text":"minus 2, and 3."},{"Start":"08:23.985 ","End":"08:25.965","Text":"Since there are 4 of them,"},{"Start":"08:25.965 ","End":"08:27.600","Text":"that\u0027s all we can expect,"},{"Start":"08:27.600 ","End":"08:30.030","Text":"there can\u0027t be anymore."},{"Start":"08:30.030 ","End":"08:33.555","Text":"These are all the roots,"},{"Start":"08:33.555 ","End":"08:36.230","Text":"the solutions of the equation,"},{"Start":"08:36.230 ","End":"08:37.459","Text":"the roots of the polynomial,"},{"Start":"08:37.459 ","End":"08:42.420","Text":"same thing, and so that\u0027s the answer."},{"Start":"08:42.670 ","End":"08:48.620","Text":"I also like to add that if they asked us to factorize the left-hand side,"},{"Start":"08:48.620 ","End":"08:51.485","Text":"which they didn\u0027t, but I\u0027m volunteering,"},{"Start":"08:51.485 ","End":"08:54.610","Text":"this would be x minus 1,"},{"Start":"08:54.610 ","End":"08:56.310","Text":"x plus 1,"},{"Start":"08:56.310 ","End":"08:58.140","Text":"just x minus each root,"},{"Start":"08:58.140 ","End":"09:00.150","Text":"x plus 2,"},{"Start":"09:00.150 ","End":"09:03.730","Text":"and x minus 3."},{"Start":"09:04.030 ","End":"09:11.145","Text":"Next one, E. Proceeding the same way,"},{"Start":"09:11.145 ","End":"09:19.440","Text":"we look for whole number divisors of 18 integers,"},{"Start":"09:19.440 ","End":"09:23.465","Text":"so here these possibilities are plus or minus 1,"},{"Start":"09:23.465 ","End":"09:25.715","Text":"this is always a possibility."},{"Start":"09:25.715 ","End":"09:31.495","Text":"Then 18 is divisible by 2."},{"Start":"09:31.495 ","End":"09:34.905","Text":"It\u0027s divisible by 3,"},{"Start":"09:34.905 ","End":"09:37.455","Text":"it\u0027s divisible by 6,"},{"Start":"09:37.455 ","End":"09:41.310","Text":"it\u0027s divisible by 9,"},{"Start":"09:41.310 ","End":"09:44.580","Text":"and it\u0027s divisible by 18."},{"Start":"09:44.580 ","End":"09:49.695","Text":"There\u0027s 12 possibilities to check,"},{"Start":"09:49.695 ","End":"09:51.570","Text":"and that\u0027s a lot of work,"},{"Start":"09:51.570 ","End":"09:54.860","Text":"and I\u0027ll do that work for you."},{"Start":"09:54.860 ","End":"10:02.570","Text":"I already did it separately on my own and I got that minus 1,"},{"Start":"10:02.570 ","End":"10:06.480","Text":"2, 3, and minus 3, all work."},{"Start":"10:06.500 ","End":"10:09.440","Text":"I\u0027ll just take 1 example,"},{"Start":"10:09.440 ","End":"10:11.120","Text":"I feel I\u0027m cheating you otherwise."},{"Start":"10:11.120 ","End":"10:13.550","Text":"Let\u0027s try the 2."},{"Start":"10:13.550 ","End":"10:19.650","Text":"If we try 2^4 minus 2 cubed minus"},{"Start":"10:19.650 ","End":"10:25.740","Text":"11 times 2 squared plus 9 times 2 plus 18, let\u0027s see what we get."},{"Start":"10:25.740 ","End":"10:29.925","Text":"16 minus 8 minus 11 times 4,"},{"Start":"10:29.925 ","End":"10:36.385","Text":"44, plus 18 plus 18."},{"Start":"10:36.385 ","End":"10:41.730","Text":"Let\u0027s see what that equals to. We take the positives separately,"},{"Start":"10:41.730 ","End":"10:44.745","Text":"18 and 18 is 36,"},{"Start":"10:44.745 ","End":"10:46.980","Text":"and 16 is 52."},{"Start":"10:46.980 ","End":"10:50.820","Text":"The negatives 8 and 44 is 52,"},{"Start":"10:50.820 ","End":"10:55.215","Text":"so we do indeed get 0."},{"Start":"10:55.215 ","End":"10:57.975","Text":"At least I verified one of them."},{"Start":"10:57.975 ","End":"11:01.460","Text":"Since there are 4 of them,"},{"Start":"11:01.460 ","End":"11:03.530","Text":"these are all that can be,"},{"Start":"11:03.530 ","End":"11:07.040","Text":"so the answer is that the solution to"},{"Start":"11:07.040 ","End":"11:12.460","Text":"this equation is that x equals any one of these 4 solutions."},{"Start":"11:12.460 ","End":"11:15.043","Text":"I\u0027ll just highlight that,"},{"Start":"11:15.043 ","End":"11:18.159","Text":"and then we\u0027ll go on to the next."},{"Start":"11:21.240 ","End":"11:24.070","Text":"This one is degree 3."},{"Start":"11:24.070 ","End":"11:27.895","Text":"Let\u0027s see, we have a minus 8 here."},{"Start":"11:27.895 ","End":"11:31.120","Text":"The values of x we have to guess."},{"Start":"11:31.120 ","End":"11:33.340","Text":"Plus or minus 1,"},{"Start":"11:33.340 ","End":"11:35.446","Text":"plus or minus 2,"},{"Start":"11:35.446 ","End":"11:37.180","Text":"plus or minus 4,"},{"Start":"11:37.180 ","End":"11:40.220","Text":"plus or minus 8."},{"Start":"11:41.550 ","End":"11:43.870","Text":"To see which of these works,"},{"Start":"11:43.870 ","End":"11:46.030","Text":"we just substitute them one at a time."},{"Start":"11:46.030 ","End":"11:47.515","Text":"I\u0027ll just check one of them."},{"Start":"11:47.515 ","End":"11:54.700","Text":"Let\u0027s try x=1 and then we get 1 minus 7 plus 14 minus 8."},{"Start":"11:54.700 ","End":"11:58.210","Text":"Actually, that works because the positives are 1 and 14,"},{"Start":"11:58.210 ","End":"12:03.925","Text":"is 15 and the negative 7 and 8 is 15, so 1 works."},{"Start":"12:03.925 ","End":"12:05.935","Text":"I\u0027ll tell you that the others,"},{"Start":"12:05.935 ","End":"12:10.290","Text":"if you try them 2"},{"Start":"12:10.350 ","End":"12:16.600","Text":"and 4 and these are all the answers that can be,"},{"Start":"12:16.600 ","End":"12:20.095","Text":"because this is degree 3 and it can have at most three roots."},{"Start":"12:20.095 ","End":"12:25.735","Text":"I would like to show you an alternative ending to this story."},{"Start":"12:25.735 ","End":"12:27.780","Text":"If we find one of them,"},{"Start":"12:27.780 ","End":"12:29.505","Text":"say x =1,"},{"Start":"12:29.505 ","End":"12:31.425","Text":"which we found in the beginning,"},{"Start":"12:31.425 ","End":"12:40.870","Text":"the other way to do it is to do a long division of this polynomial by x minus 1."},{"Start":"12:40.870 ","End":"12:43.330","Text":"Because if 1 is a root,"},{"Start":"12:43.330 ","End":"12:50.065","Text":"then we know that the left-hand side is x minus 1 times something."},{"Start":"12:50.065 ","End":"12:53.155","Text":"Then we do a long division,"},{"Start":"12:53.155 ","End":"12:56.665","Text":"so we\u0027ll just quickly do that."},{"Start":"12:56.665 ","End":"13:05.440","Text":"We put here the x cubed minus 7x squared plus 14x minus 8,"},{"Start":"13:05.440 ","End":"13:09.475","Text":"and we divide it by x minus 1."},{"Start":"13:09.475 ","End":"13:14.170","Text":"I\u0027ll tell you what, I\u0027m not going to waste time doing that."},{"Start":"13:14.170 ","End":"13:21.835","Text":"Let me just say that what you get is x^2 minus 6x plus 8."},{"Start":"13:21.835 ","End":"13:23.680","Text":"I hope you remember your long division."},{"Start":"13:23.680 ","End":"13:26.605","Text":"Then we copy that into here,"},{"Start":"13:26.605 ","End":"13:31.615","Text":"x^2 minus 6x plus 8."},{"Start":"13:31.615 ","End":"13:34.180","Text":"By the way, if you get a remainder here not 0,"},{"Start":"13:34.180 ","End":"13:39.020","Text":"then you know you\u0027ve done something wrong because it has to go in evenly."},{"Start":"13:39.270 ","End":"13:41.860","Text":"Then when we get to this point, we say,"},{"Start":"13:41.860 ","End":"13:45.130","Text":"okay, either this is 0 or this is 0."},{"Start":"13:45.130 ","End":"13:47.710","Text":"This is 0 gives us x equals 1,"},{"Start":"13:47.710 ","End":"13:50.350","Text":"which we knew already, so nothing new."},{"Start":"13:50.350 ","End":"13:55.435","Text":"But if we solve this quadratic equation using the formula or any other method,"},{"Start":"13:55.435 ","End":"13:59.575","Text":"you get x equals 2 or 4,"},{"Start":"13:59.575 ","End":"14:01.705","Text":"and then we add those to the list."},{"Start":"14:01.705 ","End":"14:06.970","Text":"That\u0027s an alternative ending and you can pick which you prefer to just keep guessing."},{"Start":"14:06.970 ","End":"14:09.670","Text":"I guess if there was a huge number of things to substitute,"},{"Start":"14:09.670 ","End":"14:14.000","Text":"you might do a division, it\u0027s up to you."},{"Start":"14:14.670 ","End":"14:22.420","Text":"Let\u0027s move on to the last one, also degree 3."},{"Start":"14:22.420 ","End":"14:30.520","Text":"We\u0027ll use the same technique to start off with."},{"Start":"14:30.520 ","End":"14:36.265","Text":"We use the guess method of seeing what goes into 15."},{"Start":"14:36.265 ","End":"14:40.495","Text":"The possibilities are plus or minus 1 as always,"},{"Start":"14:40.495 ","End":"14:43.315","Text":"and then plus or minus 3,"},{"Start":"14:43.315 ","End":"14:44.710","Text":"what else goes into 15?"},{"Start":"14:44.710 ","End":"14:48.490","Text":"5, so plus or minus 5 and 15 itself,"},{"Start":"14:48.490 ","End":"14:51.470","Text":"plus or minus 15."},{"Start":"14:51.750 ","End":"14:55.600","Text":"Let\u0027s try x equals 1."},{"Start":"14:55.600 ","End":"14:59.200","Text":"Then we get 1 plus 1 minus 17 plus 15."},{"Start":"14:59.200 ","End":"15:05.050","Text":"And yeah, first-time lucky because 1 plus 1 plus 15 is 17,"},{"Start":"15:05.050 ","End":"15:06.610","Text":"so it works out."},{"Start":"15:06.610 ","End":"15:09.820","Text":"We know that x could be equal to 1."},{"Start":"15:09.820 ","End":"15:11.440","Text":"Then there\u0027s up to two more."},{"Start":"15:11.440 ","End":"15:13.390","Text":"There might not be anymore, but up to two more."},{"Start":"15:13.390 ","End":"15:15.699","Text":"Now you have a choice of two methods."},{"Start":"15:15.699 ","End":"15:17.302","Text":"I\u0027m deliberately getting both."},{"Start":"15:17.302 ","End":"15:19.480","Text":"It\u0027s matter of personal preference."},{"Start":"15:19.480 ","End":"15:22.045","Text":"If you enjoy substituting,"},{"Start":"15:22.045 ","End":"15:23.995","Text":"then just keep guessing."},{"Start":"15:23.995 ","End":"15:30.205","Text":"I\u0027ll tell you now that the other 2 and 3 and minus 5,"},{"Start":"15:30.205 ","End":"15:31.750","Text":"I mean, there are two more."},{"Start":"15:31.750 ","End":"15:34.255","Text":"There could have been anymore,"},{"Start":"15:34.255 ","End":"15:36.745","Text":"but you also have the option,"},{"Start":"15:36.745 ","End":"15:42.055","Text":"and I\u0027ll just mention it briefly if dividing this by x minus 1."},{"Start":"15:42.055 ","End":"15:44.455","Text":"If you did a long division,"},{"Start":"15:44.455 ","End":"15:48.655","Text":"I\u0027m not going to do it fully, I\u0027ll just mention,"},{"Start":"15:48.655 ","End":"15:55.240","Text":"that you could do a long division by substituting, I mean, sorry,"},{"Start":"15:55.240 ","End":"16:02.304","Text":"copying, I meant x^3 plus x^2 minus 17x plus 15."},{"Start":"16:02.304 ","End":"16:05.140","Text":"Here you would put x minus 1."},{"Start":"16:05.140 ","End":"16:06.850","Text":"Do the long division."},{"Start":"16:06.850 ","End":"16:08.800","Text":"I\u0027m not going to do it."},{"Start":"16:08.800 ","End":"16:15.280","Text":"You get x^2 plus 2x minus 15."},{"Start":"16:15.280 ","End":"16:26.935","Text":"Then that means that this factorizes as x minus 1 x^2 plus 2x minus 15."},{"Start":"16:26.935 ","End":"16:30.130","Text":"Then if a product is 0, either this is 0,"},{"Start":"16:30.130 ","End":"16:31.450","Text":"which gives us x equals 1,"},{"Start":"16:31.450 ","End":"16:33.160","Text":"which we knew already."},{"Start":"16:33.160 ","End":"16:38.560","Text":"Or we solve this using the quadratic equation formula or any other method,"},{"Start":"16:38.560 ","End":"16:45.770","Text":"you know of solving quadratics and you would get the other two roots, 3 and -5."},{"Start":"16:46.590 ","End":"16:50.710","Text":"I\u0027m not going to remind you how to solve quadratic equations,"},{"Start":"16:50.710 ","End":"16:52.330","Text":"I could do it with the formula,"},{"Start":"16:52.330 ","End":"16:54.235","Text":"but I\u0027ll leave you to do that."},{"Start":"16:54.235 ","End":"17:04.825","Text":"These are the three solutions which I shall highlight and we are done with this exercise."},{"Start":"17:04.825 ","End":"17:11.484","Text":"Except, if you have studied complex numbers,"},{"Start":"17:11.484 ","End":"17:13.210","Text":"let\u0027s go back and see."},{"Start":"17:13.210 ","End":"17:16.075","Text":"There was one at least that we missed."},{"Start":"17:16.075 ","End":"17:19.030","Text":"I\u0027m going to scroll back up now."},{"Start":"17:19.030 ","End":"17:24.310","Text":"The first one, I said no solution if you haven\u0027t studied complex numbers,"},{"Start":"17:24.310 ","End":"17:26.125","Text":"but if you have,"},{"Start":"17:26.125 ","End":"17:28.030","Text":"then we can continue."},{"Start":"17:28.030 ","End":"17:35.485","Text":"I\u0027ll erase this no solution and this is not the final answer and we continue with this."},{"Start":"17:35.485 ","End":"17:41.335","Text":"This is a quadratic equation and I won\u0027t use the formula because it\u0027s the missing B,"},{"Start":"17:41.335 ","End":"17:42.610","Text":"the middle term is missing."},{"Start":"17:42.610 ","End":"17:48.355","Text":"It\u0027s easier to say x^2 equals minus 1."},{"Start":"17:48.355 ","End":"17:55.855","Text":"X equals plus or minus the square root of minus 1."},{"Start":"17:55.855 ","End":"17:57.790","Text":"Since you\u0027ve studied complex numbers,"},{"Start":"17:57.790 ","End":"18:00.220","Text":"the square root of minus one is i,"},{"Start":"18:00.220 ","End":"18:03.699","Text":"so x equals plus or minus i."},{"Start":"18:03.699 ","End":"18:09.700","Text":"Now we can say that there are three solutions that"},{"Start":"18:09.700 ","End":"18:17.350","Text":"x equals either 2 or i or minus i."},{"Start":"18:17.350 ","End":"18:27.560","Text":"This is the solution for those who\u0027ve studied complex numbers. We are done."}],"ID":6357},{"Watched":false,"Name":"Exercise 2","Duration":"15m 7s","ChapterTopicVideoID":6341,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6341.jpeg","UploadDate":"2016-06-22T08:43:47.3900000","DurationForVideoObject":"PT15M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.150 ","End":"00:05.725","Text":"In this exercise, there\u0027s 3 separate equations to solve."},{"Start":"00:05.725 ","End":"00:12.100","Text":"They are all cubic equations with integer coefficients."},{"Start":"00:12.100 ","End":"00:16.570","Text":"When the leading coefficient is not 1,"},{"Start":"00:16.570 ","End":"00:18.805","Text":"and in this case it isn\u0027t,"},{"Start":"00:18.805 ","End":"00:20.785","Text":"2, 2, or 4,"},{"Start":"00:20.785 ","End":"00:25.330","Text":"then we use what is called the rational root theorem."},{"Start":"00:25.330 ","End":"00:29.935","Text":"This says that if we look for rational number solutions,"},{"Start":"00:29.935 ","End":"00:31.780","Text":"rational number as in fraction,"},{"Start":"00:31.780 ","End":"00:37.240","Text":"say p/q and we also assume that this is a reduced fraction,"},{"Start":"00:37.240 ","End":"00:41.245","Text":"meaning it can\u0027t be canceled anymore."},{"Start":"00:41.245 ","End":"00:49.820","Text":"Then p goes evenly into the constant term,"},{"Start":"00:49.820 ","End":"00:57.240","Text":"that\u0027s the 3 and q goes into the 2 whole number integer times."},{"Start":"00:57.740 ","End":"01:04.845","Text":"In this case, we have possibilities for p. It could be,"},{"Start":"01:04.845 ","End":"01:07.370","Text":"well 4 possibilities,"},{"Start":"01:07.370 ","End":"01:13.040","Text":"could be plus or minus 1 and plus or minus 3."},{"Start":"01:13.040 ","End":"01:18.139","Text":"Q can be also 4 possibilities,"},{"Start":"01:18.139 ","End":"01:25.655","Text":"plus or minus 1 and plus or minus 2 in this case."},{"Start":"01:25.655 ","End":"01:33.095","Text":"Actually, that gives us a total of 16 combinations really."},{"Start":"01:33.095 ","End":"01:35.300","Text":"Because here we have 4 possibilities and here we"},{"Start":"01:35.300 ","End":"01:38.030","Text":"have 4 possibilities and that\u0027s quite a lot."},{"Start":"01:38.030 ","End":"01:44.900","Text":"Usually, the thing to do is to start looking for whole solutions first."},{"Start":"01:44.900 ","End":"01:51.455","Text":"Let\u0027s say we take the q is equal to 1 first."},{"Start":"01:51.455 ","End":"01:55.520","Text":"Then we have the possibility for x to"},{"Start":"01:55.520 ","End":"02:00.155","Text":"be if this was the case and we only have 4 things to try."},{"Start":"02:00.155 ","End":"02:04.500","Text":"X could equal q is 1,"},{"Start":"02:04.500 ","End":"02:08.325","Text":"so it\u0027s just p. We could have 1 minus 1,"},{"Start":"02:08.325 ","End":"02:10.280","Text":"3 or minus 3."},{"Start":"02:10.280 ","End":"02:14.960","Text":"That\u0027s less possibilities because that may not be whole number,"},{"Start":"02:14.960 ","End":"02:20.215","Text":"but let\u0027s start with those because they\u0027re easier to substitute and there\u0027s less of them."},{"Start":"02:20.215 ","End":"02:23.970","Text":"If we let say x=1,"},{"Start":"02:23.970 ","End":"02:25.455","Text":"we\u0027ll try that first."},{"Start":"02:25.455 ","End":"02:30.370","Text":"And then we get 2 plus 3 minus 8 plus 3."},{"Start":"02:30.370 ","End":"02:32.705","Text":"Well, we are lucky."},{"Start":"02:32.705 ","End":"02:39.245","Text":"It just happens to work because 2 plus 3 plus 3 is 8 and minus 8 here is 0."},{"Start":"02:39.245 ","End":"02:46.410","Text":"Already we have that x=1 works,"},{"Start":"02:46.410 ","End":"02:48.490","Text":"so that\u0027s one solution."},{"Start":"02:48.490 ","End":"02:57.980","Text":"Now, you could just keep plugging away with the 16 possibilities for p/q,"},{"Start":"02:57.980 ","End":"02:59.630","Text":"and we\u0027ve done one of them,"},{"Start":"02:59.630 ","End":"03:04.535","Text":"we try these 3 and then there\u0027s still another, 12 more."},{"Start":"03:04.535 ","End":"03:10.640","Text":"But the more methodical thing to do is to use polynomial long division,"},{"Start":"03:10.640 ","End":"03:14.060","Text":"say, if x=1 is a solution,"},{"Start":"03:14.060 ","End":"03:16.790","Text":"then x minus 1 goes into this."},{"Start":"03:16.790 ","End":"03:22.980","Text":"This is going to be x minus 1 times some quadratic, which equals 0."},{"Start":"03:22.980 ","End":"03:27.890","Text":"In fact I even know the first term already at 2x^2,"},{"Start":"03:27.890 ","End":"03:29.855","Text":"but okay, we\u0027re starting to do long division."},{"Start":"03:29.855 ","End":"03:32.310","Text":"Let\u0027s do it at the side."},{"Start":"03:33.280 ","End":"03:36.320","Text":"Long division."},{"Start":"03:36.320 ","End":"03:38.240","Text":"Here, we\u0027re going to put"},{"Start":"03:38.240 ","End":"03:43.770","Text":"the original polynomial 2x^3"},{"Start":"03:43.770 ","End":"03:50.070","Text":"plus 3x^2 minus 8x plus 3."},{"Start":"03:50.070 ","End":"03:54.160","Text":"Here, the x minus 1."},{"Start":"03:54.440 ","End":"03:58.590","Text":"Let\u0027s see, x into 2x3."},{"Start":"03:58.590 ","End":"04:03.255","Text":"It goes 2x^2 times multiply"},{"Start":"04:03.255 ","End":"04:11.790","Text":"2x^3 minus 2x^2 subtract 3 minus minus 2 is 5."},{"Start":"04:11.790 ","End":"04:19.400","Text":"Drop another term, x into 5^2 goes 5x times."},{"Start":"04:19.400 ","End":"04:25.620","Text":"Then we get here, multiplying out 5x^2 minus 5x."},{"Start":"04:26.410 ","End":"04:33.680","Text":"Then subtract minus 8 minus minus 5 is"},{"Start":"04:33.680 ","End":"04:41.360","Text":"minus 3x plus 3x(3x) goes minus 3 times,"},{"Start":"04:41.360 ","End":"04:46.250","Text":"multiplying minus 3x plus 3."},{"Start":"04:46.250 ","End":"04:49.615","Text":"Let\u0027s subtract completely,"},{"Start":"04:49.615 ","End":"04:52.470","Text":"there\u0027s no remainder,"},{"Start":"04:52.470 ","End":"04:56.845","Text":"and so here we put the 2x^2,"},{"Start":"04:56.845 ","End":"05:01.450","Text":"plus 5x minus 3."},{"Start":"05:01.450 ","End":"05:04.540","Text":"If this is 0,"},{"Start":"05:04.540 ","End":"05:05.710","Text":"that gives us x=1,"},{"Start":"05:05.710 ","End":"05:08.740","Text":"so there\u0027s no point in following this route."},{"Start":"05:08.740 ","End":"05:14.661","Text":"Let\u0027s go for the other route or however you pronounce it, the other."},{"Start":"05:14.661 ","End":"05:19.150","Text":"Then we get that this could be 0 also that gives us a quadratic."},{"Start":"05:19.150 ","End":"05:24.566","Text":"If this is equal to 0, we get,"},{"Start":"05:24.566 ","End":"05:34.450","Text":"that x equals minus 5 plus or"},{"Start":"05:34.450 ","End":"05:40.049","Text":"minus the square root of b^2 minus 4 times"},{"Start":"05:40.049 ","End":"05:46.590","Text":"2 times minus 3 over 2a is 4."},{"Start":"05:46.590 ","End":"05:48.780","Text":"What does that give us?"},{"Start":"05:48.780 ","End":"05:56.220","Text":"4 times 2 times 3 is 24 plus 25 is 49,"},{"Start":"05:56.220 ","End":"06:03.180","Text":"so we get minus 5 plus or minus 7/4,"},{"Start":"06:03.180 ","End":"06:07.095","Text":"5 plus 7 is 12/4 is 3."},{"Start":"06:07.095 ","End":"06:14.275","Text":"5 minus 7 is minus 2/4 is minus 1/2."},{"Start":"06:14.275 ","End":"06:19.170","Text":"In fact, it does fit this pattern."},{"Start":"06:20.890 ","End":"06:26.945","Text":"In this case, we would have that p is 3 and q is 1."},{"Start":"06:26.945 ","End":"06:28.250","Text":"Let me get the 3."},{"Start":"06:28.250 ","End":"06:29.555","Text":"For the minus 1/2,"},{"Start":"06:29.555 ","End":"06:36.220","Text":"we get p is minus 1 and q is 2."},{"Start":"06:36.220 ","End":"06:39.930","Text":"The other way round p is"},{"Start":"06:40.330 ","End":"06:44.010","Text":"1 and minus 2 or minus 1 and"},{"Start":"06:44.010 ","End":"06:47.510","Text":"2 which actually leads me to correct something I said earlier,"},{"Start":"06:47.510 ","End":"06:53.130","Text":"there\u0027s not 16 possibilities is only 8 because the plus or the minus anyway."},{"Start":"06:53.990 ","End":"06:56.869","Text":"Those are the 2 other solutions."},{"Start":"06:56.869 ","End":"07:01.895","Text":"The complete answer would be."},{"Start":"07:01.895 ","End":"07:04.820","Text":"I\u0027ll write it over here."},{"Start":"07:04.820 ","End":"07:14.420","Text":"I have some space that x=1 or 3 or minus 1/2."},{"Start":"07:14.420 ","End":"07:25.940","Text":"I\u0027ll highlight the solutions and that\u0027s it for Part A and so on to Part B."},{"Start":"07:25.940 ","End":"07:29.800","Text":"Using the same method with the rational root theorem,"},{"Start":"07:29.800 ","End":"07:34.330","Text":"we say that x is a rational number p/q,"},{"Start":"07:34.330 ","End":"07:37.760","Text":"or at least will look for rational solutions first."},{"Start":"07:38.190 ","End":"07:41.500","Text":"Could be irrational solutions,"},{"Start":"07:41.500 ","End":"07:44.950","Text":"but the rational root theorem just talks about the rational roots."},{"Start":"07:44.950 ","End":"07:51.820","Text":"Then p has to be a divisor of minus 1."},{"Start":"07:51.820 ","End":"07:59.440","Text":"P can only be plus or minus 1 and q has to go into this,"},{"Start":"07:59.440 ","End":"08:03.130","Text":"so q could be plus or minus 1,"},{"Start":"08:03.130 ","End":"08:05.960","Text":"plus or minus 2."},{"Start":"08:06.030 ","End":"08:11.440","Text":"I just want to write out all the combinations this time because there\u0027s not that many,"},{"Start":"08:11.440 ","End":"08:13.645","Text":"before there were too many."},{"Start":"08:13.645 ","End":"08:16.450","Text":"I could get that,"},{"Start":"08:16.450 ","End":"08:20.425","Text":"let\u0027s say that this was 1."},{"Start":"08:20.425 ","End":"08:25.000","Text":"Then we get 1/1,"},{"Start":"08:25.000 ","End":"08:27.850","Text":"1 over minus 1,"},{"Start":"08:27.850 ","End":"08:32.900","Text":"1/2, 1 over minus 2."},{"Start":"08:32.900 ","End":"08:36.985","Text":"The minus 1 will give the same 4 possibilities,"},{"Start":"08:36.985 ","End":"08:41.020","Text":"just in a different order because I\u0027ll take a minus here and a plus here,"},{"Start":"08:41.020 ","End":"08:42.835","Text":"anyway, you get what I mean."},{"Start":"08:42.835 ","End":"08:45.475","Text":"There\u0027s 4 possibilities for x."},{"Start":"08:45.475 ","End":"08:50.410","Text":"We need to just find one of them,"},{"Start":"08:50.410 ","End":"08:52.255","Text":"let\u0027s try them in order."},{"Start":"08:52.255 ","End":"09:00.460","Text":"x equals 1 will give us 2 plus 1, minus 2,"},{"Start":"09:00.460 ","End":"09:02.950","Text":"minus 1 and again,"},{"Start":"09:02.950 ","End":"09:10.840","Text":"we got lucky first time, and x=1 works."},{"Start":"09:10.840 ","End":"09:14.470","Text":"We have a choice of how to proceed."},{"Start":"09:14.470 ","End":"09:18.055","Text":"We can either use long division of polynomials,"},{"Start":"09:18.055 ","End":"09:20.050","Text":"which is the standard way,"},{"Start":"09:20.050 ","End":"09:25.150","Text":"or we could just keep checking these since there\u0027s not that many."},{"Start":"09:25.150 ","End":"09:30.955","Text":"I\u0027ll use polynomial division and just use this as a verification."},{"Start":"09:30.955 ","End":"09:38.110","Text":"I do division of polynomials to say if x=1 is a solution,"},{"Start":"09:38.110 ","End":"09:40.720","Text":"then x minus 1 is a factor."},{"Start":"09:40.720 ","End":"09:45.535","Text":"I get, what this is using divisions of polynomials."},{"Start":"09:45.535 ","End":"09:48.850","Text":"Now I do a long division here. But you know what?"},{"Start":"09:48.850 ","End":"09:54.950","Text":"I\u0027m going to just save time and assume you know how to do long division of polynomials."},{"Start":"09:55.670 ","End":"10:04.580","Text":"The answer is 2x^2 plus 3x plus 1."},{"Start":"10:04.580 ","End":"10:09.070","Text":"Let me denote that this was by long division, which I omitted."},{"Start":"10:09.070 ","End":"10:12.250","Text":"Then we solve that."},{"Start":"10:12.250 ","End":"10:17.500","Text":"We know that x=1 is one possibility if the first factor is 0,"},{"Start":"10:17.500 ","End":"10:21.790","Text":"so the other come from letting this thing equals 0."},{"Start":"10:21.790 ","End":"10:26.485","Text":"If I do 2x squared plus 3x plus 1 equals 0,"},{"Start":"10:26.485 ","End":"10:29.410","Text":"I get a quadratic equation."},{"Start":"10:29.410 ","End":"10:31.660","Text":"Again, to save time,"},{"Start":"10:31.660 ","End":"10:33.910","Text":"since you know how to do quadratic equations,"},{"Start":"10:33.910 ","End":"10:43.330","Text":"I\u0027ll just give you the answer that x is equal to either minus 1 or minus 1/2."},{"Start":"10:43.330 ","End":"10:46.300","Text":"I notice that they appear on the list."},{"Start":"10:46.300 ","End":"10:48.130","Text":"This is the minus 1,"},{"Start":"10:48.130 ","End":"10:52.550","Text":"this is the minus 1/2 and originally the x=1."},{"Start":"10:52.620 ","End":"10:55.870","Text":"We have 3 solutions now,"},{"Start":"10:55.870 ","End":"10:58.150","Text":"and this is a cubic, meaning degree 3."},{"Start":"10:58.150 ","End":"11:00.894","Text":"We don\u0027t need to look any further."},{"Start":"11:00.894 ","End":"11:04.570","Text":"The answer is, well,"},{"Start":"11:04.570 ","End":"11:09.040","Text":"3 possible answers, this or this, or this."},{"Start":"11:09.040 ","End":"11:11.470","Text":"That\u0027s it for Part b,"},{"Start":"11:11.470 ","End":"11:18.610","Text":"so onto Part c. Once again,"},{"Start":"11:18.610 ","End":"11:25.690","Text":"we\u0027re going to use the rational root theorem and have x as a fraction, p/q,"},{"Start":"11:25.690 ","End":"11:31.090","Text":"where p goes into minus 2,"},{"Start":"11:31.090 ","End":"11:34.495","Text":"so p could be plus or minus 1,"},{"Start":"11:34.495 ","End":"11:37.555","Text":"or plus or minus 2."},{"Start":"11:37.555 ","End":"11:43.540","Text":"Actually, I can assume q is positive because if it\u0027s negative,"},{"Start":"11:43.540 ","End":"11:45.610","Text":"I can always throw the minus to the top,"},{"Start":"11:45.610 ","End":"11:51.910","Text":"so q could equal 1, 2 or 4."},{"Start":"11:51.910 ","End":"11:54.850","Text":"Actually, that\u0027s quite a lot of combinations because there\u0027s"},{"Start":"11:54.850 ","End":"11:58.195","Text":"4 possibilities for p and 3 possibilities for q."},{"Start":"11:58.195 ","End":"12:00.325","Text":"That\u0027s 12 combinations."},{"Start":"12:00.325 ","End":"12:04.885","Text":"As usual, we start off by assuming that, well first of all,"},{"Start":"12:04.885 ","End":"12:09.189","Text":"not assuming, but looking where q is 1, the denominator."},{"Start":"12:09.189 ","End":"12:13.060","Text":"We have only 4 possibilities to try for whole numbers."},{"Start":"12:13.060 ","End":"12:15.400","Text":"1 minus 1, 2 and minus 2."},{"Start":"12:15.400 ","End":"12:19.780","Text":"Let\u0027s start with the p=1."},{"Start":"12:19.780 ","End":"12:21.955","Text":"q is 1, so x is 1."},{"Start":"12:21.955 ","End":"12:24.445","Text":"If we plug in x=1,"},{"Start":"12:24.445 ","End":"12:32.050","Text":"we get 4 plus 5 is 9 minus 7 minus 2 is minus 9 is 0,"},{"Start":"12:32.050 ","End":"12:39.380","Text":"and it works, so x=1 is one possibility."},{"Start":"12:39.870 ","End":"12:44.950","Text":"I\u0027m going to use the long division method now rather than"},{"Start":"12:44.950 ","End":"12:49.270","Text":"trying to look for other solutions by substitution,"},{"Start":"12:49.270 ","End":"12:59.540","Text":"this is a sure thing and so we divide x minus 1 into this."},{"Start":"13:02.580 ","End":"13:05.155","Text":"Using long division,"},{"Start":"13:05.155 ","End":"13:07.000","Text":"I divide this over this."},{"Start":"13:07.000 ","End":"13:09.520","Text":"I\u0027ll just give you the answer."},{"Start":"13:09.520 ","End":"13:12.040","Text":"I won\u0027t do the long division."},{"Start":"13:12.040 ","End":"13:16.615","Text":"I\u0027ll just say that if you divide this by this,"},{"Start":"13:16.615 ","End":"13:24.620","Text":"you get 4x^2 plus 9x plus 2."},{"Start":"13:25.200 ","End":"13:27.955","Text":"This is the long division you would do."},{"Start":"13:27.955 ","End":"13:29.350","Text":"Maybe I\u0027ll just do the first step."},{"Start":"13:29.350 ","End":"13:39.685","Text":"You would say x into 4x^3 goes 4x^2 times and then multiply out 4x^3 minus 4x^2."},{"Start":"13:39.685 ","End":"13:44.425","Text":"Then this minus 5 minus minus 4 is 9x^2."},{"Start":"13:44.425 ","End":"13:48.745","Text":"You can see how x goes into 9x^2 9x times and so on."},{"Start":"13:48.745 ","End":"13:51.440","Text":"I want to complete it."},{"Start":"13:51.450 ","End":"13:54.880","Text":"From this, we get that either x minus 1 is 0,"},{"Start":"13:54.880 ","End":"13:57.775","Text":"which we knew, and that gives us x=1."},{"Start":"13:57.775 ","End":"14:05.990","Text":"All we have to do now is solve the quadratic 4x^2 plus 9x plus 2 equals 0."},{"Start":"14:06.120 ","End":"14:09.460","Text":"I\u0027m assuming you know how to solve quadratic equations."},{"Start":"14:09.460 ","End":"14:10.660","Text":"I won\u0027t waste time with that."},{"Start":"14:10.660 ","End":"14:15.100","Text":"I\u0027ll give you the answer that x is either equal"},{"Start":"14:15.100 ","End":"14:22.135","Text":"to minus a 1/4 or minus 2."},{"Start":"14:22.135 ","End":"14:25.885","Text":"Just note that it fits the pattern of p/q."},{"Start":"14:25.885 ","End":"14:31.330","Text":"If I take here minus 1 and here 4, I get that."},{"Start":"14:31.330 ","End":"14:35.335","Text":"If I take here minus 2 and here 1, I get that."},{"Start":"14:35.335 ","End":"14:39.280","Text":"Altogether we have 3 solutions."},{"Start":"14:39.280 ","End":"14:43.795","Text":"This and these 2."},{"Start":"14:43.795 ","End":"14:46.450","Text":"Since we have 3 solutions and it\u0027s a cubic,"},{"Start":"14:46.450 ","End":"14:48.655","Text":"we can expect to find anymore."},{"Start":"14:48.655 ","End":"14:54.515","Text":"Well, there can\u0027t be anymore because we did it using this method."},{"Start":"14:54.515 ","End":"15:02.480","Text":"This ensures that these are the only 3 possible solutions and we\u0027re done with Part c,"},{"Start":"15:02.480 ","End":"15:04.970","Text":"and this was the last part."},{"Start":"15:04.970 ","End":"15:07.709","Text":"That\u0027s the end of the exercise."}],"ID":6358},{"Watched":false,"Name":"Exercise 3","Duration":"11m 25s","ChapterTopicVideoID":6342,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6342.jpeg","UploadDate":"2016-06-22T08:45:37.9130000","DurationForVideoObject":"PT11M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.345","Text":"This exercise is made up of 4 separate equations and each of them,"},{"Start":"00:06.345 ","End":"00:08.430","Text":"we have a polynomial = 0,"},{"Start":"00:08.430 ","End":"00:12.685","Text":"and that polynomial is of degree higher than 2,"},{"Start":"00:12.685 ","End":"00:16.125","Text":"and the leading coefficient is 1."},{"Start":"00:16.125 ","End":"00:20.820","Text":"The reason that you need some knowledge about the derivative,"},{"Start":"00:20.820 ","End":"00:23.280","Text":"which is something from Calculus,"},{"Start":"00:23.280 ","End":"00:27.810","Text":"is that it relates to the concept of a multiplicity of a root."},{"Start":"00:27.810 ","End":"00:30.810","Text":"Should be familiar with this concept."},{"Start":"00:30.810 ","End":"00:35.190","Text":"In any event I will be explaining it briefly again in each exercise."},{"Start":"00:35.190 ","End":"00:39.190","Text":"Let\u0027s start with part a."},{"Start":"00:41.080 ","End":"00:47.030","Text":"We begin by looking for whole number, an integer root."},{"Start":"00:47.030 ","End":"00:50.800","Text":"It has to be a divisor of 3."},{"Start":"00:50.800 ","End":"00:56.870","Text":"In other words, the root can only be plus or minus 1,"},{"Start":"00:56.870 ","End":"00:59.090","Text":"or plus or minus 3."},{"Start":"00:59.090 ","End":"01:01.295","Text":"There\u0027s 4 possibilities."},{"Start":"01:01.295 ","End":"01:03.260","Text":"We substitute."},{"Start":"01:03.260 ","End":"01:06.895","Text":"Let\u0027s try x= 1."},{"Start":"01:06.895 ","End":"01:10.945","Text":"These are the possibilities for x, for whole number."},{"Start":"01:10.945 ","End":"01:13.600","Text":"Let\u0027s try x= 1."},{"Start":"01:13.600 ","End":"01:20.820","Text":"That would give us 1+1-5+ 3=0."},{"Start":"01:20.820 ","End":"01:25.140","Text":"We\u0027ve already found 1 root is that x =1."},{"Start":"01:25.140 ","End":"01:30.920","Text":"Now this x=1 could be of multiplicity higher than 1,"},{"Start":"01:30.920 ","End":"01:35.525","Text":"meaning it might be a double or triple root."},{"Start":"01:35.525 ","End":"01:40.849","Text":"The way we find this out is by substituting it in the derivative."},{"Start":"01:40.849 ","End":"01:43.880","Text":"Let\u0027s first of all, compute the derivative."},{"Start":"01:43.880 ","End":"01:53.245","Text":"That would be, let\u0027s see, 3x^2 + 2x-5."},{"Start":"01:53.245 ","End":"01:56.985","Text":"Now, let\u0027s substitute this in the derivative."},{"Start":"01:56.985 ","End":"02:04.395","Text":"We get 3 + 2 -5, that is 0."},{"Start":"02:04.395 ","End":"02:09.180","Text":"This is also a root of the derivative."},{"Start":"02:09.180 ","End":"02:12.290","Text":"We can already write a second route."},{"Start":"02:12.290 ","End":"02:14.215","Text":"Also 1."},{"Start":"02:14.215 ","End":"02:16.910","Text":"Or you could say it\u0027s 1 with multiplicity 2,"},{"Start":"02:16.910 ","End":"02:19.820","Text":"but it could even have multiplicity 3."},{"Start":"02:19.820 ","End":"02:23.850","Text":"Let\u0027s derive again, and this time we get"},{"Start":"02:23.850 ","End":"02:30.065","Text":"6x + 2 and if we substitute x=1,"},{"Start":"02:30.065 ","End":"02:34.045","Text":"no it\u0027s not a root."},{"Start":"02:34.045 ","End":"02:39.870","Text":"Let\u0027s see if any other of these works as a root."},{"Start":"02:39.870 ","End":"02:43.445","Text":"Well, if you try -1, it doesn\u0027t give 0."},{"Start":"02:43.445 ","End":"02:49.399","Text":"Basically I\u0027ve tried them and the only one that works"},{"Start":"02:49.399 ","End":"02:56.000","Text":"besides 1 is -3."},{"Start":"02:56.000 ","End":"03:00.425","Text":"You can check -3^3 is -27."},{"Start":"03:00.425 ","End":"03:04.985","Text":"Then we get plus 9"},{"Start":"03:04.985 ","End":"03:13.050","Text":"plus 15 plus 3 and it works out."},{"Start":"03:13.090 ","End":"03:17.215","Text":"We also have -3."},{"Start":"03:17.215 ","End":"03:21.230","Text":"Because we found 3 roots to a cubic equation,"},{"Start":"03:21.230 ","End":"03:23.940","Text":"these are the only roots."},{"Start":"03:25.300 ","End":"03:36.275","Text":"On to part b. In part b we also have a cubic and the possibilities for a whole number x,"},{"Start":"03:36.275 ","End":"03:39.210","Text":"it\u0027s the divisors of -2."},{"Start":"03:39.210 ","End":"03:44.835","Text":"We have plus or minus 1 plus or minus 2."},{"Start":"03:44.835 ","End":"03:47.250","Text":"We try x=1."},{"Start":"03:47.250 ","End":"03:49.755","Text":"It doesn\u0027t work, 1 minus 3, minus 2,"},{"Start":"03:49.755 ","End":"03:52.755","Text":"not 0, try minus 1."},{"Start":"03:52.755 ","End":"03:57.540","Text":"Then we get minus 1 plus 3 minus 2."},{"Start":"03:57.540 ","End":"04:03.165","Text":"That will work, we have x = -1."},{"Start":"04:03.165 ","End":"04:08.495","Text":"Now let\u0027s check perhaps it\u0027s a double root or triple root,"},{"Start":"04:08.495 ","End":"04:11.000","Text":"it has maybe a multiplicity."},{"Start":"04:11.000 ","End":"04:14.970","Text":"We differentiate this polynomial,"},{"Start":"04:15.610 ","End":"04:20.870","Text":"3x^2 -3, and then substitute x"},{"Start":"04:20.870 ","End":"04:27.460","Text":"=-1 and -1^2 is 1 so it\u0027s 3-3 is 0."},{"Start":"04:27.460 ","End":"04:30.080","Text":"Yes, that\u0027s good."},{"Start":"04:30.080 ","End":"04:34.775","Text":"That means that it\u0027s already a double root, at least."},{"Start":"04:34.775 ","End":"04:38.815","Text":"Let\u0027s see, maybe it\u0027s a triple root."},{"Start":"04:38.815 ","End":"04:44.180","Text":"Differentiate again and we get 6x."},{"Start":"04:44.510 ","End":"04:48.540","Text":"Then we try and substitute -1."},{"Start":"04:48.540 ","End":"04:52.845","Text":"Nope, that doesn\u0027t do it."},{"Start":"04:52.845 ","End":"04:55.220","Text":"We have to find the third one."},{"Start":"04:55.220 ","End":"04:59.425","Text":"Let\u0027s keep looking for integer solutions."},{"Start":"04:59.425 ","End":"05:07.710","Text":"Let\u0027s try 2^3 is 8 minus 6 minus 2. That works."},{"Start":"05:07.710 ","End":"05:13.730","Text":"I have 3 then the degree of the polynomial is 3."},{"Start":"05:13.730 ","End":"05:16.204","Text":"So I need look no further."},{"Start":"05:16.204 ","End":"05:19.670","Text":"These are my 3 solutions."},{"Start":"05:19.670 ","End":"05:22.115","Text":"You can argue if it\u0027s 3 or 2,"},{"Start":"05:22.115 ","End":"05:23.690","Text":"there\u0027s really 3 solutions."},{"Start":"05:23.690 ","End":"05:25.295","Text":"2 of them happen to be the same,-1,"},{"Start":"05:25.295 ","End":"05:27.795","Text":"minus 1 and 2."},{"Start":"05:27.795 ","End":"05:32.035","Text":"Solutions to the equation or roots of the polynomial. Same thing."},{"Start":"05:32.035 ","End":"05:35.680","Text":"Now onto the last one part."},{"Start":"05:35.680 ","End":"05:39.710","Text":"We have a part the third one."},{"Start":"05:41.600 ","End":"05:49.520","Text":"This time it\u0027s degree 5 or quintic polynomial."},{"Start":"05:51.060 ","End":"05:56.335","Text":"Same idea because the leading coefficient is 1."},{"Start":"05:56.335 ","End":"05:59.530","Text":"The trailing constant."},{"Start":"05:59.530 ","End":"06:01.690","Text":"We have a lot of possibilities."},{"Start":"06:01.690 ","End":"06:04.734","Text":"X could be, if it\u0027s an integer,"},{"Start":"06:04.734 ","End":"06:07.830","Text":"could be plus or minus 1,"},{"Start":"06:07.830 ","End":"06:11.460","Text":"plus or minus 2 plus or minus 3,"},{"Start":"06:11.460 ","End":"06:13.650","Text":"12 has a lot of divisors."},{"Start":"06:13.650 ","End":"06:15.995","Text":"If you take plus or minus,"},{"Start":"06:15.995 ","End":"06:20.400","Text":"it could be plus or minus 4."},{"Start":"06:20.630 ","End":"06:27.840","Text":"It could be plus or minus 6 and it could be plus or minus 12."},{"Start":"06:27.840 ","End":"06:32.375","Text":"We have 12 possibilities just for whole numbers."},{"Start":"06:32.375 ","End":"06:34.970","Text":"Let\u0027s try them one at a time."},{"Start":"06:34.970 ","End":"06:43.420","Text":"Let\u0027s start with x =1, 1-3-5+27- 32 +12."},{"Start":"06:43.420 ","End":"06:48.860","Text":"Let\u0027s go with the pluses first we have a 1 and a 27 and a 12."},{"Start":"06:48.860 ","End":"06:53.400","Text":"That makes 40. Here we have minus 3."},{"Start":"06:53.400 ","End":"06:58.410","Text":"Well the minuses are 3,5 and 32, also 40."},{"Start":"06:58.410 ","End":"07:03.765","Text":"We got lucky first time with x = 1."},{"Start":"07:03.765 ","End":"07:07.770","Text":"Let\u0027s see if we can take this x=1 further."},{"Start":"07:07.770 ","End":"07:13.250","Text":"Maybe it\u0027s a double root or a root with multiplicity 2 or more."},{"Start":"07:13.250 ","End":"07:18.900","Text":"We differentiate and we get x, sorry, 5x^4-12x^3-15x^2+"},{"Start":"07:28.090 ","End":"07:34.165","Text":"54x- 32."},{"Start":"07:34.165 ","End":"07:38.600","Text":"Let\u0027s see if x equals 1 is a root of this."},{"Start":"07:38.600 ","End":"07:43.460","Text":"Again, all the X bits come out once or you can just throw them out."},{"Start":"07:43.460 ","End":"07:44.990","Text":"Let\u0027s take the positives,"},{"Start":"07:44.990 ","End":"07:49.370","Text":"5 and 54 is 59,"},{"Start":"07:49.370 ","End":"07:56.340","Text":"then the negatives 12 and 15 and 32 also comes out to 59."},{"Start":"07:56.340 ","End":"08:02.525","Text":"Yes, x=1 is a double root."},{"Start":"08:02.525 ","End":"08:04.160","Text":"Maybe it\u0027s a triple root."},{"Start":"08:04.160 ","End":"08:06.870","Text":"Let\u0027s differentiate again."},{"Start":"08:06.910 ","End":"08:09.900","Text":"What do we have?"},{"Start":"08:11.030 ","End":"08:20.010","Text":"20x^3-"},{"Start":"08:20.510 ","End":"08:22.850","Text":"36x^2-"},{"Start":"08:22.850 ","End":"08:26.005","Text":"30x+ 54."},{"Start":"08:26.005 ","End":"08:29.620","Text":"If I put in x = 1, no,"},{"Start":"08:29.620 ","End":"08:36.180","Text":"it does not come out 0 because the positives are 20 and 54 is 74,"},{"Start":"08:36.180 ","End":"08:43.774","Text":"but the negatives are 66 and 66 is not equal to 74, so it\u0027s not 0."},{"Start":"08:43.774 ","End":"08:46.045","Text":"Just a double root."},{"Start":"08:46.045 ","End":"08:50.060","Text":"Let\u0027s keep looking for more integer roots."},{"Start":"08:50.220 ","End":"08:57.340","Text":"I\u0027ll save you the trouble and tell you that the next 1 that works."},{"Start":"08:57.340 ","End":"09:00.910","Text":"Minus 1 doesn\u0027t work, 2 does work."},{"Start":"09:00.910 ","End":"09:03.340","Text":"If you put in 2 into here."},{"Start":"09:03.340 ","End":"09:05.475","Text":"It gives 0."},{"Start":"09:05.475 ","End":"09:08.600","Text":"Then we check maybe it\u0027s a double root,"},{"Start":"09:08.600 ","End":"09:10.610","Text":"so we put 2 in here."},{"Start":"09:10.610 ","End":"09:12.349","Text":"That also works."},{"Start":"09:12.349 ","End":"09:15.605","Text":"I\u0027m just saving you the tedious calculation."},{"Start":"09:15.605 ","End":"09:19.010","Text":"I mean, you should do this on your own.. 2 works."},{"Start":"09:19.010 ","End":"09:25.550","Text":"Then we try 2 in the second derivative and it does not come out 0."},{"Start":"09:25.550 ","End":"09:30.230","Text":"We continue looking for another maybe whole number root."},{"Start":"09:30.230 ","End":"09:35.070","Text":"Eventually we come to minus 3, which works."},{"Start":"09:35.330 ","End":"09:41.030","Text":"Once we have 5 roots and it\u0027s a degree 5 polynomial,"},{"Start":"09:41.030 ","End":"09:43.500","Text":"we know that we can stop."},{"Start":"09:43.900 ","End":"09:46.820","Text":"These are the 5 roots."},{"Start":"09:46.820 ","End":"09:52.340","Text":"We\u0027re now ready for part d, the last part."},{"Start":"09:52.340 ","End":"09:55.295","Text":"If the polynomial has an integer root,"},{"Start":"09:55.295 ","End":"10:00.560","Text":"then x can only be plus or minus 1."},{"Start":"10:00.560 ","End":"10:03.290","Text":"It has to divide the trailing constant."},{"Start":"10:03.290 ","End":"10:08.650","Text":"In other words, I only have to look at 1 or minus 1 for possibilities."},{"Start":"10:08.650 ","End":"10:17.420","Text":"I try x =1, 1-3+3-1."},{"Start":"10:17.450 ","End":"10:24.300","Text":"Yes, x=1 works."},{"Start":"10:24.300 ","End":"10:29.880","Text":"Then I go and see if I\u0027m lucky and maybe it\u0027s a double root."},{"Start":"10:31.910 ","End":"10:35.240","Text":"I substitute it in the derivative."},{"Start":"10:35.240 ","End":"10:40.660","Text":"The derivative is 3x^2-6x+3."},{"Start":"10:40.660 ","End":"10:44.040","Text":"I put in 1, I get 3 minus 6 plus 3."},{"Start":"10:44.040 ","End":"10:46.410","Text":"Yes, it is 0."},{"Start":"10:46.410 ","End":"10:52.155","Text":"I can add another 1 and maybe we can get lucky again."},{"Start":"10:52.155 ","End":"10:59.145","Text":"Let\u0027s differentiate 6x-6, substitute 1."},{"Start":"10:59.145 ","End":"11:03.735","Text":"Yet again, it is =0, 6 minus 6."},{"Start":"11:03.735 ","End":"11:06.885","Text":"We have another 1."},{"Start":"11:06.885 ","End":"11:08.750","Text":"Now we have 3 already,"},{"Start":"11:08.750 ","End":"11:12.440","Text":"so there\u0027s no need to continue and so we have"},{"Start":"11:12.440 ","End":"11:16.985","Text":"1 which is a triple root or a root with multiplicity 3."},{"Start":"11:16.985 ","End":"11:20.510","Text":"The answer is 1, 1 and 1."},{"Start":"11:20.510 ","End":"11:25.980","Text":"These are the 3 roots and we are done."}],"ID":6359},{"Watched":false,"Name":"Exercise 4","Duration":"5m 15s","ChapterTopicVideoID":6343,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6343.jpeg","UploadDate":"2016-06-22T08:46:29.3000000","DurationForVideoObject":"PT5M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"This exercise is made up of two parts."},{"Start":"00:03.300 ","End":"00:06.600","Text":"Turns out that part b is closely related to part"},{"Start":"00:06.600 ","End":"00:10.140","Text":"a but we\u0027ll see what the connection is when we get to it."},{"Start":"00:10.140 ","End":"00:14.100","Text":"Meanwhile, let\u0027s begin with part a where we just have"},{"Start":"00:14.100 ","End":"00:18.705","Text":"an equation to solve and it happens to be a cubic equation,"},{"Start":"00:18.705 ","End":"00:25.815","Text":"degree 3 and it\u0027s got integer coefficients and even a leading coefficient of 1."},{"Start":"00:25.815 ","End":"00:30.050","Text":"One way to start is to first of all look for integer solutions."},{"Start":"00:30.050 ","End":"00:36.735","Text":"The integer solutions have to divide the constant term, the trailing constant."},{"Start":"00:36.735 ","End":"00:43.205","Text":"In other words, if it\u0027s going to be a whole number it has to be 1,"},{"Start":"00:43.205 ","End":"00:46.010","Text":"minus 1, 2 or minus 2,"},{"Start":"00:46.010 ","End":"00:48.425","Text":"those are the only numbers that divide 2,"},{"Start":"00:48.425 ","End":"00:55.710","Text":"and if we try them we find that x=1 works because look, 1^3 minus 1^2."},{"Start":"00:55.710 ","End":"00:58.320","Text":"It\u0027s just 1 minus 1 minus 2 plus 2,"},{"Start":"00:58.320 ","End":"01:02.460","Text":"it is 0, so x=1 works."},{"Start":"01:02.460 ","End":"01:06.860","Text":"Unfortunately we can try the other three and none of them works."},{"Start":"01:06.860 ","End":"01:13.740","Text":"At least we have that one and if x=1 is a root or a solution,"},{"Start":"01:13.740 ","End":"01:19.405","Text":"then x minus 1 divides evenly into this polynomial."},{"Start":"01:19.405 ","End":"01:24.380","Text":"We\u0027re going to do a long division and see what it is."},{"Start":"01:24.380 ","End":"01:27.010","Text":"It\u0027ll be a quadratic and equals 0,"},{"Start":"01:27.010 ","End":"01:29.540","Text":"so I\u0027ll do that long division at the side."},{"Start":"01:29.540 ","End":"01:33.155","Text":"Let\u0027s draw a long division,"},{"Start":"01:33.155 ","End":"01:40.260","Text":"here I\u0027m going to put the x^3 minus x^2 minus 2x plus 2."},{"Start":"01:40.260 ","End":"01:42.880","Text":"Here the x minus 1,"},{"Start":"01:42.880 ","End":"01:53.105","Text":"x into x^3 goes x^2 times x^2 times x minus 1 is x^3 minus x^2."},{"Start":"01:53.105 ","End":"01:57.575","Text":"That\u0027s convenient. These dropout, it cancels."},{"Start":"01:57.575 ","End":"01:59.390","Text":"We can drop 2 more down,"},{"Start":"01:59.390 ","End":"02:01.190","Text":"minus 2x plus 2,"},{"Start":"02:01.190 ","End":"02:05.569","Text":"x into minus 2x goes minus 2 times,"},{"Start":"02:05.569 ","End":"02:12.280","Text":"minus 2 times x minus 1 is minus 2x."},{"Start":"02:12.320 ","End":"02:16.590","Text":"Oh sorry, there\u0027s a plus here, plus 2,"},{"Start":"02:16.590 ","End":"02:20.685","Text":"and this cancels completely, no remainder."},{"Start":"02:20.685 ","End":"02:26.245","Text":"What we have here is x^2 minus 2."},{"Start":"02:26.245 ","End":"02:30.950","Text":"If a product is 0 then either one must be 0."},{"Start":"02:30.950 ","End":"02:34.910","Text":"This one gives us that x=1 but we knew that already,"},{"Start":"02:34.910 ","End":"02:36.410","Text":"so that\u0027s not new."},{"Start":"02:36.410 ","End":"02:42.929","Text":"But we now get another possibility that x^2 minus 2 equals 0."},{"Start":"02:42.970 ","End":"02:45.095","Text":"We can solve this,"},{"Start":"02:45.095 ","End":"02:46.459","Text":"because there\u0027s a missing x term,"},{"Start":"02:46.459 ","End":"02:49.640","Text":"by bringing the 2 over to the right and then saying"},{"Start":"02:49.640 ","End":"02:53.525","Text":"x equals plus or minus the square root of 2."},{"Start":"02:53.525 ","End":"02:55.595","Text":"I have my three roots."},{"Start":"02:55.595 ","End":"02:57.370","Text":"They are 1,"},{"Start":"02:57.370 ","End":"03:01.800","Text":"square root of 2 and minus square root of 2,"},{"Start":"03:01.800 ","End":"03:06.000","Text":"and I\u0027ll just highlight them."},{"Start":"03:06.000 ","End":"03:13.900","Text":"There they are. Let\u0027s take a look at part b."},{"Start":"03:13.900 ","End":"03:21.365","Text":"Here\u0027s part b which asks us to find the zeros of the polynomial p of x equals."},{"Start":"03:21.365 ","End":"03:24.389","Text":"Now let\u0027s take a look at this."},{"Start":"03:27.590 ","End":"03:32.360","Text":"I think not only similar, they\u0027re identical."},{"Start":"03:32.360 ","End":"03:35.630","Text":"This exercise is trying to make a point here."},{"Start":"03:35.630 ","End":"03:37.940","Text":"It\u0027s a semantic one really,"},{"Start":"03:37.940 ","End":"03:39.320","Text":"it\u0027s a matter of definitions."},{"Start":"03:39.320 ","End":"03:43.160","Text":"The zeros of the polynomial are defined to be"},{"Start":"03:43.160 ","End":"03:49.750","Text":"the solutions of the equation p of x equals 0."},{"Start":"03:49.750 ","End":"03:52.410","Text":"That\u0027s synonymous."},{"Start":"03:52.410 ","End":"03:55.750","Text":"It\u0027s the same thing to find the solution to"},{"Start":"03:55.750 ","End":"03:59.890","Text":"the polynomial equals 0 is the same as to find the 0."},{"Start":"03:59.890 ","End":"04:03.280","Text":"There\u0027s actually an alternative word for 0 and that\u0027s root."},{"Start":"04:03.280 ","End":"04:09.029","Text":"Zeros and roots are exactly the same thing in this context."},{"Start":"04:09.029 ","End":"04:13.585","Text":"They could have said the roots of the polynomial are;"},{"Start":"04:13.585 ","End":"04:16.150","Text":"it would have meant the same thing."},{"Start":"04:16.150 ","End":"04:25.480","Text":"All we do here is copy these three answers because these are the,"},{"Start":"04:25.480 ","End":"04:30.290","Text":"I should have written this, solutions of."},{"Start":"04:32.360 ","End":"04:35.215","Text":"Once again, because it\u0027s important,"},{"Start":"04:35.215 ","End":"04:41.425","Text":"zeros of p of x are the same as solutions of p of x equals 0,"},{"Start":"04:41.425 ","End":"04:45.095","Text":"in which case they are 1,"},{"Start":"04:45.095 ","End":"04:54.280","Text":"square root of 2 and minus square root of 2 and I\u0027ll highlight them again."},{"Start":"04:54.720 ","End":"04:58.585","Text":"Just one other minor thing,"},{"Start":"04:58.585 ","End":"05:00.490","Text":"just a question of spelling."},{"Start":"05:00.490 ","End":"05:05.830","Text":"Some people put the e in zeros and some people don\u0027t."},{"Start":"05:05.830 ","End":"05:09.610","Text":"It doesn\u0027t matter, it\u0027s just a minor point,"},{"Start":"05:09.610 ","End":"05:11.690","Text":"I thought I\u0027d mention it."},{"Start":"05:12.810 ","End":"05:16.310","Text":"We are done. That\u0027s it."}],"ID":6360},{"Watched":false,"Name":"Exercise 5","Duration":"4m 45s","ChapterTopicVideoID":6344,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6344.jpeg","UploadDate":"2016-06-22T08:47:16.4570000","DurationForVideoObject":"PT4M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this exercise we have 2 parts,"},{"Start":"00:03.030 ","End":"00:05.460","Text":"and they\u0027re very closely related."},{"Start":"00:05.460 ","End":"00:09.915","Text":"In fact you\u0027ll notice that this looks very much like this,"},{"Start":"00:09.915 ","End":"00:12.630","Text":"but let\u0027s begin with Part A,"},{"Start":"00:12.630 ","End":"00:18.210","Text":"and we just have to solve a polynomial equation."},{"Start":"00:18.210 ","End":"00:20.790","Text":"The coefficients are integers."},{"Start":"00:20.790 ","End":"00:22.470","Text":"The leading coefficient is 1,"},{"Start":"00:22.470 ","End":"00:27.135","Text":"so let\u0027s look first of all for integer solutions."},{"Start":"00:27.135 ","End":"00:32.190","Text":"As we know, there is a theorem that these must be divisors of -2,"},{"Start":"00:32.190 ","End":"00:36.360","Text":"so we only have 4 possibilities for integer solutions,"},{"Start":"00:36.360 ","End":"00:37.605","Text":"and that is 1,"},{"Start":"00:37.605 ","End":"00:42.240","Text":"-1, 2, or -2."},{"Start":"00:42.240 ","End":"00:46.585","Text":"What we do is take them one at a time until we find a root."},{"Start":"00:46.585 ","End":"00:49.060","Text":"Of course it depends on what order you try them in."},{"Start":"00:49.060 ","End":"00:55.590","Text":"I\u0027ll give you a spoiler here that it turns out that 1 and 2 both the fit,"},{"Start":"00:55.590 ","End":"00:58.375","Text":"but if you were going from left to right here,"},{"Start":"00:58.375 ","End":"01:00.295","Text":"you\u0027d hit the 1 first."},{"Start":"01:00.295 ","End":"01:03.625","Text":"I\u0027m just saying that if someone happened to try 2 first,"},{"Start":"01:03.625 ","End":"01:06.070","Text":"the solution would look different."},{"Start":"01:06.070 ","End":"01:08.050","Text":"I mean, the end answer would be the same."},{"Start":"01:08.050 ","End":"01:11.050","Text":"If we try 1, which is what we usually try because it\u0027s so"},{"Start":"01:11.050 ","End":"01:15.725","Text":"easy to substitute, we get 1-4+5-2."},{"Start":"01:15.725 ","End":"01:17.610","Text":"The 1 and the 5 is 6,"},{"Start":"01:17.610 ","End":"01:21.375","Text":"the -4 and -2 is -6 and it\u0027s 0."},{"Start":"01:21.375 ","End":"01:26.800","Text":"Once we know that x=1 is one of the solutions,"},{"Start":"01:26.800 ","End":"01:32.480","Text":"then we know that we can divide this polynomial by x-1."},{"Start":"01:32.480 ","End":"01:33.950","Text":"This is the approach I want to take."},{"Start":"01:33.950 ","End":"01:35.900","Text":"I want to use division of polynomials,"},{"Start":"01:35.900 ","End":"01:40.025","Text":"so I\u0027m going to write this as x-1 times something,"},{"Start":"01:40.025 ","End":"01:43.165","Text":"which will be of degree only 2."},{"Start":"01:43.165 ","End":"01:48.475","Text":"I\u0027ll just do a little long division at the side here."},{"Start":"01:48.475 ","End":"01:56.315","Text":"Take x^3-(4x)^2+5x-2."},{"Start":"01:56.315 ","End":"01:59.765","Text":"Here I write the x-1."},{"Start":"01:59.765 ","End":"02:03.280","Text":"Let\u0027s see, x into x^3 goes x^2."},{"Start":"02:03.280 ","End":"02:05.625","Text":"You know what?"},{"Start":"02:05.625 ","End":"02:08.630","Text":"I think I don\u0027t want to waste time with the long division."},{"Start":"02:08.630 ","End":"02:10.760","Text":"You know how to do long division."},{"Start":"02:10.760 ","End":"02:13.115","Text":"I\u0027ll just give you the final answer;"},{"Start":"02:13.115 ","End":"02:18.870","Text":"it\u0027s x^2-3x+2, and that\u0027s what I"},{"Start":"02:18.870 ","End":"02:25.670","Text":"put in here, x^2-3x+2."},{"Start":"02:25.670 ","End":"02:32.155","Text":"Then we know that x-1 is 0, which gives us x=1."},{"Start":"02:32.155 ","End":"02:33.965","Text":"Well, we knew that already."},{"Start":"02:33.965 ","End":"02:38.495","Text":"We get another possibility from letting this equals 0,"},{"Start":"02:38.495 ","End":"02:45.140","Text":"so now we get a quadratic x^2-3x+2=0."},{"Start":"02:45.140 ","End":"02:49.370","Text":"Again, I don\u0027t want to waste time with solving quadratic equations."},{"Start":"02:49.370 ","End":"02:51.305","Text":"You\u0027ve done so many probably,"},{"Start":"02:51.305 ","End":"02:56.105","Text":"but the solutions happen to be 1 and 2."},{"Start":"02:56.105 ","End":"03:02.639","Text":"Now if we combine the solutions from here and here we get 3 solutions,"},{"Start":"03:02.639 ","End":"03:04.665","Text":"though it depends how you look at it."},{"Start":"03:04.665 ","End":"03:07.035","Text":"I say the solutions are 1,"},{"Start":"03:07.035 ","End":"03:08.975","Text":"1, and 2, and there\u0027s 3 of them."},{"Start":"03:08.975 ","End":"03:13.940","Text":"Some people would say there\u0027s only 2 and this one has multiplicity 2,"},{"Start":"03:13.940 ","End":"03:16.040","Text":"but only 1 and 2 are solutions."},{"Start":"03:16.040 ","End":"03:18.920","Text":"Either way, we say that there\u0027s 3 solutions,"},{"Start":"03:18.920 ","End":"03:22.170","Text":"and 2 of them happen to be the same."},{"Start":"03:22.170 ","End":"03:24.270","Text":"I\u0027ll just highlight that,"},{"Start":"03:24.270 ","End":"03:27.610","Text":"and we move on to Part B."},{"Start":"03:27.680 ","End":"03:34.455","Text":"In Part B, which is very related to Part A,"},{"Start":"03:34.455 ","End":"03:39.140","Text":"we\u0027re asked to find the zeros of this polynomial."},{"Start":"03:39.140 ","End":"03:44.270","Text":"The definition of a 0 of a polynomial is the same"},{"Start":"03:44.270 ","End":"03:49.190","Text":"as the solution to the equation where the polynomial is 0."},{"Start":"03:49.190 ","End":"03:54.530","Text":"In other words, the zeros of f are the same as"},{"Start":"03:54.530 ","End":"04:01.520","Text":"the solutions of f(x)=0,"},{"Start":"04:01.520 ","End":"04:04.760","Text":"but we already have the solutions of f(x)= 0."},{"Start":"04:04.760 ","End":"04:06.980","Text":"I mean, this is f(x),"},{"Start":"04:06.980 ","End":"04:08.570","Text":"so it\u0027s just the same thing."},{"Start":"04:08.570 ","End":"04:11.040","Text":"I just copy 1, 1,"},{"Start":"04:11.040 ","End":"04:16.980","Text":"and 2, and highlight it."},{"Start":"04:17.600 ","End":"04:23.030","Text":"I just mentioned that zeros is the same thing as roots."},{"Start":"04:23.030 ","End":"04:26.990","Text":"In this context, the roots of a polynomial,"},{"Start":"04:26.990 ","End":"04:28.760","Text":"the zeros of a polynomial,"},{"Start":"04:28.760 ","End":"04:32.585","Text":"they\u0027re both the solutions of that polynomial equal to 0."},{"Start":"04:32.585 ","End":"04:35.435","Text":"The final remark is that on spelling,"},{"Start":"04:35.435 ","End":"04:38.740","Text":"some people spell zeros within an E,"},{"Start":"04:38.740 ","End":"04:41.175","Text":"and some spell it without the E,"},{"Start":"04:41.175 ","End":"04:44.590","Text":"no big deal. We\u0027re done."}],"ID":6361},{"Watched":false,"Name":"Exercise 6","Duration":"13m 3s","ChapterTopicVideoID":6345,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6345.jpeg","UploadDate":"2016-06-22T08:49:26.6570000","DurationForVideoObject":"PT13M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.070","Text":"This exercise consists of two separate not really related equations."},{"Start":"00:08.070 ","End":"00:09.653","Text":"They have something in common,"},{"Start":"00:09.653 ","End":"00:12.240","Text":"they\u0027re both a polynomial equals 0."},{"Start":"00:12.240 ","End":"00:17.760","Text":"The two of them have integer coefficients and a leading coefficient of 1,"},{"Start":"00:17.760 ","End":"00:20.805","Text":"so we use the same methods."},{"Start":"00:20.805 ","End":"00:23.384","Text":"Let\u0027s start with the first."},{"Start":"00:23.384 ","End":"00:25.980","Text":"The first one is a cubic equation,"},{"Start":"00:25.980 ","End":"00:30.210","Text":"it\u0027s of degree 3 and the second one happens to be degree 4,"},{"Start":"00:30.210 ","End":"00:32.640","Text":"also known as a quartic equation."},{"Start":"00:32.640 ","End":"00:36.000","Text":"Quadratic, cubic, quartic, quantic."},{"Start":"00:36.000 ","End":"00:40.090","Text":"Anyway, you don\u0027t need to know those terms."},{"Start":"00:41.740 ","End":"00:45.560","Text":"The usual way of doing it is to look for integer solution"},{"Start":"00:45.560 ","End":"00:49.130","Text":"first and if it\u0027s an integer solution,"},{"Start":"00:49.130 ","End":"00:54.215","Text":"it has to divide the trailing constant, this minus 2."},{"Start":"00:54.215 ","End":"00:58.575","Text":"We have four possibilities: 1, minus 1,"},{"Start":"00:58.575 ","End":"01:01.665","Text":"2, or minus 2."},{"Start":"01:01.665 ","End":"01:03.960","Text":"Try x=1."},{"Start":"01:03.960 ","End":"01:06.945","Text":"1 minus 1 minus 1 minus 2."},{"Start":"01:06.945 ","End":"01:09.760","Text":"No good, not 0, minus 1."},{"Start":"01:09.760 ","End":"01:15.290","Text":"That will give us actually minus 1,"},{"Start":"01:15.290 ","End":"01:17.850","Text":"minus 1, plus 1,"},{"Start":"01:17.850 ","End":"01:20.445","Text":"minus 2, still not 0."},{"Start":"01:20.445 ","End":"01:26.145","Text":"Try 2. 2^3 is 8 minus 4 minus 2, minus 2."},{"Start":"01:26.145 ","End":"01:27.465","Text":"Yeah, that\u0027ll work,"},{"Start":"01:27.465 ","End":"01:30.195","Text":"so 2 is a solution."},{"Start":"01:30.195 ","End":"01:32.830","Text":"We have x=2."},{"Start":"01:32.830 ","End":"01:36.815","Text":"Now we use the technique of polynomial division."},{"Start":"01:36.815 ","End":"01:41.429","Text":"I write this polynomial as x minus this,"},{"Start":"01:41.429 ","End":"01:43.625","Text":"and then I need to find some other."},{"Start":"01:43.625 ","End":"01:45.080","Text":"We\u0027ll have degree 1 less."},{"Start":"01:45.080 ","End":"01:46.129","Text":"It\u0027ll be a quadratic,"},{"Start":"01:46.129 ","End":"01:48.080","Text":"in other words, equals 0,"},{"Start":"01:48.080 ","End":"01:52.655","Text":"so I do a long division over here where we"},{"Start":"01:52.655 ","End":"01:58.655","Text":"copy here the x^3 minus x^2 minus x minus 2,"},{"Start":"01:58.655 ","End":"02:00.800","Text":"also called the dividend."},{"Start":"02:00.800 ","End":"02:05.240","Text":"The divisor will be the x minus 2."},{"Start":"02:05.240 ","End":"02:10.785","Text":"Let\u0027s see, x into x^3 is x^2 times."},{"Start":"02:10.785 ","End":"02:12.450","Text":"This one, I\u0027ll do."},{"Start":"02:12.450 ","End":"02:17.160","Text":"Don\u0027t worry, I don\u0027t always do the long division. Do a practice one."},{"Start":"02:17.160 ","End":"02:22.800","Text":"x^2 times x minus 2 is x^3 minus 2x^2."},{"Start":"02:22.800 ","End":"02:27.690","Text":"Then we subtract x^3 minus x^3, nothing."},{"Start":"02:27.690 ","End":"02:33.315","Text":"Minus x^2 minus minus 2x^2 is plus x^2."},{"Start":"02:33.315 ","End":"02:34.740","Text":"Drop another one."},{"Start":"02:34.740 ","End":"02:38.430","Text":"x into x^2 goes x times."},{"Start":"02:38.430 ","End":"02:42.505","Text":"x times x minus 2 is x^2 minus 2x."},{"Start":"02:42.505 ","End":"02:44.228","Text":"Again, subtract."},{"Start":"02:44.228 ","End":"02:45.815","Text":"This minus this is nothing."},{"Start":"02:45.815 ","End":"02:50.760","Text":"Minus x minus minus 2x is plus x minus"},{"Start":"02:50.760 ","End":"02:56.380","Text":"2. x goes into x 1,1 times x minus 2 is x minus 2."},{"Start":"02:56.380 ","End":"02:59.660","Text":"It goes in evenly, remainder of 0."},{"Start":"02:59.660 ","End":"03:02.270","Text":"This quadratic here,"},{"Start":"03:02.270 ","End":"03:08.250","Text":"the quotient is x^2 plus x plus 1."},{"Start":"03:10.130 ","End":"03:13.040","Text":"Either x minus 2 is 0,"},{"Start":"03:13.040 ","End":"03:14.600","Text":"which gives us that x equals 2,"},{"Start":"03:14.600 ","End":"03:15.905","Text":"which we knew already."},{"Start":"03:15.905 ","End":"03:19.930","Text":"But we might get something new from this one being 0."},{"Start":"03:19.930 ","End":"03:27.090","Text":"Let\u0027s see if we try x^2 plus x plus 1 equals 0 and we solve it,"},{"Start":"03:27.090 ","End":"03:28.730","Text":"let\u0027s say using the formula,"},{"Start":"03:28.730 ","End":"03:34.340","Text":"we would get x equals minus 1, minus b,"},{"Start":"03:34.340 ","End":"03:38.930","Text":"plus or minus the square root of b^2 minus 4ac,"},{"Start":"03:38.930 ","End":"03:40.970","Text":"4 times 1 times 1,"},{"Start":"03:40.970 ","End":"03:46.050","Text":"over 2a is 2."},{"Start":"03:48.800 ","End":"03:53.545","Text":"Here we have the square root of minus 3,"},{"Start":"03:53.545 ","End":"04:01.580","Text":"minus 1, plus or minus the square root of minus 3 over 2."},{"Start":"04:03.150 ","End":"04:08.845","Text":"How we proceed depends on whether or not you\u0027ve studied complex numbers."},{"Start":"04:08.845 ","End":"04:18.460","Text":"If you\u0027ve studied complex numbers like the square root of minus things and, for example,"},{"Start":"04:18.460 ","End":"04:19.960","Text":"if this makes sense to you,"},{"Start":"04:19.960 ","End":"04:22.440","Text":"the square root of minus 1 is i,"},{"Start":"04:22.440 ","End":"04:23.880","Text":"and it doesn\u0027t look like gibberish,"},{"Start":"04:23.880 ","End":"04:32.600","Text":"you\u0027ve probably studied complex number and I\u0027ll deal with that in a moment."},{"Start":"04:32.600 ","End":"04:37.490","Text":"But let\u0027s say you haven\u0027t and then you would just stop here and say, okay,"},{"Start":"04:37.490 ","End":"04:40.700","Text":"you would say no solution because there is no square root of"},{"Start":"04:40.700 ","End":"04:45.720","Text":"minus 3 and you would say no solution."},{"Start":"04:51.560 ","End":"04:57.215","Text":"Take 2. You would declare that this thing,"},{"Start":"04:57.215 ","End":"05:03.020","Text":"this equation, the quadratic has no solution."},{"Start":"05:03.020 ","End":"05:06.110","Text":"But we still already have a solution from here."},{"Start":"05:06.110 ","End":"05:13.640","Text":"You would say that the answer is just x equals 2 and highlight it and say,"},{"Start":"05:13.640 ","End":"05:16.680","Text":"yes, we\u0027re done with part a."},{"Start":"05:16.750 ","End":"05:20.720","Text":"However, you might have studied complex numbers,"},{"Start":"05:20.720 ","End":"05:24.060","Text":"in which case we have an alternative ending."},{"Start":"05:24.190 ","End":"05:30.440","Text":"Let\u0027s go back in time to when we were at this point and you\u0027ve studied"},{"Start":"05:30.440 ","End":"05:38.480","Text":"complex numbers so we can attempt to figure out the square root of minus 3."},{"Start":"05:38.480 ","End":"05:40.369","Text":"I\u0027ll just do that at the side."},{"Start":"05:40.369 ","End":"05:47.510","Text":"The square root of minus 3 is the square root of 3 times the square root of"},{"Start":"05:47.510 ","End":"05:50.960","Text":"minus 1 because this is the product of"},{"Start":"05:50.960 ","End":"05:56.900","Text":"minus 1 and 3 and this is the square root of 3 times i."},{"Start":"05:56.900 ","End":"06:01.850","Text":"Back here, what we would get is minus 1 plus or"},{"Start":"06:01.850 ","End":"06:09.360","Text":"minus the square root of 3 times i over 2."},{"Start":"06:09.400 ","End":"06:17.405","Text":"In this case, we would write our answer as three possibilities."},{"Start":"06:17.405 ","End":"06:26.075","Text":"Either x is 2 from there or if I take the plus and I\u0027ll also divide each bit,"},{"Start":"06:26.075 ","End":"06:28.985","Text":"the real and imaginary parts by 2 so we\u0027d get"},{"Start":"06:28.985 ","End":"06:36.145","Text":"minus a half plus the square root of 3 over 2i."},{"Start":"06:36.145 ","End":"06:40.205","Text":"The third solution from taking the minus would be"},{"Start":"06:40.205 ","End":"06:46.380","Text":"minus a half minus the square root of 3 over 2i."},{"Start":"06:46.600 ","End":"06:54.910","Text":"Then we\u0027d get three solutions to a cubic equation and that\u0027s the end of part a."},{"Start":"06:54.910 ","End":"06:58.360","Text":"Let\u0027s go on to part b."},{"Start":"06:59.660 ","End":"07:02.370","Text":"Using the same method as before,"},{"Start":"07:02.370 ","End":"07:07.880","Text":"we look first for integer solutions and the integer solutions for x have"},{"Start":"07:07.880 ","End":"07:14.045","Text":"to be divisors of the trailing constant, the 6 here."},{"Start":"07:14.045 ","End":"07:19.775","Text":"So a few possibilities could be plus or minus 1,"},{"Start":"07:19.775 ","End":"07:21.725","Text":"could be plus or minus 2,"},{"Start":"07:21.725 ","End":"07:23.750","Text":"could be plus or minus 3,"},{"Start":"07:23.750 ","End":"07:25.190","Text":"plus or minus 6,"},{"Start":"07:25.190 ","End":"07:29.210","Text":"8 possibilities to try."},{"Start":"07:29.210 ","End":"07:32.090","Text":"Let\u0027s try x=1."},{"Start":"07:32.090 ","End":"07:33.866","Text":"That\u0027s the easiest to substitute."},{"Start":"07:33.866 ","End":"07:38.555","Text":"You get 1 minus 1 minus 5 plus 3 plus 6."},{"Start":"07:38.555 ","End":"07:44.880","Text":"That\u0027s not good because the positives are 1 and 3 and 6,"},{"Start":"07:44.880 ","End":"07:48.180","Text":"is 10, negatives 1 and 5 is minus 6."},{"Start":"07:48.180 ","End":"07:49.275","Text":"It comes out to be 4,"},{"Start":"07:49.275 ","End":"07:51.135","Text":"not 0 anyway."},{"Start":"07:51.135 ","End":"07:57.750","Text":"1 is no."},{"Start":"07:57.750 ","End":"08:08.040","Text":"Let\u0027s try next minus 1 and we will get a plus 1."},{"Start":"08:08.040 ","End":"08:10.935","Text":"Then minus minus 1 is 2,"},{"Start":"08:10.935 ","End":"08:15.585","Text":"minus 5, so we\u0027re up to minus 3."},{"Start":"08:15.585 ","End":"08:18.270","Text":"Then another minus 3 plus 6."},{"Start":"08:18.270 ","End":"08:20.730","Text":"Yes, minus 1 worked."},{"Start":"08:20.730 ","End":"08:26.070","Text":"Yes."},{"Start":"08:26.070 ","End":"08:33.105","Text":"That means that we can divide x minus 1,"},{"Start":"08:33.105 ","End":"08:34.920","Text":"sorry, x minus minus 1,"},{"Start":"08:34.920 ","End":"08:36.990","Text":"which is x plus 1 into"},{"Start":"08:36.990 ","End":"08:44.765","Text":"this using long division and I\u0027m just going to give you the answer."},{"Start":"08:44.765 ","End":"08:55.710","Text":"It comes out x^3 minus 2x^2 minus 3x plus 6."},{"Start":"08:55.710 ","End":"08:59.035","Text":"That\u0027s also equal 0."},{"Start":"08:59.035 ","End":"09:01.010","Text":"Now this is a cubic,"},{"Start":"09:01.010 ","End":"09:04.190","Text":"so we\u0027re going to have to again guess another root."},{"Start":"09:04.190 ","End":"09:07.931","Text":"When I say, I guess, I mean educated guess according to this."},{"Start":"09:07.931 ","End":"09:11.555","Text":"We still have to find a divisor of 6."},{"Start":"09:11.555 ","End":"09:15.210","Text":"We still have the same list of possibilities."},{"Start":"09:15.500 ","End":"09:21.590","Text":"Now, 1 is not going to be a solution to this because if 1 made this 0,"},{"Start":"09:21.590 ","End":"09:25.610","Text":"it would make the product 0 and it would make this 0, so 1 is no good."},{"Start":"09:25.610 ","End":"09:28.470","Text":"We could try minus 1 again."},{"Start":"09:29.360 ","End":"09:33.390","Text":"I\u0027ve checked it, minus 1 does not do the trick."},{"Start":"09:33.390 ","End":"09:35.670","Text":"Let\u0027s go on to the next one."},{"Start":"09:35.670 ","End":"09:38.050","Text":"Lets try 2."},{"Start":"09:38.200 ","End":"09:40.999","Text":"This one turns out it does work."},{"Start":"09:40.999 ","End":"09:46.370","Text":"2^3 is 8 and here we have minus 8,"},{"Start":"09:46.370 ","End":"09:50.700","Text":"then minus 6 plus 6 of 2 is another yes."},{"Start":"09:51.530 ","End":"09:59.850","Text":"What I can do is this x plus 1 just stays on here because 2 is a solution,"},{"Start":"09:59.850 ","End":"10:03.420","Text":"x minus 2 will divide into this and again,"},{"Start":"10:03.420 ","End":"10:06.330","Text":"we\u0027ll do long division."},{"Start":"10:06.330 ","End":"10:09.640","Text":"I guess I should\u0027ve made a note that from here to here"},{"Start":"10:09.640 ","End":"10:14.365","Text":"I\u0027m using long division of polynomials."},{"Start":"10:14.365 ","End":"10:18.130","Text":"Again, long division of polynomials."},{"Start":"10:18.130 ","End":"10:22.480","Text":"I\u0027m just not wasting time by doing it."},{"Start":"10:22.480 ","End":"10:26.560","Text":"Just to emphasize what is the long division that we\u0027re doing?"},{"Start":"10:26.560 ","End":"10:29.110","Text":"In this case, we would be taking"},{"Start":"10:29.110 ","End":"10:38.400","Text":"the x^3 minus 2x^2 minus 3x"},{"Start":"10:38.400 ","End":"10:43.030","Text":"plus 6 and dividing it by this x minus 2."},{"Start":"10:43.030 ","End":"10:46.105","Text":"This is the long division I\u0027m talking about and if we do it,"},{"Start":"10:46.105 ","End":"10:52.825","Text":"the answer comes out to be x^2 minus 3."},{"Start":"10:52.825 ","End":"10:56.965","Text":"Here I write x^2 minus 3."},{"Start":"10:56.965 ","End":"10:58.660","Text":"Of course in each of these cases,"},{"Start":"10:58.660 ","End":"11:00.910","Text":"we could check by multiplying out."},{"Start":"11:00.910 ","End":"11:02.470","Text":"For example, in this one,"},{"Start":"11:02.470 ","End":"11:03.490","Text":"x times x^2 is x^3."},{"Start":"11:03.490 ","End":"11:08.020","Text":"Minus 2x^2, minus 3x."},{"Start":"11:08.020 ","End":"11:09.820","Text":"Minus 3 minus 3 is plus 6."},{"Start":"11:09.820 ","End":"11:12.985","Text":"Now we are aware this is correct"},{"Start":"11:12.985 ","End":"11:20.885","Text":"and then we solve this using quadratic equations."},{"Start":"11:20.885 ","End":"11:27.243","Text":"Actually, none of the rest of these whole number solutions fit."},{"Start":"11:27.243 ","End":"11:36.350","Text":"What we can see is that x^2 minus 3 equals 0 will give us,"},{"Start":"11:36.350 ","End":"11:37.675","Text":"if we solve it,"},{"Start":"11:37.675 ","End":"11:39.760","Text":"x^2 equals 3,"},{"Start":"11:39.760 ","End":"11:45.215","Text":"not using the formula because it\u0027s like a special quadratic with missing x term."},{"Start":"11:45.215 ","End":"11:48.890","Text":"x is just plus or minus 3."},{"Start":"11:48.890 ","End":"11:52.205","Text":"Now we already had two of the solutions."},{"Start":"11:52.205 ","End":"12:00.860","Text":"I should have emphasized them that we got this from saying that x"},{"Start":"12:00.860 ","End":"12:05.970","Text":"equals minus 1 was a solution and then we got"},{"Start":"12:05.970 ","End":"12:13.230","Text":"this by saying that x equals 2 is a solution."},{"Start":"12:13.230 ","End":"12:16.800","Text":"Now we\u0027ve got plus or minus, did I say 3?"},{"Start":"12:16.800 ","End":"12:18.450","Text":"I meant square root of 3."},{"Start":"12:18.450 ","End":"12:21.250","Text":"Sorry. Of course."},{"Start":"12:22.280 ","End":"12:29.045","Text":"We gather this bit, this bit,"},{"Start":"12:29.045 ","End":"12:37.335","Text":"and these two, and together we get that x is equal to minus 1, maybe underline it."},{"Start":"12:37.335 ","End":"12:42.800","Text":"Here we had the possibility of 2 and"},{"Start":"12:42.800 ","End":"12:48.950","Text":"here we have also square root of 3 and minus the square root of 3."},{"Start":"12:48.950 ","End":"12:51.710","Text":"It\u0027s a quartic equation of degree 4."},{"Start":"12:51.710 ","End":"12:53.960","Text":"So we can expect four solutions,"},{"Start":"12:53.960 ","End":"12:59.605","Text":"and these are the solutions."},{"Start":"12:59.605 ","End":"13:02.790","Text":"We are done."}],"ID":6362},{"Watched":false,"Name":"Exercise 7","Duration":"15m 43s","ChapterTopicVideoID":6346,"CourseChapterTopicPlaylistID":56153,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6346.jpeg","UploadDate":"2016-06-22T08:52:03.8300000","DurationForVideoObject":"PT15M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"Here we have a pair of exercises."},{"Start":"00:02.550 ","End":"00:08.700","Text":"In each case we have a polynomial and we have 2 listed zeros,"},{"Start":"00:08.700 ","End":"00:13.170","Text":"including the multiplicity of the 0."},{"Start":"00:13.170 ","End":"00:17.070","Text":"The 0 appears more than once and it has a multiplicity of 2,"},{"Start":"00:17.070 ","End":"00:19.800","Text":"or 3, or so on."},{"Start":"00:19.800 ","End":"00:21.915","Text":"We\u0027ll start with the first 1,"},{"Start":"00:21.915 ","End":"00:25.050","Text":"which is a degree 4 polynomial."},{"Start":"00:25.050 ","End":"00:31.620","Text":"To find it\u0027s zeros means to solve the equation that the polynomial is equal to 0."},{"Start":"00:31.620 ","End":"00:39.375","Text":"We do it by this semi guesswork method where we use a theorem that,"},{"Start":"00:39.375 ","End":"00:41.810","Text":"if this is an integer polynomial,"},{"Start":"00:41.810 ","End":"00:45.140","Text":"which it is and the leading coefficient is 1,"},{"Start":"00:45.140 ","End":"00:53.230","Text":"then any integer roots have to divide the constant term, the minus 6."},{"Start":"00:53.230 ","End":"01:03.300","Text":"The only place to look for integer x would be plus or minus 1,"},{"Start":"01:03.300 ","End":"01:05.335","Text":"plus or minus 2,"},{"Start":"01:05.335 ","End":"01:07.130","Text":"plus or minus 3,"},{"Start":"01:07.130 ","End":"01:08.630","Text":"and plus or minus 6."},{"Start":"01:08.630 ","End":"01:13.970","Text":"These are all the divisors that go in an integer number of times into minus 6."},{"Start":"01:13.970 ","End":"01:16.265","Text":"Has already 8 possibilities,"},{"Start":"01:16.265 ","End":"01:17.975","Text":"which is quite a lot."},{"Start":"01:17.975 ","End":"01:22.110","Text":"Let\u0027s try the first one, x=1."},{"Start":"01:22.110 ","End":"01:25.410","Text":"Substitute 1 plus 3, plus 3,"},{"Start":"01:25.410 ","End":"01:29.040","Text":"minus 1, minus 6, it comes out 0."},{"Start":"01:29.040 ","End":"01:31.725","Text":"There is a 1 minus 1 and there is a 3 plus 3 minus 6,"},{"Start":"01:31.725 ","End":"01:32.940","Text":"it comes out 0,"},{"Start":"01:32.940 ","End":"01:39.310","Text":"so we know that x=1 is 1 of the solutions."},{"Start":"01:39.710 ","End":"01:42.440","Text":"If x equals 1 is a solution,"},{"Start":"01:42.440 ","End":"01:50.495","Text":"that means that we can divide this polynomial evenly by x minus 1. Let\u0027s do that."},{"Start":"01:50.495 ","End":"01:56.735","Text":"Write it as x minus 1 times something which will be only cubic,"},{"Start":"01:56.735 ","End":"02:00.045","Text":"not quartic, degree 3 instead of 4."},{"Start":"02:00.045 ","End":"02:02.330","Text":"We do that with long division,"},{"Start":"02:02.330 ","End":"02:05.630","Text":"but I\u0027m going to save the long division and just tell you the answer of"},{"Start":"02:05.630 ","End":"02:09.445","Text":"what happens if we get this divided by this,"},{"Start":"02:09.445 ","End":"02:18.140","Text":"it comes out to x cubed minus 4x^2 plus,"},{"Start":"02:18.140 ","End":"02:20.855","Text":"oops, there should be a plus, I\u0027m sorry,"},{"Start":"02:20.855 ","End":"02:26.340","Text":"plus 7x plus 6."},{"Start":"02:26.930 ","End":"02:30.470","Text":"We have to solve either x minus 1 is 0,"},{"Start":"02:30.470 ","End":"02:35.015","Text":"which we already know just gives us x equals 1, or this thing is 0."},{"Start":"02:35.015 ","End":"02:40.600","Text":"Now a cubic we don\u0027t know how to solve algebraically except by the guessing method."},{"Start":"02:40.600 ","End":"02:46.690","Text":"Once again, the 6 at the end means that a whole number solution has to be 1 of these,"},{"Start":"02:46.690 ","End":"02:48.430","Text":"and if we keep trying,"},{"Start":"02:48.430 ","End":"02:51.000","Text":"I won\u0027t waste time trying each."},{"Start":"02:51.000 ","End":"02:56.940","Text":"I\u0027ll let you know that the next one that works is minus 2."},{"Start":"02:56.940 ","End":"03:01.110","Text":"One doesn\u0027t work here and minus 1 doesn\u0027t work."},{"Start":"03:01.110 ","End":"03:05.105","Text":"Minus 2 works because then we get minus 8."},{"Start":"03:05.105 ","End":"03:09.895","Text":"Let\u0027s do the negatives. Minus 8 and minus 14 is minus 22."},{"Start":"03:09.895 ","End":"03:12.910","Text":"For the positives, we get 4 times 4 is 16 plus 6,"},{"Start":"03:12.910 ","End":"03:14.785","Text":"also 22, 22 minus 22."},{"Start":"03:14.785 ","End":"03:17.415","Text":"In other words, minus 2 works."},{"Start":"03:17.415 ","End":"03:21.710","Text":"Which means that we can divide this by x minus minus 2."},{"Start":"03:21.710 ","End":"03:25.370","Text":"In other words, we can take x plus 2 out as a factor."},{"Start":"03:25.370 ","End":"03:30.815","Text":"Again, if you do a long division of this by this and I won\u0027t do it to save time,"},{"Start":"03:30.815 ","End":"03:33.625","Text":"then we are left with,"},{"Start":"03:33.625 ","End":"03:35.900","Text":"I calculated this somewhere."},{"Start":"03:35.900 ","End":"03:41.675","Text":"It comes out to x^2 plus 2x plus 3."},{"Start":"03:41.675 ","End":"03:45.530","Text":"Now, we want this to be 0."},{"Start":"03:45.530 ","End":"03:52.260","Text":"To find the solutions of the equation is the same as the zeros of the polynomial."},{"Start":"03:53.170 ","End":"03:56.750","Text":"To look for more roots besides 1 and minus 2,"},{"Start":"03:56.750 ","End":"04:00.950","Text":"we need to set this quadratic polynomial to be 0."},{"Start":"04:00.950 ","End":"04:03.080","Text":"In other words, we get the quadratic equation,"},{"Start":"04:03.080 ","End":"04:08.070","Text":"x^2 plus 2x plus 3 equals 0."},{"Start":"04:08.120 ","End":"04:14.143","Text":"What we get is that if you use the formula that x equals"},{"Start":"04:14.143 ","End":"04:20.590","Text":"minus 2 plus or minus the square root of 2^2 minus 4 times 1 times 3,"},{"Start":"04:20.590 ","End":"04:22.305","Text":"4 times 3,"},{"Start":"04:22.305 ","End":"04:25.090","Text":"all this over 2."},{"Start":"04:25.340 ","End":"04:28.260","Text":"If I just look at this square root part,"},{"Start":"04:28.260 ","End":"04:32.555","Text":"what I have under the square root sign is 4 minus 12,"},{"Start":"04:32.555 ","End":"04:35.370","Text":"which is minus 8."},{"Start":"04:36.470 ","End":"04:41.375","Text":"I want to say that if you\u0027ve studied complex numbers,"},{"Start":"04:41.375 ","End":"04:46.295","Text":"then I\u0027ll give a complex solution at the end."},{"Start":"04:46.295 ","End":"04:48.440","Text":"But if you haven\u0027t, at this point,"},{"Start":"04:48.440 ","End":"04:54.020","Text":"we say that this quadratic has no solution."},{"Start":"04:54.020 ","End":"04:59.855","Text":"If the quadratic has no solution,"},{"Start":"04:59.855 ","End":"05:03.605","Text":"then the only 2 solutions are from here."},{"Start":"05:03.605 ","End":"05:06.740","Text":"Just to put it in the framework of the question,"},{"Start":"05:06.740 ","End":"05:09.600","Text":"to list it with the multiplicities."},{"Start":"05:09.940 ","End":"05:13.860","Text":"Maybe draw it as a table."},{"Start":"05:16.720 ","End":"05:20.225","Text":"This column will be the 0,"},{"Start":"05:20.225 ","End":"05:25.350","Text":"and in this column I\u0027ll write the word multiplicity."},{"Start":"05:26.650 ","End":"05:31.230","Text":"The zeros are 1 and minus 2,"},{"Start":"05:31.230 ","End":"05:32.970","Text":"but each of them occurs only once,"},{"Start":"05:32.970 ","End":"05:34.500","Text":"so the multiplicity is 1,"},{"Start":"05:34.500 ","End":"05:39.430","Text":"which is actually not very interesting, but there it is."},{"Start":"05:40.310 ","End":"05:45.050","Text":"I would say that we\u0027re done except that if you have"},{"Start":"05:45.050 ","End":"05:51.290","Text":"studied complex numbers, then I\u0027ll continue."},{"Start":"05:51.290 ","End":"05:55.265","Text":"I\u0027m assuming now that you\u0027ve studied complex numbers and I\u0027ll continue,"},{"Start":"05:55.265 ","End":"05:56.690","Text":"I\u0027ll use a different color."},{"Start":"05:56.690 ","End":"06:04.300","Text":"This is equal to the square root of 8 times the square root of minus 1,"},{"Start":"06:04.300 ","End":"06:11.205","Text":"which is the square root of 8 times i."},{"Start":"06:11.205 ","End":"06:15.875","Text":"Actually, the square root of 8 can be simplified."},{"Start":"06:15.875 ","End":"06:20.120","Text":"In general, if you see a square root and there\u0027s a square number that goes into it,"},{"Start":"06:20.120 ","End":"06:23.170","Text":"like 4 is perfect square that goes into 8,"},{"Start":"06:23.170 ","End":"06:25.760","Text":"then we can break it up further and say,"},{"Start":"06:25.760 ","End":"06:27.985","Text":"this would be the square root of 4,"},{"Start":"06:27.985 ","End":"06:32.625","Text":"the square root of 2 times i,"},{"Start":"06:32.625 ","End":"06:34.785","Text":"and the square root of 4 is 2,"},{"Start":"06:34.785 ","End":"06:38.790","Text":"so it\u0027s 2 square root of 2i."},{"Start":"06:38.790 ","End":"06:42.850","Text":"Now I go and plug it in here."},{"Start":"06:43.550 ","End":"06:47.999","Text":"I get that x is equal to"},{"Start":"06:47.999 ","End":"06:56.835","Text":"minus 2 plus or minus 2 square root of 2i over 2."},{"Start":"06:56.835 ","End":"07:03.155","Text":"I can divide top and bottom by 2 and get minus 1,"},{"Start":"07:03.155 ","End":"07:09.930","Text":"plus or minus the square root of 2 times i."},{"Start":"07:09.980 ","End":"07:12.920","Text":"Now I\u0027ve got 2 more solutions,"},{"Start":"07:12.920 ","End":"07:18.500","Text":"although they are complex numbers and add them to the table of zeros and multiplicities."},{"Start":"07:18.500 ","End":"07:21.890","Text":"I have minus 1 plus square root of 2i,"},{"Start":"07:21.890 ","End":"07:25.565","Text":"and minus 1 minus square root of 2i."},{"Start":"07:25.565 ","End":"07:27.860","Text":"Each of them occurs only once."},{"Start":"07:27.860 ","End":"07:29.750","Text":"There\u0027s no repeated root,"},{"Start":"07:29.750 ","End":"07:33.890","Text":"there\u0027s no multiplicities, which means that the multiplicity is 1."},{"Start":"07:33.890 ","End":"07:36.560","Text":"Like I said, that\u0027s the less interesting case."},{"Start":"07:36.560 ","End":"07:39.560","Text":"I think in Part b we might get something more interesting."},{"Start":"07:39.560 ","End":"07:41.360","Text":"Anyway, I\u0027m done with Part a,"},{"Start":"07:41.360 ","End":"07:45.635","Text":"and now let\u0027s do Part b."},{"Start":"07:45.635 ","End":"07:49.250","Text":"To find the zeros of this polynomial,"},{"Start":"07:49.250 ","End":"07:53.945","Text":"we want to solve the equation where this equals 0."},{"Start":"07:53.945 ","End":"07:59.555","Text":"The zeros of the polynomial is the same as the solutions of the equation."},{"Start":"07:59.555 ","End":"08:07.580","Text":"We\u0027re going to use the usual semi guess trial and error method where we say that"},{"Start":"08:07.580 ","End":"08:15.535","Text":"the integer roots have to be divisors of 4, or minus 4."},{"Start":"08:15.535 ","End":"08:18.574","Text":"In other words, we have to look for x."},{"Start":"08:18.574 ","End":"08:24.355","Text":"If it\u0027s going to be an integer amongst the numbers plus or minus 1,"},{"Start":"08:24.355 ","End":"08:26.565","Text":"plus or minus 2,"},{"Start":"08:26.565 ","End":"08:30.841","Text":"plus or minus 4, 6 possibilities."},{"Start":"08:30.841 ","End":"08:34.510","Text":"Let\u0027s try the first one, x=1,"},{"Start":"08:34.510 ","End":"08:40.330","Text":"and then we get 1 minus 5 is minus 4."},{"Start":"08:40.330 ","End":"08:44.710","Text":"Then plus and minus 9 doesn\u0027t change anything."},{"Start":"08:44.710 ","End":"08:47.110","Text":"Minus 4 plus 8 minus 4."},{"Start":"08:47.110 ","End":"08:49.555","Text":"We take minus 4 minus 4 it is 0."},{"Start":"08:49.555 ","End":"08:57.820","Text":"In other words, x=1 is one of the zeros already."},{"Start":"08:57.820 ","End":"08:59.290","Text":"If that\u0027s the case,"},{"Start":"08:59.290 ","End":"09:04.405","Text":"then we know that this thing divides by x minus 1."},{"Start":"09:04.405 ","End":"09:13.680","Text":"We can say that x minus 1 times something is equal to 0."},{"Start":"09:13.680 ","End":"09:16.845","Text":"This something is what you get when you divide"},{"Start":"09:16.845 ","End":"09:22.600","Text":"this polynomial by x minus 1 using long division."},{"Start":"09:22.600 ","End":"09:25.510","Text":"I\u0027m not going to spend the time doing it,"},{"Start":"09:25.510 ","End":"09:27.070","Text":"you know long division."},{"Start":"09:27.070 ","End":"09:29.545","Text":"I\u0027m just going to quote the answer,"},{"Start":"09:29.545 ","End":"09:32.034","Text":"it is"},{"Start":"09:32.034 ","End":"09:45.460","Text":"x^4-4x^3+5x^2-4x+4."},{"Start":"09:45.460 ","End":"09:51.280","Text":"Now we still have a degree for a quartic equation."},{"Start":"09:51.280 ","End":"09:56.020","Text":"We need to find another route."},{"Start":"09:56.020 ","End":"10:02.094","Text":"Again, we have 4 at the end of the plus or minus so would have to be one of these."},{"Start":"10:02.094 ","End":"10:04.000","Text":"We try 1 again,"},{"Start":"10:04.000 ","End":"10:11.300","Text":"we find that it doesn\u0027t work x=1 does not provide 0."},{"Start":"10:11.940 ","End":"10:14.800","Text":"In fact, if you try minus 1,"},{"Start":"10:14.800 ","End":"10:16.330","Text":"it also doesn\u0027t give 0."},{"Start":"10:16.330 ","End":"10:20.230","Text":"I\u0027ve done this already at the side somewhere."},{"Start":"10:20.230 ","End":"10:24.100","Text":"But x=2 is where we get lucky."},{"Start":"10:24.100 ","End":"10:25.870","Text":"If we put x=2,"},{"Start":"10:25.870 ","End":"10:31.240","Text":"we\u0027ve got 16 minus 4 times 8 is 32 minus"},{"Start":"10:31.240 ","End":"10:37.780","Text":"16 plus 20 is plus 4,"},{"Start":"10:37.780 ","End":"10:41.575","Text":"and then minus 8 plus 4, its zero."},{"Start":"10:41.575 ","End":"10:44.105","Text":"We also have x=2."},{"Start":"10:44.105 ","End":"10:47.470","Text":"When x=2 is a solution,"},{"Start":"10:47.470 ","End":"10:51.490","Text":"that means we can take out x minus 2 from this."},{"Start":"10:51.490 ","End":"10:55.960","Text":"Now we get x minus 1, x minus 2."},{"Start":"10:55.960 ","End":"10:58.705","Text":"If we do a long division of this,"},{"Start":"10:58.705 ","End":"11:02.770","Text":"by this, I\u0027ll just quote the answer,"},{"Start":"11:02.770 ","End":"11:13.100","Text":"we get x^3 minus 2x^2 plus x minus 2."},{"Start":"11:14.640 ","End":"11:19.885","Text":"We\u0027ve gone down from quintic to a quartic to a cubic,"},{"Start":"11:19.885 ","End":"11:21.970","Text":"from 5 to 4 to 3."},{"Start":"11:21.970 ","End":"11:29.455","Text":"We still need another root because we don\u0027t know how to solve cubics in general."},{"Start":"11:29.455 ","End":"11:37.960","Text":"This time it can\u0027t be the plus or minus 4 anymore,"},{"Start":"11:37.960 ","End":"11:41.200","Text":"it has to be plus or minus 1,"},{"Start":"11:41.200 ","End":"11:43.940","Text":"plus or minus 2."},{"Start":"11:45.120 ","End":"11:49.390","Text":"It\u0027s not going to be plus or minus 1 because then we would"},{"Start":"11:49.390 ","End":"11:53.485","Text":"have found the extra plus or minus 1 here."},{"Start":"11:53.485 ","End":"11:59.590","Text":"Let me just tell you that it turns out that 2 is a solution."},{"Start":"11:59.590 ","End":"12:03.295","Text":"Just mentally check that 2^3 is 8,"},{"Start":"12:03.295 ","End":"12:05.560","Text":"and 2 times 4 is 8, so that\u0027s 0."},{"Start":"12:05.560 ","End":"12:07.960","Text":"Another 2 minus 2 is also 0."},{"Start":"12:07.960 ","End":"12:12.925","Text":"So we have another 2 and that means we can do another division"},{"Start":"12:12.925 ","End":"12:19.180","Text":"and get x minus 1, x minus 2."},{"Start":"12:19.180 ","End":"12:28.165","Text":"Again, x minus 2 and if you divide this by x minus 2,"},{"Start":"12:28.165 ","End":"12:33.260","Text":"what you get is x^2 plus 1."},{"Start":"12:33.720 ","End":"12:37.120","Text":"I\u0027ll just verify that last one."},{"Start":"12:37.120 ","End":"12:41.095","Text":"X times x^2 is x^3 here we have minus 2x^2."},{"Start":"12:41.095 ","End":"12:43.210","Text":"Here we have plus x and this,"},{"Start":"12:43.210 ","End":"12:45.565","Text":"and this gives minus 2."},{"Start":"12:45.565 ","End":"12:48.820","Text":"So we already have the 1,"},{"Start":"12:48.820 ","End":"12:50.575","Text":"2, and 2 from here."},{"Start":"12:50.575 ","End":"12:54.490","Text":"The only other possibility for roots is from the quadratic."},{"Start":"12:54.490 ","End":"12:59.420","Text":"So we can try x^2 plus 1 equals 0."},{"Start":"13:00.950 ","End":"13:06.674","Text":"How I continue depends on whether or not you\u0027ve studied complex numbers."},{"Start":"13:06.674 ","End":"13:10.339","Text":"If you haven\u0027t studied complex numbers,"},{"Start":"13:10.339 ","End":"13:17.500","Text":"we would say that there is no solution to this because it gives us that x^2 is"},{"Start":"13:17.500 ","End":"13:25.090","Text":"minus 1 and there is no real number as opposed to complex."},{"Start":"13:25.090 ","End":"13:33.205","Text":"The numbers that you know that squared give minus 1 so we say here, no solution."},{"Start":"13:33.205 ","End":"13:35.650","Text":"All we\u0027re left with is 1,"},{"Start":"13:35.650 ","End":"13:37.735","Text":"2, and 2."},{"Start":"13:37.735 ","End":"13:43.675","Text":"The question asked to list each one with a multiplicity so I\u0027ll make a table."},{"Start":"13:43.675 ","End":"13:47.480","Text":"I just copied the table from Part A."},{"Start":"13:48.120 ","End":"13:51.430","Text":"If we\u0027re talking about zeros and multiplicities,"},{"Start":"13:51.430 ","End":"13:54.610","Text":"there really are only two different zeros."},{"Start":"13:54.610 ","End":"13:57.805","Text":"There\u0027s 1 and there\u0027s 2."},{"Start":"13:57.805 ","End":"14:00.535","Text":"But 1 appears only once,"},{"Start":"14:00.535 ","End":"14:07.660","Text":"and 2 appears twice so 2 has a multiplicity of 2 that makes it interesting,"},{"Start":"14:07.660 ","End":"14:14.020","Text":"and we would stop here."},{"Start":"14:14.020 ","End":"14:16.180","Text":"In fact, we are done."},{"Start":"14:16.180 ","End":"14:20.959","Text":"Unless you\u0027ve studied complex numbers,"},{"Start":"14:23.910 ","End":"14:28.360","Text":"you\u0027re still here, which means that you\u0027ve studied complex numbers."},{"Start":"14:28.360 ","End":"14:32.350","Text":"I\u0027m going to erase this no solution and continue,"},{"Start":"14:32.350 ","End":"14:34.945","Text":"x^2 is minus 1."},{"Start":"14:34.945 ","End":"14:40.765","Text":"So x has to be plus or minus the square root of minus 1."},{"Start":"14:40.765 ","End":"14:45.070","Text":"The square root of minus 1 is i."},{"Start":"14:45.070 ","End":"14:48.655","Text":"So we have plus or minus i."},{"Start":"14:48.655 ","End":"14:52.675","Text":"That gives us 2 extra solutions to put into the table."},{"Start":"14:52.675 ","End":"14:59.620","Text":"That gives us i and minus i."},{"Start":"14:59.620 ","End":"15:02.410","Text":"If I didn\u0027t care about multiplicities."},{"Start":"15:02.410 ","End":"15:03.520","Text":"I would just say, okay,"},{"Start":"15:03.520 ","End":"15:05.485","Text":"we have 5 roots, 1, 2,"},{"Start":"15:05.485 ","End":"15:07.900","Text":"2, i and minus i,"},{"Start":"15:07.900 ","End":"15:10.540","Text":"but we\u0027re asked to organize them this way."},{"Start":"15:10.540 ","End":"15:14.380","Text":"This way there\u0027s only 4 different zeros because i"},{"Start":"15:14.380 ","End":"15:18.520","Text":"and minus i occur only once each of them multiplicity of 1."},{"Start":"15:18.520 ","End":"15:24.320","Text":"This is the only interesting line."},{"Start":"15:25.140 ","End":"15:29.050","Text":"Depends what you mean by interesting anyway, this is the table."},{"Start":"15:29.050 ","End":"15:33.625","Text":"There\u0027s 4 different zeros with their multiplicities."},{"Start":"15:33.625 ","End":"15:36.970","Text":"But if you don\u0027t insist on different than there are 5 zeros,"},{"Start":"15:36.970 ","End":"15:39.295","Text":"which is the most you can get for a Degree 5."},{"Start":"15:39.295 ","End":"15:42.980","Text":"Then anyway, we\u0027re done here."}],"ID":6363}],"Thumbnail":null,"ID":56153},{"Name":"Partial Fractions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"5m 39s","ChapterTopicVideoID":5283,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5283.jpeg","UploadDate":"2020-09-30T13:24:13.3070000","DurationForVideoObject":"PT5M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this clip, I\u0027m going to be talking about"},{"Start":"00:02.730 ","End":"00:08.835","Text":"partial fraction decomposition and that sounds very fancy and mysterious."},{"Start":"00:08.835 ","End":"00:14.145","Text":"Let me start with an introduction and an example."},{"Start":"00:14.145 ","End":"00:17.235","Text":"Here\u0027s a simple exercise."},{"Start":"00:17.235 ","End":"00:23.984","Text":"4 over x minus 1 plus 10 over"},{"Start":"00:23.984 ","End":"00:31.460","Text":"x minus 11 and let\u0027s see if we can compute this as something over something."},{"Start":"00:31.460 ","End":"00:34.640","Text":"What we would normally do is take a common denominator,"},{"Start":"00:34.640 ","End":"00:38.480","Text":"just like with numerical fractions and in this case,"},{"Start":"00:38.480 ","End":"00:40.730","Text":"as a common denominator,"},{"Start":"00:40.730 ","End":"00:42.910","Text":"we could take just the product,"},{"Start":"00:42.910 ","End":"00:48.360","Text":"so x minus 1 times x minus 11."},{"Start":"00:48.360 ","End":"00:50.630","Text":"This 4 over x minus 1,"},{"Start":"00:50.630 ","End":"00:54.845","Text":"I have to multiply top and bottom by the missing factor x minus 11,"},{"Start":"00:54.845 ","End":"01:01.535","Text":"so here I get 4 times x minus 11 and similarly on the other one,"},{"Start":"01:01.535 ","End":"01:07.250","Text":"I get 10 times x minus 1 because I\u0027m multiplying top and bottom by x minus"},{"Start":"01:07.250 ","End":"01:14.975","Text":"1 and then we just open brackets and start simplifying."},{"Start":"01:14.975 ","End":"01:26.165","Text":"On the top we have 4x minus 44 plus 10x minus 10."},{"Start":"01:26.165 ","End":"01:30.862","Text":"I just copied the denominator here, hasn\u0027t changed."},{"Start":"01:30.862 ","End":"01:33.480","Text":"Now let\u0027s collect like terms on the numerator,"},{"Start":"01:33.480 ","End":"01:40.980","Text":"so it\u0027s 4x plus 10x is 14x minus 44 minus 10 minus 54."},{"Start":"01:40.980 ","End":"01:47.555","Text":"On the denominator, I can write the product of these 2x times x is x^2,"},{"Start":"01:47.555 ","End":"01:50.675","Text":"minus x minus 11x is minus 12x,"},{"Start":"01:50.675 ","End":"01:54.530","Text":"minus 1 times minus 11 is plus 11."},{"Start":"01:54.530 ","End":"01:56.855","Text":"This is a straightforward exercise."},{"Start":"01:56.855 ","End":"01:59.495","Text":"Now what does this got to do with partial fractions?"},{"Start":"01:59.495 ","End":"02:03.905","Text":"Well, one of the questions partial fractions can help you answer is,"},{"Start":"02:03.905 ","End":"02:05.270","Text":"if I had the opposite,"},{"Start":"02:05.270 ","End":"02:09.500","Text":"suppose I started with this and I gave you the following exercise,"},{"Start":"02:09.500 ","End":"02:20.170","Text":"14x minus 54 over x^2 minus 12x plus 11,"},{"Start":"02:20.170 ","End":"02:24.005","Text":"and I ask you to write it somehow in simpler terms,"},{"Start":"02:24.005 ","End":"02:26.900","Text":"something that will look in the end like this."},{"Start":"02:26.900 ","End":"02:30.920","Text":"Partial fraction decomposition will take an expression like this"},{"Start":"02:30.920 ","End":"02:35.305","Text":"and decompose it into the sum of simpler forms."},{"Start":"02:35.305 ","End":"02:37.280","Text":"I\u0027m going to do this exercise,"},{"Start":"02:37.280 ","End":"02:42.530","Text":"but a bit later I just want to introduce a little bit of theory next."},{"Start":"02:42.530 ","End":"02:47.795","Text":"I\u0027d like to take something like this, but more general."},{"Start":"02:47.795 ","End":"02:56.810","Text":"Suppose I had mx plus n"},{"Start":"02:56.810 ","End":"03:03.050","Text":"over x^2 plus bx"},{"Start":"03:03.050 ","End":"03:10.355","Text":"plus c. This will be one of the basic cases that we\u0027ll be working on in decomposing."},{"Start":"03:10.355 ","End":"03:13.250","Text":"In general we\u0027ll be decomposing more complex stuff,"},{"Start":"03:13.250 ","End":"03:16.340","Text":"pretty much any polynomial over any other polynomial,"},{"Start":"03:16.340 ","End":"03:18.035","Text":"what\u0027s called a rational function."},{"Start":"03:18.035 ","End":"03:23.225","Text":"But this particular form occurs so often we\u0027re going to deal with it first."},{"Start":"03:23.225 ","End":"03:26.613","Text":"In fact, it\u0027s so important I think I\u0027ll change it to red."},{"Start":"03:26.613 ","End":"03:30.635","Text":"We\u0027re going to learn how to decompose it into partial fractions,"},{"Start":"03:30.635 ","End":"03:34.130","Text":"which I\u0027ll just call something simpler."},{"Start":"03:34.130 ","End":"03:40.025","Text":"For example, if I start off with this,"},{"Start":"03:40.025 ","End":"03:43.984","Text":"then I hope that at the end I will get this."},{"Start":"03:43.984 ","End":"03:51.310","Text":"Notice that the general form is a linear expression over a quadratic expression."},{"Start":"03:51.310 ","End":"03:53.800","Text":"Any linear expression over any quadratic."},{"Start":"03:53.800 ","End":"03:57.050","Text":"Well, you might say a quadratic expression has an a here"},{"Start":"03:57.050 ","End":"04:01.460","Text":"also and in a sense you\u0027d be right,"},{"Start":"04:01.460 ","End":"04:02.810","Text":"but for it to be a quadratic,"},{"Start":"04:02.810 ","End":"04:07.880","Text":"a would have to be non-zero and then we could divide both top and bottom by a,"},{"Start":"04:07.880 ","End":"04:12.815","Text":"so I can always assume it\u0027s a 1 here and I\u0027ll just erase that a."},{"Start":"04:12.815 ","End":"04:15.680","Text":"It turns out that we have to subdivide into"},{"Start":"04:15.680 ","End":"04:21.300","Text":"3 cases depending on what kind of quadratic this is."},{"Start":"04:22.180 ","End":"04:26.340","Text":"What we\u0027re going to do is distinguish,"},{"Start":"04:27.080 ","End":"04:31.410","Text":"as I say 3 cases, case 1,"},{"Start":"04:31.410 ","End":"04:33.855","Text":"and I\u0027ll say what it is in a moment,"},{"Start":"04:33.855 ","End":"04:41.205","Text":"and case 2 and case 3."},{"Start":"04:41.205 ","End":"04:47.584","Text":"If I take this quadratic polynomial and then assign it to 0,"},{"Start":"04:47.584 ","End":"04:50.195","Text":"I get a quadratic equation."},{"Start":"04:50.195 ","End":"04:53.060","Text":"We know that such an equation can have"},{"Start":"04:53.060 ","End":"04:58.925","Text":"either 2 solutions or just 1 solution or no solutions."},{"Start":"04:58.925 ","End":"05:06.030","Text":"In case 1 this equation has 2 different solutions."},{"Start":"05:07.630 ","End":"05:10.220","Text":"Did a bit of a copy paste here,"},{"Start":"05:10.220 ","End":"05:13.139","Text":"so let me just erase some stuff."},{"Start":"05:13.480 ","End":"05:18.890","Text":"Case 2 is when the quadratic equation only has"},{"Start":"05:18.890 ","End":"05:27.665","Text":"1 solution and the last case is when the quadratic equation has no solutions,"},{"Start":"05:27.665 ","End":"05:31.194","Text":"zero solutions or no solutions."},{"Start":"05:31.194 ","End":"05:37.430","Text":"That gives us our 3 cases and I\u0027ll start each one on a clean page."},{"Start":"05:37.430 ","End":"05:39.900","Text":"Let\u0027s start with case 1."}],"ID":5264},{"Watched":false,"Name":"Basic Case I","Duration":"7m 52s","ChapterTopicVideoID":5284,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5284.jpeg","UploadDate":"2020-09-30T13:57:00.7730000","DurationForVideoObject":"PT7M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"Continuing with partial fraction decomposition,"},{"Start":"00:03.630 ","End":"00:06.225","Text":"here we are with Case 1."},{"Start":"00:06.225 ","End":"00:11.220","Text":"This is our basic form of polynomial over polynomial,"},{"Start":"00:11.220 ","End":"00:12.930","Text":"linear over a quadratic."},{"Start":"00:12.930 ","End":"00:15.330","Text":"In Case 1, the denominator here,"},{"Start":"00:15.330 ","End":"00:20.705","Text":"the quadratic has 2 solutions to the equation."},{"Start":"00:20.705 ","End":"00:31.010","Text":"Let\u0027s call those 2 solutions x_1 and x_2."},{"Start":"00:31.010 ","End":"00:33.750","Text":"The theory is then,"},{"Start":"00:33.750 ","End":"00:35.700","Text":"and I just copied this over here,"},{"Start":"00:35.700 ","End":"00:42.005","Text":"is that we can decompose this into the general form of some number,"},{"Start":"00:42.005 ","End":"00:47.935","Text":"call it a for now over x minus x_1 that\u0027s one of the roots."},{"Start":"00:47.935 ","End":"00:53.765","Text":"B another number over x minus the other roots."},{"Start":"00:53.765 ","End":"01:00.455","Text":"That\u0027s the general form of the solution and then we use algebra to find what A and B are."},{"Start":"01:00.455 ","End":"01:04.845","Text":"This is best illustrated with an example."},{"Start":"01:04.845 ","End":"01:07.935","Text":"I\u0027ll write example,"},{"Start":"01:07.935 ","End":"01:09.635","Text":"and as an example,"},{"Start":"01:09.635 ","End":"01:13.129","Text":"I\u0027ll take the expression from before."},{"Start":"01:13.129 ","End":"01:16.820","Text":"Let\u0027s first check that we are indeed in Case 1,"},{"Start":"01:16.820 ","End":"01:20.870","Text":"so we solve the equation and I\u0027ll do it at the side here."},{"Start":"01:20.870 ","End":"01:27.170","Text":"X squared minus 12x plus 11 equals 0."},{"Start":"01:27.170 ","End":"01:30.785","Text":"I\u0027m not going to waste time solving quadratic equations."},{"Start":"01:30.785 ","End":"01:34.789","Text":"I\u0027ll just assume you know how to do that and I\u0027ll write down the solutions."},{"Start":"01:34.789 ","End":"01:36.395","Text":"There are indeed 2 of them,"},{"Start":"01:36.395 ","End":"01:38.000","Text":"the first one,"},{"Start":"01:38.000 ","End":"01:43.135","Text":"1 and the second 11."},{"Start":"01:43.135 ","End":"01:46.805","Text":"This being so we are indeed in Case 1,"},{"Start":"01:46.805 ","End":"01:56.750","Text":"and so we are guaranteed that this will indeed have a partial fraction decomposition of"},{"Start":"01:56.750 ","End":"02:00.995","Text":"the form A over x minus"},{"Start":"02:00.995 ","End":"02:08.545","Text":"1 plus B over x minus 11."},{"Start":"02:08.545 ","End":"02:14.210","Text":"Next thing to do is to expand this to give it a common denominator."},{"Start":"02:14.210 ","End":"02:22.189","Text":"The denominator will be x minus 1 times x minus 11."},{"Start":"02:22.189 ","End":"02:26.015","Text":"Then the numerator will get this times this plus this times this,"},{"Start":"02:26.015 ","End":"02:34.235","Text":"A times x minus 11 plus B times x minus 1."},{"Start":"02:34.235 ","End":"02:38.510","Text":"I\u0027d like to compute this product at the side here."},{"Start":"02:38.510 ","End":"02:44.236","Text":"What is x minus 1 times x minus 11?"},{"Start":"02:44.236 ","End":"02:48.080","Text":"This is equal to x squared."},{"Start":"02:48.080 ","End":"02:52.580","Text":"This times this minus x minus 11x is minus 12x,"},{"Start":"02:52.580 ","End":"02:57.049","Text":"and minus 1 times minus 11 is plus 11."},{"Start":"02:57.049 ","End":"03:03.320","Text":"Now, look at this quadratic x squared minus 12x plus 11."},{"Start":"03:03.320 ","End":"03:07.220","Text":"Isn\u0027t that exactly the same as what\u0027s written here?"},{"Start":"03:07.220 ","End":"03:10.355","Text":"This is no coincidence."},{"Start":"03:10.355 ","End":"03:18.605","Text":"It turns out that this will always be so and so this has the same denominator as this."},{"Start":"03:18.605 ","End":"03:21.455","Text":"I just copied this over here."},{"Start":"03:21.455 ","End":"03:28.910","Text":"What we have here is 2 rational expressions with the same denominator,"},{"Start":"03:28.910 ","End":"03:31.520","Text":"this equals this as we see from here,"},{"Start":"03:31.520 ","End":"03:33.530","Text":"and if the denominators are the same,"},{"Start":"03:33.530 ","End":"03:36.550","Text":"then we can compare the numerators."},{"Start":"03:36.550 ","End":"03:40.100","Text":"At this point I\u0027d just like to mention a variation."},{"Start":"03:40.100 ","End":"03:49.160","Text":"If you can factorize this x squared plus bx plus c into a product of say x minus x_1,"},{"Start":"03:49.160 ","End":"03:51.560","Text":"x minus x_2,"},{"Start":"03:51.560 ","End":"03:58.865","Text":"then you can skip solving the quadratic equation and straightaway use the 2 factors here,"},{"Start":"03:58.865 ","End":"04:00.785","Text":"as the denominators here."},{"Start":"04:00.785 ","End":"04:03.440","Text":"You might have started out in our example"},{"Start":"04:03.440 ","End":"04:08.535","Text":"with x^2 minus 12x plus 11, let\u0027s say from here,"},{"Start":"04:08.535 ","End":"04:12.080","Text":"and you factorized it into this times this using"},{"Start":"04:12.080 ","End":"04:16.235","Text":"the trinomial method or any other method that you know,"},{"Start":"04:16.235 ","End":"04:18.380","Text":"you take one of the factors and put it here,"},{"Start":"04:18.380 ","End":"04:20.600","Text":"and one of the factors and put it here."},{"Start":"04:20.600 ","End":"04:23.900","Text":"This is just something I\u0027m mentioning that it may be easier to"},{"Start":"04:23.900 ","End":"04:29.000","Text":"factorize the quadratic rather than solve the quadratic equation."},{"Start":"04:29.000 ","End":"04:32.285","Text":"In any event, we get to this point here."},{"Start":"04:32.285 ","End":"04:35.240","Text":"Once we get to this point here, like I said,"},{"Start":"04:35.240 ","End":"04:38.795","Text":"we\u0027re going to be comparing the numerators."},{"Start":"04:38.795 ","End":"04:44.630","Text":"From this side I get 14x minus 54."},{"Start":"04:44.630 ","End":"04:54.180","Text":"On this side, a times x minus 11 plus b times x minus 1."},{"Start":"04:54.310 ","End":"05:01.180","Text":"The important thing to notice that this is not an equation in x. I\u0027m not looking for x."},{"Start":"05:01.180 ","End":"05:05.550","Text":"What I\u0027m looking for is A and B. I\u0027m looking for"},{"Start":"05:05.550 ","End":"05:10.460","Text":"numbers A and B such that this expression will be the same as this expression."},{"Start":"05:10.460 ","End":"05:15.485","Text":"This is true for all x and we have to find A and B."},{"Start":"05:15.485 ","End":"05:20.255","Text":"If it\u0027s true for all x and we can substitute any value of x we want,"},{"Start":"05:20.255 ","End":"05:23.715","Text":"and we can choose convenient values of x."},{"Start":"05:23.715 ","End":"05:25.840","Text":"What do I mean by convenient?"},{"Start":"05:25.840 ","End":"05:28.449","Text":"Well, if I let x equal 11,"},{"Start":"05:28.449 ","End":"05:30.560","Text":"this thing will be 0,"},{"Start":"05:30.560 ","End":"05:32.400","Text":"and if I let x equal 1,"},{"Start":"05:32.400 ","End":"05:33.615","Text":"then this will be 0."},{"Start":"05:33.615 ","End":"05:35.739","Text":"This look like good values to substitute."},{"Start":"05:35.739 ","End":"05:39.685","Text":"Let\u0027s try substituting x equals 11."},{"Start":"05:39.685 ","End":"05:43.029","Text":"Then we get on the left-hand side,"},{"Start":"05:43.029 ","End":"05:51.370","Text":"14 times 11 is 154 minus 54 is 100,"},{"Start":"05:51.370 ","End":"05:54.510","Text":"so I get 100 equals,"},{"Start":"05:54.510 ","End":"05:56.400","Text":"now if x is 11,"},{"Start":"05:56.400 ","End":"06:06.650","Text":"this thing is 0 and times A is still 0 is equal to b times 11 minus 1 is 10."},{"Start":"06:06.770 ","End":"06:12.340","Text":"I immediately get a 100 equals 10b,"},{"Start":"06:12.340 ","End":"06:15.655","Text":"so b equals 10."},{"Start":"06:15.655 ","End":"06:20.780","Text":"On the other hand, if I substitute x equals 1 in this,"},{"Start":"06:20.780 ","End":"06:28.450","Text":"I get 14 times 1 minus 54 is 14 minus 54 is minus 40."},{"Start":"06:28.450 ","End":"06:35.955","Text":"This will equal A times 1 minus 11 is minus 10."},{"Start":"06:35.955 ","End":"06:37.890","Text":"Here, nothing."},{"Start":"06:37.890 ","End":"06:41.780","Text":"Pepsa should just emphasize it that I haven\u0027t forgotten this and here"},{"Start":"06:41.780 ","End":"06:46.535","Text":"I had 0 plus this and here I have this plus 0,"},{"Start":"06:46.535 ","End":"06:51.630","Text":"just to show that this or this 0 respectively."},{"Start":"06:51.630 ","End":"06:56.375","Text":"From here I get A is minus 40 over minus 10,"},{"Start":"06:56.375 ","End":"07:00.490","Text":"which gives me that A equals 4."},{"Start":"07:00.490 ","End":"07:04.965","Text":"Finally, what we do is we,"},{"Start":"07:04.965 ","End":"07:09.260","Text":"and I just copied this expression down here,"},{"Start":"07:09.260 ","End":"07:12.020","Text":"is to fill in the values of A and B,"},{"Start":"07:12.020 ","End":"07:14.035","Text":"which we now know,"},{"Start":"07:14.035 ","End":"07:19.305","Text":"4 where A was and where B was I write 10,"},{"Start":"07:19.305 ","End":"07:23.220","Text":"and this is actually the answer."},{"Start":"07:23.220 ","End":"07:27.380","Text":"I just highlighted it and if you check back in the beginning of"},{"Start":"07:27.380 ","End":"07:33.080","Text":"the lesson you\u0027ll see that this is the expression we started off with,"},{"Start":"07:33.080 ","End":"07:34.850","Text":"and we expanded it to get this,"},{"Start":"07:34.850 ","End":"07:37.825","Text":"so we know that this is the correct answer."},{"Start":"07:37.825 ","End":"07:44.990","Text":"Now, there are plenty of solved examples after the tutorial,"},{"Start":"07:44.990 ","End":"07:49.445","Text":"so we\u0027ll settle for this solved example here for Case 1,"},{"Start":"07:49.445 ","End":"07:52.440","Text":"and let\u0027s move on to Case 2."}],"ID":5265},{"Watched":false,"Name":"Basic Case II","Duration":"9m 17s","ChapterTopicVideoID":5285,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5285.jpeg","UploadDate":"2020-09-30T13:29:32.3400000","DurationForVideoObject":"PT9M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"We just finished case 1,"},{"Start":"00:02.460 ","End":"00:04.710","Text":"and now we want to move on to case 2."},{"Start":"00:04.710 ","End":"00:07.590","Text":"I just kept the highlights from case 1."},{"Start":"00:07.590 ","End":"00:09.570","Text":"Let\u0027s see what changes we have to make."},{"Start":"00:09.570 ","End":"00:12.850","Text":"Well, first of all, we\u0027ll call it case 2."},{"Start":"00:14.330 ","End":"00:16.560","Text":"Case 2, if you remember,"},{"Start":"00:16.560 ","End":"00:19.260","Text":"is when this quadratic here has"},{"Start":"00:19.260 ","End":"00:27.705","Text":"only one solution and I need to get rid of that plural."},{"Start":"00:27.705 ","End":"00:29.700","Text":"If there\u0027s only one solution,"},{"Start":"00:29.700 ","End":"00:32.590","Text":"then it\u0027s just x_1."},{"Start":"00:33.260 ","End":"00:38.910","Text":"What it also means is that if I don\u0027t have x_1 and x_2,"},{"Start":"00:38.910 ","End":"00:40.275","Text":"there\u0027s only one solution,"},{"Start":"00:40.275 ","End":"00:45.685","Text":"then that solution is a repeated solution, a double solution."},{"Start":"00:45.685 ","End":"00:51.365","Text":"It factorizes as x minus the solution squared."},{"Start":"00:51.365 ","End":"01:00.020","Text":"Now the only thing that\u0027s left to change is the basic format or template of the solution."},{"Start":"01:00.020 ","End":"01:04.160","Text":"Well, there\u0027s no x_2 so if we were making"},{"Start":"01:04.160 ","End":"01:07.070","Text":"an educated guess then we\u0027d say maybe it\u0027s of"},{"Start":"01:07.070 ","End":"01:11.945","Text":"the form a over x minus x_1 plus b over x minus x_1."},{"Start":"01:11.945 ","End":"01:15.440","Text":"But now that can\u0027t be right because I combine these"},{"Start":"01:15.440 ","End":"01:20.180","Text":"together and just call it c over x minus x_1."},{"Start":"01:20.180 ","End":"01:21.650","Text":"That\u0027s not going to do."},{"Start":"01:21.650 ","End":"01:24.865","Text":"This is not the right formula."},{"Start":"01:24.865 ","End":"01:31.115","Text":"Turns out the right thing to do is to put this thing squared."},{"Start":"01:31.115 ","End":"01:34.140","Text":"I need 2 constants."},{"Start":"01:34.140 ","End":"01:36.510","Text":"I need for x minus x1,"},{"Start":"01:36.510 ","End":"01:39.905","Text":"to be represented as a power of 1,"},{"Start":"01:39.905 ","End":"01:42.800","Text":"and as a power of 2."},{"Start":"01:42.800 ","End":"01:46.075","Text":"This is the basic starting point for"},{"Start":"01:46.075 ","End":"01:51.360","Text":"the one solution case and best illustrated with an example."},{"Start":"01:51.580 ","End":"01:54.275","Text":"Here\u0027s the example."},{"Start":"01:54.275 ","End":"01:59.030","Text":"X plus 1 over"},{"Start":"01:59.030 ","End":"02:05.530","Text":"x squared minus 2x plus 1."},{"Start":"02:05.530 ","End":"02:08.015","Text":"It\u0027s certainly of this form,"},{"Start":"02:08.015 ","End":"02:13.190","Text":"a linear over a quadratic and the leading coefficient is 1."},{"Start":"02:13.190 ","End":"02:14.690","Text":"As I said, if it isn\u0027t 1,"},{"Start":"02:14.690 ","End":"02:18.905","Text":"we just divide top and bottom so we can always make sure that there\u0027s a 1 here."},{"Start":"02:18.905 ","End":"02:21.950","Text":"Now, this is the solution we\u0027re looking for."},{"Start":"02:21.950 ","End":"02:24.950","Text":"For, I can say, well it is what x_1 is,"},{"Start":"02:24.950 ","End":"02:26.359","Text":"what is the solution?"},{"Start":"02:26.359 ","End":"02:28.550","Text":"We either solve the quadratic."},{"Start":"02:28.550 ","End":"02:31.130","Text":"Suppose I try to solve the equation,"},{"Start":"02:31.130 ","End":"02:35.575","Text":"x squared minus 2x plus 1 equals 0."},{"Start":"02:35.575 ","End":"02:37.935","Text":"What I\u0027ll do it this time,"},{"Start":"02:37.935 ","End":"02:39.877","Text":"I\u0027ll do it by the formula method,"},{"Start":"02:39.877 ","End":"02:46.775","Text":"so minus b plus or minus the square root of b squared,"},{"Start":"02:46.775 ","End":"02:50.970","Text":"which is 4 minus 4c plus 4ac,"},{"Start":"02:50.970 ","End":"02:55.980","Text":"but a is 1 minus 4 times c is 1,"},{"Start":"02:55.980 ","End":"03:00.730","Text":"all over 2a, which is 2."},{"Start":"03:01.850 ","End":"03:04.680","Text":"This is what x is going to equal."},{"Start":"03:04.680 ","End":"03:06.680","Text":"Now 4 minus 4 is 0,"},{"Start":"03:06.680 ","End":"03:10.355","Text":"so it\u0027s the square root of 0 is 0."},{"Start":"03:10.355 ","End":"03:18.320","Text":"We get basically 2 plus or minus 0 over 2 because it\u0027s plus or minus 0."},{"Start":"03:18.320 ","End":"03:20.645","Text":"That\u0027s what gives us the only one solution."},{"Start":"03:20.645 ","End":"03:23.495","Text":"Because plus 0 minus 0 is still just 2 over 2,"},{"Start":"03:23.495 ","End":"03:25.980","Text":"which is 1 or 1."},{"Start":"03:25.980 ","End":"03:30.190","Text":"Sometimes we say there\u0027s only one solution, x equals 1,"},{"Start":"03:30.190 ","End":"03:34.720","Text":"and sometimes we say that the solution x equals 1 appears twice."},{"Start":"03:34.720 ","End":"03:38.215","Text":"But in this regard this is what I\u0027m talking about."},{"Start":"03:38.215 ","End":"03:40.180","Text":"However you call it,"},{"Start":"03:40.180 ","End":"03:43.510","Text":"whether it\u0027s a double solution or only one solution,"},{"Start":"03:43.510 ","End":"03:45.220","Text":"this is what I mean."},{"Start":"03:45.220 ","End":"03:49.525","Text":"Our x_1 is equal to 1 in this case."},{"Start":"03:49.525 ","End":"03:52.045","Text":"Once we\u0027ve got that x1 equals 1,"},{"Start":"03:52.045 ","End":"03:57.580","Text":"then we can write this as a over x minus"},{"Start":"03:57.580 ","End":"04:05.775","Text":"1 plus b over x minus 1 squared."},{"Start":"04:05.775 ","End":"04:12.420","Text":"We\u0027re going to use algebraic manipulation to find a and b."},{"Start":"04:12.420 ","End":"04:17.960","Text":"As before, I want to point out that we didn\u0027t have to solve the equation."},{"Start":"04:17.960 ","End":"04:26.405","Text":"The other way is to factorize x squared minus 2x plus 1."},{"Start":"04:26.405 ","End":"04:29.780","Text":"If you factorize this by whatever method,"},{"Start":"04:29.780 ","End":"04:32.960","Text":"the trinomial, however you do it,"},{"Start":"04:32.960 ","End":"04:37.340","Text":"you will get that this is equal to x minus 1 squared."},{"Start":"04:37.340 ","End":"04:40.220","Text":"If you factorize it,"},{"Start":"04:40.220 ","End":"04:42.245","Text":"then instead of using the formula,"},{"Start":"04:42.245 ","End":"04:45.499","Text":"then you just take this thing and see that it\u0027s something squared."},{"Start":"04:45.499 ","End":"04:48.120","Text":"You put it here and here, one squared,"},{"Start":"04:48.120 ","End":"04:52.625","Text":"once without the square and constants on the top."},{"Start":"04:52.625 ","End":"04:56.210","Text":"This is just minor variations of getting to this point."},{"Start":"04:56.210 ","End":"05:00.729","Text":"Either by solving and getting x1,"},{"Start":"05:00.729 ","End":"05:02.645","Text":"and putting it here and here,"},{"Start":"05:02.645 ","End":"05:07.066","Text":"or by factorizing and factorizing gives you this thing,"},{"Start":"05:07.066 ","End":"05:08.690","Text":"and then you put it also over here,"},{"Start":"05:08.690 ","End":"05:10.675","Text":"but without the square."},{"Start":"05:10.675 ","End":"05:17.840","Text":"I think that part is clear now we actually have to find a and b as actual numbers."},{"Start":"05:17.840 ","End":"05:23.675","Text":"We need to do some work with fractions and find a common denominator."},{"Start":"05:23.675 ","End":"05:27.575","Text":"This is our common denominator because it\u0027s the same as this,"},{"Start":"05:27.575 ","End":"05:29.120","Text":"as we see over here."},{"Start":"05:29.120 ","End":"05:31.445","Text":"This part I just copied as is,"},{"Start":"05:31.445 ","End":"05:32.855","Text":"then the other side,"},{"Start":"05:32.855 ","End":"05:38.330","Text":"we need to put it over x minus 1^2."},{"Start":"05:38.330 ","End":"05:43.700","Text":"What do we get? The second part is the easy part that\u0027s going to be plus b."},{"Start":"05:43.700 ","End":"05:45.890","Text":"This part doesn\u0027t change,"},{"Start":"05:45.890 ","End":"05:48.710","Text":"but here we\u0027re missing an x minus 1."},{"Start":"05:48.710 ","End":"05:53.290","Text":"It\u0027s just going to be a times x minus 1,"},{"Start":"05:53.290 ","End":"05:55.755","Text":"and then plus b over this."},{"Start":"05:55.755 ","End":"05:58.535","Text":"Now because the denominators are equal,"},{"Start":"05:58.535 ","End":"06:00.110","Text":"I know they look a bit different,"},{"Start":"06:00.110 ","End":"06:02.090","Text":"but remember, they are equal,"},{"Start":"06:02.090 ","End":"06:04.700","Text":"we can compare the numerators."},{"Start":"06:04.700 ","End":"06:10.940","Text":"Let\u0027s do that and say that x plus"},{"Start":"06:10.940 ","End":"06:20.250","Text":"1 is equal to a x minus 1 plus b."},{"Start":"06:20.780 ","End":"06:24.710","Text":"Remember, we\u0027re not solving for x,."},{"Start":"06:24.710 ","End":"06:26.615","Text":"It\u0027s not an equation in x."},{"Start":"06:26.615 ","End":"06:30.200","Text":"These have got to be exactly the same expression."},{"Start":"06:30.200 ","End":"06:35.930","Text":"We\u0027re looking for a and b that will make it be x plus 1 here also."},{"Start":"06:35.930 ","End":"06:39.619","Text":"In fact, if they\u0027re the same expression,"},{"Start":"06:39.619 ","End":"06:41.810","Text":"I can substitute any value of x i want,"},{"Start":"06:41.810 ","End":"06:42.935","Text":"and it should be equal."},{"Start":"06:42.935 ","End":"06:45.080","Text":"That\u0027s how we\u0027re going to solve it."},{"Start":"06:45.080 ","End":"06:47.150","Text":"As in the previous case,"},{"Start":"06:47.150 ","End":"06:51.110","Text":"we looked for convenient values to substitute and the most convenient will"},{"Start":"06:51.110 ","End":"06:56.240","Text":"be x equals 1 because then it\u0027ll get rid of this term altogether."},{"Start":"06:56.240 ","End":"06:59.225","Text":"Substituting x equals 1,"},{"Start":"06:59.225 ","End":"07:02.980","Text":"I get 1 plus 1 is 2."},{"Start":"07:02.980 ","End":"07:09.225","Text":"Then a times 0 plus b,"},{"Start":"07:09.225 ","End":"07:13.110","Text":"I wrote that to just express that it is 0,"},{"Start":"07:13.110 ","End":"07:15.855","Text":"and of course this just drops off."},{"Start":"07:15.855 ","End":"07:20.670","Text":"We get that b is equal to 2."},{"Start":"07:20.670 ","End":"07:24.195","Text":"Now how do we find a."},{"Start":"07:24.195 ","End":"07:28.720","Text":"There\u0027s no magic value that makes something disappear,"},{"Start":"07:28.720 ","End":"07:33.745","Text":"but just use any other value besides one that we haven\u0027t used."},{"Start":"07:33.745 ","End":"07:37.510","Text":"Zero\u0027s often, good because calculating with it is easy."},{"Start":"07:37.510 ","End":"07:40.195","Text":"Let\u0027s try x equals 0."},{"Start":"07:40.195 ","End":"07:41.539","Text":"This is just arbitrary,"},{"Start":"07:41.539 ","End":"07:43.270","Text":"just use any other value."},{"Start":"07:43.270 ","End":"07:46.795","Text":"Then we get 0 plus 1 is 1,"},{"Start":"07:46.795 ","End":"07:51.620","Text":"is equal to a times 0 minus 1,"},{"Start":"07:51.620 ","End":"07:57.490","Text":"is minus 1 plus b. I\u0027m substituting it in here,"},{"Start":"07:57.490 ","End":"08:00.280","Text":"of course both of them are in this equation."},{"Start":"08:00.280 ","End":"08:02.790","Text":"What does that give me?"},{"Start":"08:02.790 ","End":"08:07.020","Text":"Because I know that b is 2,"},{"Start":"08:07.040 ","End":"08:09.990","Text":"then what it says is that,"},{"Start":"08:09.990 ","End":"08:13.635","Text":"1 equals minus a plus 2."},{"Start":"08:13.635 ","End":"08:16.650","Text":"Bring the a here and the 1 here,"},{"Start":"08:16.650 ","End":"08:19.965","Text":"and I\u0027ve got a equals minus 1 plus 2,"},{"Start":"08:19.965 ","End":"08:24.555","Text":"so a equals 1."},{"Start":"08:24.555 ","End":"08:32.480","Text":"Then lastly, I just did a copy paste of this over here."},{"Start":"08:32.480 ","End":"08:34.475","Text":"I don\u0027t need this."},{"Start":"08:34.475 ","End":"08:40.384","Text":"What I\u0027m going to do is erase the a and the b and put in the actual values that I found."},{"Start":"08:40.384 ","End":"08:45.525","Text":"This was a, so it\u0027s 1 this was b, so it\u0027s 2."},{"Start":"08:45.525 ","End":"08:48.900","Text":"I think I\u0027ll just highlight it."},{"Start":"08:48.900 ","End":"08:54.840","Text":"This is our partial fraction decomposition of this."},{"Start":"08:55.270 ","End":"08:58.580","Text":"I only did 1 example,"},{"Start":"08:58.580 ","End":"09:05.255","Text":"but there are plenty of examples exercises at the end of the tutorial."},{"Start":"09:05.255 ","End":"09:09.305","Text":"There\u0027s quite a lot of those, so you should go and take a look at them."},{"Start":"09:09.305 ","End":"09:16.920","Text":"Meanwhile, we\u0027ll say that we\u0027re done with case 2 for now and so onto case 3."}],"ID":5266},{"Watched":false,"Name":"Basic Case III","Duration":"3m 23s","ChapterTopicVideoID":5286,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5286.jpeg","UploadDate":"2020-09-30T13:31:36.6870000","DurationForVideoObject":"PT3M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"Here we are about to discuss Case 3,"},{"Start":"00:03.240 ","End":"00:08.220","Text":"but I\u0027d like to just briefly summarize Case 1 and Case 2."},{"Start":"00:08.220 ","End":"00:13.425","Text":"All this revolves around this particular rational expression,"},{"Start":"00:13.425 ","End":"00:17.280","Text":"a linear over a quadratic polynomial."},{"Start":"00:17.280 ","End":"00:23.130","Text":"This is the only building block that we\u0027ve been discussing up till now."},{"Start":"00:23.130 ","End":"00:27.060","Text":"Let me just emphasize this here."},{"Start":"00:27.060 ","End":"00:31.845","Text":"We distinguish three cases depending on the denominator."},{"Start":"00:31.845 ","End":"00:37.240","Text":"We said that if the denominator has two solutions,"},{"Start":"00:37.240 ","End":"00:43.650","Text":"let\u0027s call them x1 and x2."},{"Start":"00:43.650 ","End":"00:48.470","Text":"It turns out that the left-hand side or the denominator here can be"},{"Start":"00:48.470 ","End":"00:54.425","Text":"factorized as x minus 1, x minus x2."},{"Start":"00:54.425 ","End":"01:02.195","Text":"In this case, the partial fraction decomposition of this is of the form a over"},{"Start":"01:02.195 ","End":"01:11.400","Text":"x minus x1 plus b over x minus x2."},{"Start":"01:11.400 ","End":"01:15.960","Text":"In Case 2 where this only has 1 solution,"},{"Start":"01:15.960 ","End":"01:18.250","Text":"call that solution x1,"},{"Start":"01:18.250 ","End":"01:24.500","Text":"then this factors into x minus x1^2,"},{"Start":"01:24.500 ","End":"01:26.457","Text":"the left-hand side,"},{"Start":"01:26.457 ","End":"01:33.090","Text":"and the partial fraction decomposition of this is a over x"},{"Start":"01:33.090 ","End":"01:41.450","Text":"minus x1 plus b over x minus x1^2."},{"Start":"01:41.450 ","End":"01:46.905","Text":"We did a numerical example for each one of Case 1 and Case 2."},{"Start":"01:46.905 ","End":"01:48.810","Text":"Now Case 3."},{"Start":"01:48.810 ","End":"01:52.625","Text":"Well, Case 3 is the easiest because it turns out"},{"Start":"01:52.625 ","End":"01:56.645","Text":"that there is no simpler way of decomposing this."},{"Start":"01:56.645 ","End":"01:59.675","Text":"In Case 3 you just leave it as is,"},{"Start":"01:59.675 ","End":"02:03.470","Text":"just leave it alone or if you like,"},{"Start":"02:03.470 ","End":"02:12.785","Text":"just leave it as mx plus n over the same x^2 plus bx plus c. Nothing can be done."},{"Start":"02:12.785 ","End":"02:17.930","Text":"An important question is how do we know it has no solutions?"},{"Start":"02:17.930 ","End":"02:21.200","Text":"Well you just try and solve this using"},{"Start":"02:21.200 ","End":"02:25.640","Text":"the quadratic formula and find something negative under the square root."},{"Start":"02:25.640 ","End":"02:28.700","Text":"In other words, if b^2 minus 4ac,"},{"Start":"02:28.700 ","End":"02:31.400","Text":"or in this case b^2 minus 4c,"},{"Start":"02:31.400 ","End":"02:35.045","Text":"because our a is 1 is less than 0,"},{"Start":"02:35.045 ","End":"02:37.535","Text":"then you know it has no solutions."},{"Start":"02:37.535 ","End":"02:41.750","Text":"Or by any other method you conclude that there\u0027s no solutions,"},{"Start":"02:41.750 ","End":"02:47.060","Text":"then this is as basic as it gets and there is no decomposition."},{"Start":"02:48.050 ","End":"02:53.330","Text":"All we\u0027re done with now is this particular basic case."},{"Start":"02:53.330 ","End":"02:55.879","Text":"I will continue with partial fractions."},{"Start":"02:55.879 ","End":"03:04.145","Text":"We will expand beyond this linear over quadratic and go into more general cases."},{"Start":"03:04.145 ","End":"03:08.510","Text":"Ideally, we should be able to do every polynomial over polynomial."},{"Start":"03:08.510 ","End":"03:13.519","Text":"Well almost, there is a question of the degree of the numerator and denominator,"},{"Start":"03:13.519 ","End":"03:15.725","Text":"but I don\u0027t want to get into that now."},{"Start":"03:15.725 ","End":"03:20.810","Text":"In the next clip we\u0027ll change this to some other basic form."},{"Start":"03:20.810 ","End":"03:23.190","Text":"Meanwhile, we\u0027re done."}],"ID":5267},{"Watched":false,"Name":"General Case","Duration":"8m 34s","ChapterTopicVideoID":5287,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5287.jpeg","UploadDate":"2020-09-30T13:36:08.3870000","DurationForVideoObject":"PT8M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"Continuing with partial fraction decomposition,"},{"Start":"00:03.900 ","End":"00:06.935","Text":"I want to generalize what we\u0027ve done so far."},{"Start":"00:06.935 ","End":"00:12.795","Text":"Up till now, we took 1 specific form of a rational function,"},{"Start":"00:12.795 ","End":"00:17.605","Text":"and this was what it looked like and it\u0027s a linear over a quadratic."},{"Start":"00:17.605 ","End":"00:23.115","Text":"I want to generalize it to a rational function,"},{"Start":"00:23.115 ","End":"00:30.090","Text":"say P(x) divided by Q(x),"},{"Start":"00:30.090 ","End":"00:32.700","Text":"where P and Q are polynomials."},{"Start":"00:32.700 ","End":"00:35.550","Text":"I\u0027m going to write down 1 condition and"},{"Start":"00:35.550 ","End":"00:40.550","Text":"3 steps in finding the partial fraction decomposition."},{"Start":"00:40.550 ","End":"00:42.050","Text":"I won\u0027t write anything yet,"},{"Start":"00:42.050 ","End":"00:45.890","Text":"I\u0027ll just say what is going on and then I\u0027ll write it up."},{"Start":"00:45.890 ","End":"00:48.455","Text":"The first thing is just like here,"},{"Start":"00:48.455 ","End":"00:50.155","Text":"the degree of P,"},{"Start":"00:50.155 ","End":"00:52.460","Text":"that\u0027s got to be less than the degree of Q."},{"Start":"00:52.460 ","End":"00:54.695","Text":"Like here is a degree 2 polynomial,"},{"Start":"00:54.695 ","End":"00:56.870","Text":"and this is a degree 1 polynomial,"},{"Start":"00:56.870 ","End":"01:02.660","Text":"or maybe even a degree 0 polynomial if I don\u0027t have an M. That\u0027s the first condition,"},{"Start":"01:02.660 ","End":"01:05.390","Text":"degree on top, lower than the degree on the bottom,"},{"Start":"01:05.390 ","End":"01:08.590","Text":"otherwise, we can\u0027t do partial fractions."},{"Start":"01:08.590 ","End":"01:11.680","Text":"Then we\u0027re going to break it up into 3 steps."},{"Start":"01:11.680 ","End":"01:15.950","Text":"First of all, we\u0027re going to factorize the denominator where possible."},{"Start":"01:15.950 ","End":"01:17.630","Text":"In our case, for example,"},{"Start":"01:17.630 ","End":"01:20.240","Text":"we factorized it and there were 3 possibilities,"},{"Start":"01:20.240 ","End":"01:23.435","Text":"either factorized as 2 separate factors,"},{"Start":"01:23.435 ","End":"01:27.440","Text":"or we got the same factor twice,"},{"Start":"01:27.440 ","End":"01:35.570","Text":"or we weren\u0027t able to factorize it at all that x^2 plus bx plus c just wasn\u0027t factorable."},{"Start":"01:35.570 ","End":"01:39.470","Text":"But in any event, according to how we factored the denominator,"},{"Start":"01:39.470 ","End":"01:43.445","Text":"we decided the general shape of the partial fraction."},{"Start":"01:43.445 ","End":"01:46.200","Text":"That\u0027s going to be step 2,"},{"Start":"01:46.200 ","End":"01:50.620","Text":"is deciding the general shape using constants,"},{"Start":"01:50.620 ","End":"01:53.560","Text":"which will use capital letters for."},{"Start":"01:53.560 ","End":"01:57.850","Text":"The third step will be to use algebra like we did in the examples,"},{"Start":"01:57.850 ","End":"02:00.550","Text":"common denominators and substituting values and"},{"Start":"02:00.550 ","End":"02:03.820","Text":"so on to find the actual values of the constants."},{"Start":"02:03.820 ","End":"02:06.500","Text":"Let me first write that."},{"Start":"02:07.470 ","End":"02:16.500","Text":"Partial fraction decomposition applies to rational expression."},{"Start":"02:16.500 ","End":"02:21.450","Text":"A rational expression is a polynomial over a polynomial,"},{"Start":"02:21.450 ","End":"02:24.160","Text":"so this is a polynomial,"},{"Start":"02:25.780 ","End":"02:28.370","Text":"and this is a polynomial."},{"Start":"02:28.370 ","End":"02:30.760","Text":"The condition that we need is that the degree"},{"Start":"02:30.760 ","End":"02:33.505","Text":"on top is less than the degree on the bottom."},{"Start":"02:33.505 ","End":"02:41.225","Text":"In other words, the degree of P must be less"},{"Start":"02:41.225 ","End":"02:49.500","Text":"than the degree of Q. P is the numerator Q is the denominator."},{"Start":"02:49.500 ","End":"02:53.265","Text":"That\u0027s the setup."},{"Start":"02:53.265 ","End":"02:55.290","Text":"Then there are 3 steps,"},{"Start":"02:55.290 ","End":"03:00.080","Text":"one is you completely factorize the denominator."},{"Start":"03:02.270 ","End":"03:06.670","Text":"It turns out that every polynomial can be factorized as"},{"Start":"03:06.670 ","End":"03:10.315","Text":"a product of 2 kinds of factor, either linear,"},{"Start":"03:10.315 ","End":"03:13.670","Text":"something like x plus a,"},{"Start":"03:13.670 ","End":"03:19.355","Text":"and the other building blocks of the factors could be irreducible quadratics,"},{"Start":"03:19.355 ","End":"03:25.370","Text":"like x^2 plus bx plus c. But when I say irreducible,"},{"Start":"03:25.370 ","End":"03:28.820","Text":"I mean that this has no roots,"},{"Start":"03:28.820 ","End":"03:33.245","Text":"or you could say the equation where this equals 0 has no solutions."},{"Start":"03:33.245 ","End":"03:35.720","Text":"That\u0027s what we mean by irreducible."},{"Start":"03:35.720 ","End":"03:42.245","Text":"Now I\u0027ll just write the word irreducible."},{"Start":"03:42.245 ","End":"03:47.900","Text":"There\u0027s also the way of checking if it\u0027s irreducible using the formula,"},{"Start":"03:47.900 ","End":"03:51.840","Text":"the square root of b^2 minus 4ac,"},{"Start":"03:52.390 ","End":"03:57.360","Text":"b^2 minus 4c is negative,"},{"Start":"03:57.360 ","End":"03:58.915","Text":"a is 1 here."},{"Start":"03:58.915 ","End":"04:03.060","Text":"Which also leads me to point out that we are"},{"Start":"04:03.060 ","End":"04:07.970","Text":"assuming that Q has a leading coefficient of 1,"},{"Start":"04:07.970 ","End":"04:10.060","Text":"just like we had earlier."},{"Start":"04:10.060 ","End":"04:14.110","Text":"We had x^2 plus bx plus c leading coefficient 1."},{"Start":"04:14.110 ","End":"04:16.000","Text":"The 1 doesn\u0027t appear of course."},{"Start":"04:16.000 ","End":"04:19.810","Text":"But also I want to point out that these factors,"},{"Start":"04:19.810 ","End":"04:21.130","Text":"if they appear more than once,"},{"Start":"04:21.130 ","End":"04:23.610","Text":"could contribute an exponent here,"},{"Start":"04:23.610 ","End":"04:26.275","Text":"so this could be to the power of some number k,"},{"Start":"04:26.275 ","End":"04:28.915","Text":"could be 1, it could be higher than 1."},{"Start":"04:28.915 ","End":"04:35.510","Text":"Now, I\u0027ll give you an example of what I mean by building something from these factors."},{"Start":"04:35.540 ","End":"04:46.505","Text":"I\u0027ll give you an example pretty complicated more than you would get of Q(x) factorized,"},{"Start":"04:46.505 ","End":"04:57.970","Text":"you could get (x-2)^3(x+4)(x-1)^2 may"},{"Start":"04:58.760 ","End":"05:07.030","Text":"be ("},{"Start":"05:07.030 ","End":"05:11.270","Text":"x^2+9),"},{"Start":"05:11.270 ","End":"05:18.200","Text":"and (x^2+2x+5)^3."},{"Start":"05:18.200 ","End":"05:27.560","Text":"What we have here is expressions of this form x plus or minus a constant, like here,"},{"Start":"05:27.560 ","End":"05:28.895","Text":"x plus 4,"},{"Start":"05:28.895 ","End":"05:32.810","Text":"or we could have x plus or minus something to a power with a k,"},{"Start":"05:32.810 ","End":"05:36.440","Text":"or we could have an irreducible quadratic,"},{"Start":"05:36.440 ","End":"05:40.130","Text":"I\u0027ll show you in a moment why I know this is irreducible,"},{"Start":"05:40.130 ","End":"05:43.654","Text":"and this is also irreducible to a power."},{"Start":"05:43.654 ","End":"05:49.880","Text":"In general, we can factorize each polynomial into this kind of thing."},{"Start":"05:49.880 ","End":"05:51.630","Text":"Why these are irreducible?"},{"Start":"05:51.630 ","End":"05:55.170","Text":"Just check the b^2 minus 4c for them,"},{"Start":"05:55.170 ","End":"05:58.215","Text":"or try and solve x^2 plus 9 equals 0,"},{"Start":"05:58.215 ","End":"06:00.540","Text":"x^2 plus 2x plus 5 equals 0,"},{"Start":"06:00.540 ","End":"06:03.340","Text":"and you\u0027ll see there\u0027s no solution."},{"Start":"06:03.560 ","End":"06:06.625","Text":"That\u0027s step 1,"},{"Start":"06:06.625 ","End":"06:11.720","Text":"factorized into linears and irreducible quadratics,"},{"Start":"06:11.720 ","End":"06:14.315","Text":"productive several of these may be,"},{"Start":"06:14.315 ","End":"06:17.400","Text":"and some with exponents on them."},{"Start":"06:17.810 ","End":"06:20.155","Text":"Then step 2,"},{"Start":"06:20.155 ","End":"06:23.640","Text":"we decide the general shape."},{"Start":"06:26.130 ","End":"06:29.380","Text":"I\u0027ll say a lot more about this later,"},{"Start":"06:29.380 ","End":"06:33.860","Text":"I\u0027ll just show you what I mean in general terms."},{"Start":"06:33.860 ","End":"06:38.200","Text":"I just jump back to a previous page that we"},{"Start":"06:38.200 ","End":"06:42.600","Text":"said that if the denominator was x minus x_1,"},{"Start":"06:42.600 ","End":"06:43.745","Text":"x minus x_2,"},{"Start":"06:43.745 ","End":"06:47.035","Text":"then the general form of the partial fraction was this,"},{"Start":"06:47.035 ","End":"06:49.225","Text":"unless the denominator was this,"},{"Start":"06:49.225 ","End":"06:51.985","Text":"then the general form was this."},{"Start":"06:51.985 ","End":"06:57.955","Text":"In other words, we decided on a general shape involving constants like A and B,"},{"Start":"06:57.955 ","End":"06:59.785","Text":"use capital letters for these,"},{"Start":"06:59.785 ","End":"07:03.780","Text":"depending on the denominator."},{"Start":"07:03.780 ","End":"07:06.320","Text":"Let\u0027s go back to where we were."},{"Start":"07:06.320 ","End":"07:09.270","Text":"You\u0027ve seen what I meant by the general shape,"},{"Start":"07:09.270 ","End":"07:11.240","Text":"and I\u0027ll just add that it"},{"Start":"07:11.240 ","End":"07:20.550","Text":"depends only on the denominator Q and how it factorizes."},{"Start":"07:20.550 ","End":"07:23.990","Text":"I\u0027ll just say that we\u0027ll talk about this"},{"Start":"07:23.990 ","End":"07:28.370","Text":"more later because this is a whole art and this is the essence of it."},{"Start":"07:28.370 ","End":"07:31.130","Text":"I just want to give an overview of the steps."},{"Start":"07:31.130 ","End":"07:36.605","Text":"The third step,"},{"Start":"07:36.605 ","End":"07:43.145","Text":"we use algebra to find the constants that were in step 2,"},{"Start":"07:43.145 ","End":"07:46.258","Text":"remember we had constants like A,"},{"Start":"07:46.258 ","End":"07:48.755","Text":"B, C, capital letters, etc."},{"Start":"07:48.755 ","End":"07:54.380","Text":"The main techniques we used where we"},{"Start":"07:54.380 ","End":"07:56.750","Text":"did the common denominators and we"},{"Start":"07:56.750 ","End":"07:59.780","Text":"substituted the values of x and we eventually found A,"},{"Start":"07:59.780 ","End":"08:02.585","Text":"B, C, and whoever many constants we had."},{"Start":"08:02.585 ","End":"08:06.560","Text":"I just wanted to give an overview of the general steps which were,"},{"Start":"08:06.560 ","End":"08:10.205","Text":"first of all, factorization of the denominator,"},{"Start":"08:10.205 ","End":"08:13.670","Text":"and then deciding on the general shape or"},{"Start":"08:13.670 ","End":"08:17.735","Text":"format of the partial fraction involving constants,"},{"Start":"08:17.735 ","End":"08:22.880","Text":"and then using algebra to find those constants."},{"Start":"08:22.880 ","End":"08:26.300","Text":"In a few seconds, we\u0027ll take a break and when we come back,"},{"Start":"08:26.300 ","End":"08:28.460","Text":"we\u0027ll return to this part 2,"},{"Start":"08:28.460 ","End":"08:33.990","Text":"which is the heart of the matter to the part where I say more later."}],"ID":5268},{"Watched":false,"Name":"General Case (continued)","Duration":"14m 29s","ChapterTopicVideoID":5288,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5288.jpeg","UploadDate":"2016-03-07T09:24:49.3670000","DurationForVideoObject":"PT14M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.145","Text":"Back from the break,"},{"Start":"00:02.145 ","End":"00:05.880","Text":"we were going to talk more about Part 2,"},{"Start":"00:05.880 ","End":"00:08.580","Text":"which is how to decide the general shape of"},{"Start":"00:08.580 ","End":"00:13.870","Text":"the partial fraction depending on how we factorize Q."},{"Start":"00:13.970 ","End":"00:16.995","Text":"There is a formal system."},{"Start":"00:16.995 ","End":"00:22.190","Text":"But I\u0027m actually going to start with an example and see if that might make it clear."},{"Start":"00:22.190 ","End":"00:25.035","Text":"Then I\u0027ll put the formal rules for it."},{"Start":"00:25.035 ","End":"00:28.020","Text":"I\u0027ll take this example right here."},{"Start":"00:28.020 ","End":"00:31.650","Text":"I erased everything we don\u0027t need and let me just move this"},{"Start":"00:31.650 ","End":"00:34.710","Text":"up here just to remind you what\u0027s going on."},{"Start":"00:34.710 ","End":"00:40.910","Text":"We assume that we have the rational expression 1 polynomial over another."},{"Start":"00:40.910 ","End":"00:44.400","Text":"That we took as an example,"},{"Start":"00:44.400 ","End":"00:48.394","Text":"Q(x), which factorizes as follows."},{"Start":"00:48.394 ","End":"00:51.215","Text":"We assume that Q is factorized completely."},{"Start":"00:51.215 ","End":"00:52.910","Text":"When you factorize it completely,"},{"Start":"00:52.910 ","End":"00:55.389","Text":"what you\u0027re going to get linear terms,"},{"Start":"00:55.389 ","End":"01:02.030","Text":"possibly with a power like here and here, and irreducible quadratics."},{"Start":"01:02.030 ","End":"01:04.730","Text":"We decided that these 2 are both irreducible quadratics,"},{"Start":"01:04.730 ","End":"01:07.040","Text":"again, with or without an exponent."},{"Start":"01:07.040 ","End":"01:08.280","Text":"Well, there\u0027s always an exponent,"},{"Start":"01:08.280 ","End":"01:10.008","Text":"who could be an exponent of 1,"},{"Start":"01:10.008 ","End":"01:12.480","Text":"and here it\u0027s 3."},{"Start":"01:12.480 ","End":"01:19.825","Text":"This tells us how to write this P(x)/Q(x)."},{"Start":"01:19.825 ","End":"01:23.080","Text":"I\u0027ll try to explain just by this example"},{"Start":"01:23.080 ","End":"01:26.185","Text":"without giving you the theory and later I\u0027ll show you the theory."},{"Start":"01:26.185 ","End":"01:30.040","Text":"Let\u0027s start with the first one, (x minus 2)^3."},{"Start":"01:30.040 ","End":"01:32.155","Text":"When it\u0027s a linear to the power,"},{"Start":"01:32.155 ","End":"01:36.845","Text":"we put a constant and we put on the denominator,"},{"Start":"01:36.845 ","End":"01:40.695","Text":"each of the powers up to 3."},{"Start":"01:40.695 ","End":"01:44.430","Text":"I have A over x minus 2 plus"},{"Start":"01:44.430 ","End":"01:52.140","Text":"B/(x minus 2)^2 plus C over (x"},{"Start":"01:52.140 ","End":"01:54.780","Text":"minus 2)^3."},{"Start":"01:54.780 ","End":"01:56.910","Text":"When you see A_3, like A_k,"},{"Start":"01:56.910 ","End":"02:00.042","Text":"you take all the powers from 1-k,"},{"Start":"02:00.042 ","End":"02:02.050","Text":"in this case from 1-3."},{"Start":"02:02.050 ","End":"02:03.500","Text":"That\u0027s how it works."},{"Start":"02:03.500 ","End":"02:07.595","Text":"That\u0027s just the first term is going to be a long expression."},{"Start":"02:07.595 ","End":"02:10.390","Text":"Also, let me tell you not to worry,"},{"Start":"02:10.390 ","End":"02:14.600","Text":"it\u0027s not the thing that you will get on a question."},{"Start":"02:14.600 ","End":"02:18.295","Text":"I\u0027m just doing it to throw everything into 1 example."},{"Start":"02:18.295 ","End":"02:21.142","Text":"Next, we have just x plus 4,"},{"Start":"02:21.142 ","End":"02:24.884","Text":"when it\u0027s just x plus 4 not to a power or power of 1,"},{"Start":"02:24.884 ","End":"02:31.610","Text":"it\u0027s just a single constant over x plus 4."},{"Start":"02:31.610 ","End":"02:34.695","Text":"Next I have (x minus 1)^2."},{"Start":"02:34.695 ","End":"02:36.610","Text":"The same rule applies us with the 3."},{"Start":"02:36.610 ","End":"02:38.810","Text":"I need another constant."},{"Start":"02:38.810 ","End":"02:41.060","Text":"We typically use letters of the alphabet,"},{"Start":"02:41.060 ","End":"02:44.450","Text":"capital letters just in order as they come."},{"Start":"02:44.450 ","End":"02:54.380","Text":"I\u0027m going to have E/(x minus 1) plus F/(x minus 1)^2."},{"Start":"02:54.380 ","End":"03:02.415","Text":"I need both the 1 and the 2. No room here."},{"Start":"03:02.415 ","End":"03:07.350","Text":"Continue over here."},{"Start":"03:07.350 ","End":"03:13.130","Text":"The x^2 plus 9."},{"Start":"03:13.130 ","End":"03:16.955","Text":"You put a linear term when it\u0027s an irreducible quadratic."},{"Start":"03:16.955 ","End":"03:21.590","Text":"I need a linear term, Gx plus H,"},{"Start":"03:21.590 ","End":"03:28.260","Text":"just G and H being the next letters in the alphabet over x^2 plus 9."},{"Start":"03:28.900 ","End":"03:32.870","Text":"Here I have another irreducible quadratic,"},{"Start":"03:32.870 ","End":"03:38.105","Text":"but this time cubed so I need to have this,"},{"Start":"03:38.105 ","End":"03:40.460","Text":"first of all to the power of 1,"},{"Start":"03:40.460 ","End":"03:44.510","Text":"just as is x^2 plus 2x plus 5."},{"Start":"03:44.510 ","End":"03:47.180","Text":"You know what? I\u0027ll just write the denominators first."},{"Start":"03:47.180 ","End":"03:55.220","Text":"Then I\u0027ll need to have (x^2 plus 2x plus 5)^2."},{"Start":"03:55.220 ","End":"04:03.140","Text":"Then I\u0027ll need x^2 plus 2x plus 5^3."},{"Start":"04:03.140 ","End":"04:06.540","Text":"Then each of them, I need a linear expression."},{"Start":"04:06.540 ","End":"04:08.220","Text":"Let\u0027s see up to H,"},{"Start":"04:08.220 ","End":"04:12.930","Text":"I_x plus J,"},{"Start":"04:12.930 ","End":"04:21.360","Text":"K_x plus L,"},{"Start":"04:21.360 ","End":"04:27.455","Text":"M_x plus N. We\u0027ve gone through 14 letters of the alphabet, that\u0027s 14 unknowns."},{"Start":"04:27.455 ","End":"04:32.345","Text":"It\u0027s quite a bit of trouble to actually solve."},{"Start":"04:32.345 ","End":"04:34.490","Text":"We haven\u0027t even got the expression."},{"Start":"04:34.490 ","End":"04:36.335","Text":"We don\u0027t know what P(x) is."},{"Start":"04:36.335 ","End":"04:40.970","Text":"All I wanted to show you is the rules for how we convert."},{"Start":"04:40.970 ","End":"04:43.700","Text":"Once we factorize the denominator,"},{"Start":"04:43.700 ","End":"04:49.500","Text":"the general form of the partial fraction with the constants."},{"Start":"04:50.080 ","End":"04:54.980","Text":"Now I\u0027d like to try and formalize this a little bit."},{"Start":"04:54.980 ","End":"04:56.990","Text":"I\u0027ll get myself some space,"},{"Start":"04:56.990 ","End":"05:03.080","Text":"but I want to keep an eye on this example as far as possible."},{"Start":"05:03.960 ","End":"05:06.650","Text":"What I can say is this,"},{"Start":"05:06.650 ","End":"05:14.540","Text":"each one of these factors with the power contributes a bit of the partial fraction,"},{"Start":"05:14.540 ","End":"05:20.570","Text":"like I\u0027d say that this contributes these 3 terms and the x plus 4 contributes this,"},{"Start":"05:20.570 ","End":"05:25.820","Text":"and this contributes these 2 and I\u0027ll use the word contributes."},{"Start":"05:25.820 ","End":"05:30.405","Text":"I\u0027ll write down the general form for example,"},{"Start":"05:30.405 ","End":"05:34.445","Text":"let me take the x plus 4 as an example."},{"Start":"05:34.445 ","End":"05:42.155","Text":"Or in general, if I have x plus a in the denominator,"},{"Start":"05:42.155 ","End":"05:47.015","Text":"it will contribute in the partial fractions,"},{"Start":"05:47.015 ","End":"05:48.979","Text":"something of the form,"},{"Start":"05:48.979 ","End":"05:50.960","Text":"not the same area as this."},{"Start":"05:50.960 ","End":"05:59.570","Text":"I\u0027ll just say A/x plus a, some constant."},{"Start":"05:59.570 ","End":"06:05.230","Text":"If I have something like (x"},{"Start":"06:05.230 ","End":"06:13.490","Text":"plus a)^k in the denominator."},{"Start":"06:13.490 ","End":"06:18.190","Text":"As I do like in the first one it will contribute these 3."},{"Start":"06:18.190 ","End":"06:22.900","Text":"Now, I could do it with a general k,"},{"Start":"06:22.900 ","End":"06:29.440","Text":"but I\u0027ll start off with a particular example."},{"Start":"06:29.440 ","End":"06:30.940","Text":"Then we\u0027ll see if we can generalize it."},{"Start":"06:30.940 ","End":"06:34.120","Text":"Let\u0027s make k=3."},{"Start":"06:34.120 ","End":"06:36.985","Text":"If I have (x plus a)^3."},{"Start":"06:36.985 ","End":"06:38.500","Text":"These are separate scenarios."},{"Start":"06:38.500 ","End":"06:41.740","Text":"These are not to be confused and let me put it maybe a little asterisk on each one."},{"Start":"06:41.740 ","End":"06:43.690","Text":"This is first scenario,"},{"Start":"06:43.690 ","End":"06:47.000","Text":"the second scenario, and it\u0027ll be 4 of these."},{"Start":"06:47.000 ","End":"06:51.300","Text":"I\u0027ll stick with the"},{"Start":"06:51.300 ","End":"06:55.240","Text":"3 and then I\u0027ll mention that we can generalize it to k or any whole number."},{"Start":"06:55.240 ","End":"06:58.780","Text":"Then we have 3 constants."},{"Start":"06:58.780 ","End":"07:01.165","Text":"Like I do here, A, B, and C,"},{"Start":"07:01.165 ","End":"07:05.560","Text":"but in general, I don\u0027t know how many they\u0027ll be."},{"Start":"07:05.560 ","End":"07:12.295","Text":"I\u0027m going to write it as A_1/x plus a and another constant,"},{"Start":"07:12.295 ","End":"07:17.630","Text":"A_2 over (x plus a)^2,"},{"Start":"07:17.630 ","End":"07:24.738","Text":"and another constant, A_3/(x_a)^3."},{"Start":"07:24.738 ","End":"07:31.710","Text":"I\u0027d like to be able to say and generalize this for instead of 3,"},{"Start":"07:31.710 ","End":"07:35.490","Text":"you have k. Maybe I\u0027ll do that now."},{"Start":"07:35.490 ","End":"07:38.880","Text":"I just wanted you to see it in a concrete example with 3."},{"Start":"07:38.880 ","End":"07:48.100","Text":"Now, I\u0027m going to change that 3 to a k. I\u0027ll erase that and I\u0027ll erase the last term."},{"Start":"07:48.950 ","End":"07:51.840","Text":"If we have k here,"},{"Start":"07:51.840 ","End":"07:56.160","Text":"then we need to take k times A1,"},{"Start":"07:56.160 ","End":"08:04.470","Text":"A2, and then I need to use dot dot dot notation because I don\u0027t know what k is."},{"Start":"08:04.470 ","End":"08:08.745","Text":"But I go up to some other constant,"},{"Start":"08:08.745 ","End":"08:16.230","Text":"Ak and x plus a to the power of k. This dot dot dot means that"},{"Start":"08:16.230 ","End":"08:23.700","Text":"I take all powers of x plus a from 1 up to k. In practice,"},{"Start":"08:23.700 ","End":"08:26.310","Text":"I don\u0027t have to write A1, A2, A3."},{"Start":"08:26.310 ","End":"08:27.690","Text":"I could write B, C,"},{"Start":"08:27.690 ","End":"08:30.240","Text":"D, or whatever letters are available."},{"Start":"08:30.240 ","End":"08:33.720","Text":"Here because I don\u0027t know how many letters I need,"},{"Start":"08:33.720 ","End":"08:36.120","Text":"I\u0027m writing it like this."},{"Start":"08:36.120 ","End":"08:46.185","Text":"The next step thing case to consider would be like here if I have the x^2 plus 9."},{"Start":"08:46.185 ","End":"08:50.115","Text":"In general, we would have"},{"Start":"08:50.115 ","End":"08:57.000","Text":"x^2 plus bx plus C. B could be 0 like here,"},{"Start":"08:57.000 ","End":"08:59.025","Text":"and C could be 9."},{"Start":"08:59.025 ","End":"09:00.630","Text":"This will"},{"Start":"09:00.630 ","End":"09:10.380","Text":"contribute Ax plus B or any other 2 letters"},{"Start":"09:10.380 ","End":"09:12.120","Text":"that don\u0027t clash."},{"Start":"09:12.120 ","End":"09:17.580","Text":"In our case, it came out to be Gx plus H over"},{"Start":"09:17.580 ","End":"09:20.939","Text":"that same x^2 plus bx"},{"Start":"09:20.939 ","End":"09:26.460","Text":"plus C. I\u0027m just reminding you that we assume this is an irreducible,"},{"Start":"09:26.460 ","End":"09:28.680","Text":"that it can\u0027t be factorized,"},{"Start":"09:28.680 ","End":"09:31.090","Text":"that it has no roots."},{"Start":"09:33.620 ","End":"09:37.080","Text":"The final case, I mean,"},{"Start":"09:37.080 ","End":"09:41.550","Text":"I guess I\u0027m just going to have to yeah."},{"Start":"09:41.550 ","End":"09:44.415","Text":"If I scroll, I\u0027m just going to go off the board, never mind."},{"Start":"09:44.415 ","End":"09:50.910","Text":"Just look at the x^2 plus 2x plus 5^3."},{"Start":"09:50.910 ","End":"09:57.390","Text":"If we have x^2 plus bx plus C,"},{"Start":"09:57.390 ","End":"10:01.290","Text":"and this time I won\u0027t use 3 for instance,"},{"Start":"10:01.290 ","End":"10:05.385","Text":"I\u0027ll use an actual variable,"},{"Start":"10:05.385 ","End":"10:07.840","Text":"I\u0027ll choose k again."},{"Start":"10:07.840 ","End":"10:15.840","Text":"This contributes Ax plus"},{"Start":"10:15.840 ","End":"10:22.200","Text":"B over x^2 plus bx plus C. Not quite."},{"Start":"10:22.200 ","End":"10:24.390","Text":"Let me just put subscripts A_1,"},{"Start":"10:24.390 ","End":"10:28.680","Text":"B_1 because I need to represent all powers."},{"Start":"10:28.680 ","End":"10:30.570","Text":"Just like here, I had power of 1,"},{"Start":"10:30.570 ","End":"10:32.250","Text":"power of 2, power of 3."},{"Start":"10:32.250 ","End":"10:34.559","Text":"Next one would be another constant,"},{"Start":"10:34.559 ","End":"10:41.790","Text":"A_2 plus B_2,"},{"Start":"10:41.790 ","End":"10:47.220","Text":"and then x^2 plus bx plus C to the power of 2, and then dot dot dot."},{"Start":"10:47.220 ","End":"10:53.145","Text":"All powers from 1 to k the last one will be a pair of constants,"},{"Start":"10:53.145 ","End":"10:57.210","Text":"A_k times x and another constant B_k,"},{"Start":"10:57.210 ","End":"11:06.460","Text":"another linear term with x^2 plus bx plus C to the power of k."},{"Start":"11:16.520 ","End":"11:17.610","Text":"I wanted to have this table written down and I should have really labeled the columns."},{"Start":"11:17.610 ","End":"11:19.515","Text":"I cleared a bit of space here"},{"Start":"11:19.515 ","End":"11:22.080","Text":"and I wish I had"},{"Start":"11:22.080 ","End":"11:26.820","Text":"the expression for Q(x). Look there it is."},{"Start":"11:26.820 ","End":"11:28.440","Text":"I guess I got my wish."},{"Start":"11:28.440 ","End":"11:34.530","Text":"Now I\u0027m going to put some headers on these columns and try and make it a bit more formal."},{"Start":"11:34.530 ","End":"11:40.530","Text":"I\u0027ll call this one the factor in the denominator,"},{"Start":"11:40.530 ","End":"11:44.190","Text":"then the arrow will stand for contributes."},{"Start":"11:44.190 ","End":"11:49.640","Text":"This, I\u0027ll call the term or"},{"Start":"11:49.640 ","End":"11:56.375","Text":"terms in the partial fraction decomposition."},{"Start":"11:56.375 ","End":"11:59.364","Text":"P. F is partial fraction."},{"Start":"11:59.364 ","End":"12:02.130","Text":"Another way, I could\u0027ve shortened the table."},{"Start":"12:02.130 ","End":"12:07.170","Text":"I mean, x plus a is like x plus a to the k with k=1."},{"Start":"12:07.170 ","End":"12:11.880","Text":"Similarly, this is the same as this if k is 1 special case,"},{"Start":"12:11.880 ","End":"12:18.900","Text":"but it\u0027s more convenient to separate the case when k=1 as a special case."},{"Start":"12:18.900 ","End":"12:20.580","Text":"We really have 4 cases."},{"Start":"12:20.580 ","End":"12:27.390","Text":"I\u0027d like to show you how we would\u0027ve got to this form using this table."},{"Start":"12:27.390 ","End":"12:29.940","Text":"Remember, this is Q(x),"},{"Start":"12:29.940 ","End":"12:33.165","Text":"and this is the denominator."},{"Start":"12:33.165 ","End":"12:36.345","Text":"I think I\u0027ll use colors."},{"Start":"12:36.345 ","End":"12:41.070","Text":"I\u0027ll color the 4 cases,"},{"Start":"12:41.070 ","End":"12:46.710","Text":"each in a different color like so."},{"Start":"12:46.710 ","End":"12:53.520","Text":"Then I\u0027ll look at the factors in the denominator and decide which type each one is."},{"Start":"12:53.520 ","End":"12:55.320","Text":"Then I\u0027ll look at each one and say,"},{"Start":"12:55.320 ","End":"13:00.015","Text":"this one is one of these so I need"},{"Start":"13:00.015 ","End":"13:06.570","Text":"some constants over well x minus 2,"},{"Start":"13:06.570 ","End":"13:09.690","Text":"in this case, up to x minus 2^3."},{"Start":"13:09.690 ","End":"13:14.020","Text":"That gives me this bit here."},{"Start":"13:14.180 ","End":"13:18.120","Text":"Then I look at this and I see that it\u0027s one of those and I just"},{"Start":"13:18.120 ","End":"13:21.900","Text":"need a single constant over x plus 4."},{"Start":"13:21.900 ","End":"13:24.495","Text":"That gives me this."},{"Start":"13:24.495 ","End":"13:26.760","Text":"Then I see, and I have another one of those,"},{"Start":"13:26.760 ","End":"13:27.900","Text":"but to the power of 2,"},{"Start":"13:27.900 ","End":"13:33.060","Text":"so I only need two terms like so."},{"Start":"13:33.060 ","End":"13:36.345","Text":"Then I see this which is a gray one,"},{"Start":"13:36.345 ","End":"13:40.725","Text":"which just needs a linear over the irreducible quadratic."},{"Start":"13:40.725 ","End":"13:43.290","Text":"That\u0027s this bit here."},{"Start":"13:43.290 ","End":"13:49.635","Text":"Then finally, I have one of these which is an irreducible quadratic to a power,"},{"Start":"13:49.635 ","End":"13:52.440","Text":"and I need all powers."},{"Start":"13:52.440 ","End":"13:56.410","Text":"That gives me these here."},{"Start":"13:57.410 ","End":"13:59.775","Text":"I think you get the idea."},{"Start":"13:59.775 ","End":"14:05.320","Text":"I hope you do and that\u0027s basically it."},{"Start":"14:05.600 ","End":"14:09.330","Text":"All we need now is to see how it all fits together."},{"Start":"14:09.330 ","End":"14:13.470","Text":"This was just step 2 of a 3 step program,"},{"Start":"14:13.470 ","End":"14:20.310","Text":"how to take a rational expression and reduce it to partial fractions."},{"Start":"14:20.310 ","End":"14:22.140","Text":"To see how it all fits together,"},{"Start":"14:22.140 ","End":"14:24.930","Text":"all we need are some more examples,"},{"Start":"14:24.930 ","End":"14:29.110","Text":"and we\u0027ll do that after a break."}],"ID":5269},{"Watched":false,"Name":"Worked Example","Duration":"21m 24s","ChapterTopicVideoID":5289,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5289.jpeg","UploadDate":"2020-09-30T13:49:44.8900000","DurationForVideoObject":"PT21M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.745","Text":"In this exercise, we have to"},{"Start":"00:02.745 ","End":"00:06.450","Text":"determine the partial fraction decomposition of the following."},{"Start":"00:06.450 ","End":"00:08.460","Text":"It\u0027s a rational expression."},{"Start":"00:08.460 ","End":"00:11.580","Text":"It\u0027s a polynomial over polynomial."},{"Start":"00:11.580 ","End":"00:16.155","Text":"The degree in the numerator is less than that in the denominator."},{"Start":"00:16.155 ","End":"00:20.128","Text":"The first thing we want to do is factorize the denominator."},{"Start":"00:20.128 ","End":"00:22.545","Text":"Clearly, x can be taken out."},{"Start":"00:22.545 ","End":"00:25.755","Text":"If we do that, then we\u0027re left with x^2 minus 1."},{"Start":"00:25.755 ","End":"00:27.903","Text":"This is the difference of squares,"},{"Start":"00:27.903 ","End":"00:32.430","Text":"so the x^2 minus 1 can be written as x minus 1 x plus 1."},{"Start":"00:32.430 ","End":"00:35.520","Text":"Now the denominator is fully factored."},{"Start":"00:35.520 ","End":"00:39.650","Text":"Notice that there\u0027s three different linear factors,"},{"Start":"00:39.650 ","End":"00:42.110","Text":"and that\u0027s the simplest case."},{"Start":"00:42.110 ","End":"00:46.040","Text":"We simply have a letter A,"},{"Start":"00:46.040 ","End":"00:47.690","Text":"B, and C. Well, I\u0027ll just show you."},{"Start":"00:47.690 ","End":"00:51.230","Text":"A over the x factor,"},{"Start":"00:51.230 ","End":"00:54.110","Text":"B over the x minus 1 factor,"},{"Start":"00:54.110 ","End":"00:56.675","Text":"and C over the x plus 1."},{"Start":"00:56.675 ","End":"01:04.955","Text":"We get rid of fractions by multiplying by the common denominator, which is this."},{"Start":"01:04.955 ","End":"01:07.445","Text":"What we get then is,"},{"Start":"01:07.445 ","End":"01:11.330","Text":"A is multiplied by the missing factors,"},{"Start":"01:11.330 ","End":"01:13.040","Text":"which is everything without the x,"},{"Start":"01:13.040 ","End":"01:14.930","Text":"x minus 1 x plus 1,"},{"Start":"01:14.930 ","End":"01:19.850","Text":"B goes with the missing ones of this and this,"},{"Start":"01:19.850 ","End":"01:24.420","Text":"and C goes with this one and this one."},{"Start":"01:24.420 ","End":"01:25.880","Text":"At this point,"},{"Start":"01:25.880 ","End":"01:30.550","Text":"we can already start substituting values."},{"Start":"01:30.550 ","End":"01:35.090","Text":"What we\u0027ll choose is things that make one of these 0."},{"Start":"01:35.090 ","End":"01:38.280","Text":"We have x, we have x minus 1, and x plus 1."},{"Start":"01:38.280 ","End":"01:45.090","Text":"We start with x=0, that will make this term 0 and this one 0."},{"Start":"01:45.090 ","End":"01:48.375","Text":"So what we\u0027re left with, really,"},{"Start":"01:48.375 ","End":"01:55.115","Text":"is, this one and this one are 0, this one is left."},{"Start":"01:55.115 ","End":"01:59.075","Text":"If x is 0, we get minus 1 times plus 1 is minus 1."},{"Start":"01:59.075 ","End":"02:02.420","Text":"So basically, we just get minus 1 is minus A."},{"Start":"02:02.420 ","End":"02:06.050","Text":"These two are 0 and can be ignored, and A=1."},{"Start":"02:06.050 ","End":"02:09.925","Text":"Next, we should let x either be 1 or minus 1."},{"Start":"02:09.925 ","End":"02:12.330","Text":"I\u0027ll go with the 1 first."},{"Start":"02:12.330 ","End":"02:16.670","Text":"That 1 will make this 0 and this 0,"},{"Start":"02:16.670 ","End":"02:18.020","Text":"the x minus 1 terms."},{"Start":"02:18.020 ","End":"02:20.165","Text":"All we\u0027re left with is the B."},{"Start":"02:20.165 ","End":"02:23.160","Text":"If x is 1, 1 times 1 plus 1 is 2,"},{"Start":"02:23.160 ","End":"02:24.757","Text":"so we\u0027ve got 2B."},{"Start":"02:24.757 ","End":"02:27.390","Text":"On the left we have,"},{"Start":"02:27.390 ","End":"02:30.630","Text":"if we put in 1,"},{"Start":"02:30.630 ","End":"02:32.910","Text":"we have 1 plus 1 minus 1 is 1,"},{"Start":"02:32.910 ","End":"02:35.175","Text":"so 2B=1B is a half."},{"Start":"02:35.175 ","End":"02:37.910","Text":"It\u0027s just very simple algebra."},{"Start":"02:37.910 ","End":"02:40.485","Text":"Then we\u0027re going to let x equal minus 1."},{"Start":"02:40.485 ","End":"02:44.685","Text":"If we do that, it\u0027s fairly straightforward."},{"Start":"02:44.685 ","End":"02:50.820","Text":"Just substitute the minus 1 here and then these first two"},{"Start":"02:50.820 ","End":"02:56.720","Text":"become 0 and we just can conclude what C is from here, is minus a half."},{"Start":"02:56.720 ","End":"02:59.460","Text":"The last thing we do is to take A, B,"},{"Start":"02:59.460 ","End":"03:02.240","Text":"and C, and put them here instead of A, B,"},{"Start":"03:02.240 ","End":"03:05.330","Text":"and C. This gives us our final answer,"},{"Start":"03:05.330 ","End":"03:10.500","Text":"which is the partial fraction decomposition. We\u0027re done."}],"ID":5270},{"Watched":false,"Name":"Exercise 1","Duration":"2m 15s","ChapterTopicVideoID":5265,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5265.jpeg","UploadDate":"2016-03-07T09:05:23.4700000","DurationForVideoObject":"PT2M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this exercise, we have to determine"},{"Start":"00:02.550 ","End":"00:06.720","Text":"the partial fraction decomposition of the following expression."},{"Start":"00:06.720 ","End":"00:10.780","Text":"It\u0027s a rational expression polynomial over polynomial."},{"Start":"00:12.800 ","End":"00:17.340","Text":"What we first of all do is factorize whatever we can."},{"Start":"00:17.340 ","End":"00:19.530","Text":"The denominator is a difference of squares."},{"Start":"00:19.530 ","End":"00:21.330","Text":"So it\u0027s X minus 2,"},{"Start":"00:21.330 ","End":"00:24.000","Text":"X plus 2 because 4 is 2^2."},{"Start":"00:24.000 ","End":"00:28.335","Text":"Then we know that the general form of this,"},{"Start":"00:28.335 ","End":"00:36.490","Text":"when it\u0027s all linear factors is something over X minus 2 plus something over X plus 2."},{"Start":"00:37.370 ","End":"00:43.000","Text":"We have to find out what these constants A and B are."},{"Start":"00:43.370 ","End":"00:50.375","Text":"We multiply both sides by the denominator to get rid of the fractions."},{"Start":"00:50.375 ","End":"00:58.505","Text":"Basically here it comes out to be X plus 2 is the missing factor,"},{"Start":"00:58.505 ","End":"01:04.470","Text":"and here X minus 2 is what\u0027s missing so we get the following equation."},{"Start":"01:04.470 ","End":"01:06.440","Text":"Actually, it\u0027s not an equation,"},{"Start":"01:06.440 ","End":"01:08.840","Text":"it\u0027s an identity. It\u0027s true for all x."},{"Start":"01:08.840 ","End":"01:11.315","Text":"If it\u0027s true for all x,"},{"Start":"01:11.315 ","End":"01:14.660","Text":"that means we can substitute whatever value we want."},{"Start":"01:14.660 ","End":"01:19.685","Text":"Now, the good values to substitute where possible is to get something to be 0."},{"Start":"01:19.685 ","End":"01:21.860","Text":"Now if we put x=2,"},{"Start":"01:21.860 ","End":"01:24.150","Text":"this bit will come out to be 0."},{"Start":"01:24.150 ","End":"01:26.760","Text":"We\u0027ll get that 1 from here,"},{"Start":"01:26.760 ","End":"01:32.910","Text":"equals A times 2 plus 2 and b time 0."},{"Start":"01:32.910 ","End":"01:36.195","Text":"We get that 4A is 1,"},{"Start":"01:36.195 ","End":"01:38.755","Text":"A is a 1/4."},{"Start":"01:38.755 ","End":"01:42.395","Text":"Then a similar thing if we put x= minus 2,"},{"Start":"01:42.395 ","End":"01:49.510","Text":"this thing becomes 0 and we get that 1= minus 2,"},{"Start":"01:49.510 ","End":"01:51.830","Text":"minus 2, which is minus 4."},{"Start":"01:51.830 ","End":"01:54.380","Text":"If 1 is minus 4 times B,"},{"Start":"01:54.380 ","End":"01:57.395","Text":"then B is minus 1/4."},{"Start":"01:57.395 ","End":"02:04.310","Text":"Finally, we just place these constants A and B back in this expression,"},{"Start":"02:04.310 ","End":"02:06.800","Text":"and then we get the partial decomposition,"},{"Start":"02:06.800 ","End":"02:11.179","Text":"which is 1/4 over x minus 2,"},{"Start":"02:11.179 ","End":"02:16.390","Text":"plus minus1/4 over x plus 2."}],"ID":5271},{"Watched":false,"Name":"Exercise 2","Duration":"2m 8s","ChapterTopicVideoID":5266,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5266.jpeg","UploadDate":"2016-03-07T09:05:45.3430000","DurationForVideoObject":"PT2M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this exercise we have to determine"},{"Start":"00:02.460 ","End":"00:06.420","Text":"the partial fraction decomposition of the following expression."},{"Start":"00:06.420 ","End":"00:09.570","Text":"It\u0027s a rational expression polynomial over polynomial."},{"Start":"00:09.570 ","End":"00:12.840","Text":"The first thing we want to to do is factorize."},{"Start":"00:12.840 ","End":"00:16.590","Text":"The numerator is already factorized."},{"Start":"00:16.590 ","End":"00:23.520","Text":"The denominator could be factorized by taking x out the brackets and we get x, x plus 5."},{"Start":"00:23.520 ","End":"00:27.645","Text":"Now we have the denominator as a product of linear terms,"},{"Start":"00:27.645 ","End":"00:30.780","Text":"so we know the general form of the decomposition,"},{"Start":"00:30.780 ","End":"00:36.220","Text":"something over x and something else over x plus 5."},{"Start":"00:37.420 ","End":"00:41.580","Text":"Just get some more space here."},{"Start":"00:42.080 ","End":"00:45.820","Text":"What we can do is multiply both sides by x,"},{"Start":"00:45.820 ","End":"00:48.460","Text":"x plus 5 to get rid of fractions."},{"Start":"00:48.460 ","End":"00:54.445","Text":"Here we just get the 5 minus x and here A is multiplied by the missing factor x plus 5,"},{"Start":"00:54.445 ","End":"00:59.950","Text":"and B is missing the x so this is what we get."},{"Start":"00:59.950 ","End":"01:02.080","Text":"This is not an equation,"},{"Start":"01:02.080 ","End":"01:04.765","Text":"it\u0027s an identity, meaning it\u0027s true for all x."},{"Start":"01:04.765 ","End":"01:07.735","Text":"We can substitute whatever value of x we like."},{"Start":"01:07.735 ","End":"01:09.640","Text":"A and B are what we\u0027re looking for."},{"Start":"01:09.640 ","End":"01:15.740","Text":"If we let x equal say 0,"},{"Start":"01:15.740 ","End":"01:18.120","Text":"then this thing disappears."},{"Start":"01:18.120 ","End":"01:23.275","Text":"That\u0027s a general strategy to substitute something that will make a term disappear."},{"Start":"01:23.275 ","End":"01:27.570","Text":"Then we get 5 minus 0,"},{"Start":"01:27.570 ","End":"01:33.450","Text":"which is 5 is A times 0 plus 5 plus B times 0."},{"Start":"01:33.450 ","End":"01:35.010","Text":"This of course, is nothing,"},{"Start":"01:35.010 ","End":"01:37.050","Text":"so we have 5 equals 5A,"},{"Start":"01:37.050 ","End":"01:38.790","Text":"which makes A equals 1."},{"Start":"01:38.790 ","End":"01:44.280","Text":"To get this 0 we let x equal minus 5 then we get 5 minus,"},{"Start":"01:44.280 ","End":"01:45.855","Text":"minus 5 is 10,"},{"Start":"01:45.855 ","End":"01:47.595","Text":"and this one is 0,"},{"Start":"01:47.595 ","End":"01:49.470","Text":"and this time x is minus 5,"},{"Start":"01:49.470 ","End":"01:51.390","Text":"so minus 5B is 10,"},{"Start":"01:51.390 ","End":"01:52.860","Text":"B is minus 2."},{"Start":"01:52.860 ","End":"01:59.900","Text":"The last thing is to put A and B back here with the actual values and then we get"},{"Start":"01:59.900 ","End":"02:07.890","Text":"that the answer is the partial decomposition is this. We\u0027re done."}],"ID":5272},{"Watched":false,"Name":"Exercise 3","Duration":"2m 56s","ChapterTopicVideoID":5267,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5267.jpeg","UploadDate":"2016-03-07T09:06:15.1500000","DurationForVideoObject":"PT2M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"In this exercise, we have to do a partial fraction decomposition."},{"Start":"00:04.725 ","End":"00:08.295","Text":"This here is a rational expression."},{"Start":"00:08.295 ","End":"00:14.160","Text":"The first thing we want to do is to factorize both numerator and denominator."},{"Start":"00:14.160 ","End":"00:15.870","Text":"The numerator\u0027s already factorized,"},{"Start":"00:15.870 ","End":"00:21.775","Text":"so we need to factorize the denominator as a quadratic and what we\u0027re going to"},{"Start":"00:21.775 ","End":"00:29.735","Text":"do is use the principle that if you have a quadratic with leading coefficient 1,"},{"Start":"00:29.735 ","End":"00:32.285","Text":"that\u0027s the a, a=1."},{"Start":"00:32.285 ","End":"00:35.930","Text":"Then if we know the 2 roots,"},{"Start":"00:35.930 ","End":"00:39.515","Text":"the solutions of the quadratic equation equal 0,"},{"Start":"00:39.515 ","End":"00:40.820","Text":"then we can put them here,"},{"Start":"00:40.820 ","End":"00:44.790","Text":"and this is how it factorizes."},{"Start":"00:45.290 ","End":"00:47.790","Text":"We let this x^2 plus 5,"},{"Start":"00:47.790 ","End":"00:51.890","Text":"x plus 6 equals 0 solve the quadratic equation,"},{"Start":"00:51.890 ","End":"00:58.460","Text":"this bit I am omitting because you already know how to solve quadratic equations."},{"Start":"00:58.460 ","End":"01:02.750","Text":"I\u0027m just telling you the answer is minus 3 or minus 2."},{"Start":"01:02.750 ","End":"01:08.500","Text":"Then I put those in here and what I get of course,"},{"Start":"01:08.500 ","End":"01:18.040","Text":"is plus 3 and plus 2 because it\u0027s x minus minus 3 and x minus minus 2."},{"Start":"01:18.200 ","End":"01:21.640","Text":"Now that I have it factorized,"},{"Start":"01:21.640 ","End":"01:24.400","Text":"then I can write"},{"Start":"01:24.400 ","End":"01:31.765","Text":"the denominator as this and now I know the general shape of the decomposition."},{"Start":"01:31.765 ","End":"01:37.970","Text":"It\u0027s going to be something over x plus 3 and something over x plus 2."},{"Start":"01:37.970 ","End":"01:39.885","Text":"We have to find now,"},{"Start":"01:39.885 ","End":"01:42.310","Text":"what are a and b."},{"Start":"01:42.310 ","End":"01:46.540","Text":"We multiply both sides by this denominator, x plus 3,"},{"Start":"01:46.540 ","End":"01:51.200","Text":"x plus 2 and we get on the left just the x."},{"Start":"01:51.200 ","End":"01:56.840","Text":"Here we multiply a by the x plus 2 and b by x plus 3,"},{"Start":"01:56.840 ","End":"01:59.345","Text":"in each case the missing factor."},{"Start":"01:59.345 ","End":"02:02.810","Text":"This equation here is really an identity,"},{"Start":"02:02.810 ","End":"02:04.230","Text":"which means it\u0027s true for all x."},{"Start":"02:04.230 ","End":"02:06.275","Text":"We can substitute whatever we want."},{"Start":"02:06.275 ","End":"02:08.870","Text":"Good values would be minus 2 and minus"},{"Start":"02:08.870 ","End":"02:11.870","Text":"3 because then we\u0027d get a 0 and that makes it easy."},{"Start":"02:11.870 ","End":"02:20.400","Text":"First off we\u0027ll substitute the minus 3 that will make the b coefficient 0."},{"Start":"02:20.400 ","End":"02:22.980","Text":"We just get minus 3,"},{"Start":"02:22.980 ","End":"02:26.460","Text":"which is x, is a times minus 1,"},{"Start":"02:26.460 ","End":"02:32.680","Text":"which is x plus 2 minus 3 plus 2 is minus 1 and that gives us a equals 3."},{"Start":"02:32.680 ","End":"02:36.095","Text":"Similarly, if we do the minus 2,"},{"Start":"02:36.095 ","End":"02:40.845","Text":"this thing comes out 0 minus 2 plus 3 is 1."},{"Start":"02:40.845 ","End":"02:45.815","Text":"We get this equation and the solution is b equals minus 2."},{"Start":"02:45.815 ","End":"02:49.091","Text":"Then we put a and b in here,"},{"Start":"02:49.091 ","End":"02:51.949","Text":"and so that gives us our answer,"},{"Start":"02:51.949 ","End":"02:54.133","Text":"which is this."},{"Start":"02:54.133 ","End":"02:56.440","Text":"We are done."}],"ID":5273},{"Watched":false,"Name":"Exercise 4","Duration":"4m 40s","ChapterTopicVideoID":5268,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5268.jpeg","UploadDate":"2016-03-07T09:07:00.6570000","DurationForVideoObject":"PT4M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.999","Text":"In this exercise, we want to decompose this expression into partial fractions."},{"Start":"00:05.999 ","End":"00:09.000","Text":"It\u0027s a rational expression, polynomial over polynomial."},{"Start":"00:09.000 ","End":"00:13.000","Text":"The first thing to do is to factorize."},{"Start":"00:15.650 ","End":"00:20.430","Text":"What I\u0027m going to do is to factorize a quadratic,"},{"Start":"00:20.430 ","End":"00:24.030","Text":"I\u0027m going to use the theorem or principle"},{"Start":"00:24.030 ","End":"00:28.035","Text":"that if you have a quadratic with leading coefficient 1,"},{"Start":"00:28.035 ","End":"00:31.365","Text":"which I do here now that I\u0027ve taken the 2 out,"},{"Start":"00:31.365 ","End":"00:36.375","Text":"then the idea is to find the root of this quadratic,"},{"Start":"00:36.375 ","End":"00:37.830","Text":"let\u0027s say x_1 and x_2,"},{"Start":"00:37.830 ","End":"00:42.840","Text":"and then you write it as x minus x_1, x minus x_2."},{"Start":"00:42.840 ","End":"00:47.810","Text":"We want to solve this quadratic equation by setting it."},{"Start":"00:47.810 ","End":"00:51.755","Text":"It\u0027s a polynomial, but we make it an equation by setting it equal to 0,"},{"Start":"00:51.755 ","End":"00:53.915","Text":"and then we solve it this part,"},{"Start":"00:53.915 ","End":"00:57.440","Text":"omitting the solution of a quadratic equation."},{"Start":"00:57.440 ","End":"01:00.120","Text":"You can do it on your own."},{"Start":"01:00.610 ","End":"01:03.440","Text":"Don\u0027t want to waste time here."},{"Start":"01:03.440 ","End":"01:08.655","Text":"The solutions come out to be minus 1/2 and 2."},{"Start":"01:08.655 ","End":"01:11.500","Text":"Working in decimals today."},{"Start":"01:11.500 ","End":"01:13.430","Text":"Now that I have these two,"},{"Start":"01:13.430 ","End":"01:15.440","Text":"I plug that in here."},{"Start":"01:15.440 ","End":"01:21.985","Text":"what we get is that this quadratic polynomial is equal to this."},{"Start":"01:21.985 ","End":"01:25.590","Text":"I then substitute that in here."},{"Start":"01:25.590 ","End":"01:28.660","Text":"Scroll down first."},{"Start":"01:28.660 ","End":"01:30.800","Text":"When I put it in here,"},{"Start":"01:30.800 ","End":"01:33.670","Text":"we then get 1/2,"},{"Start":"01:33.670 ","End":"01:37.445","Text":"8x minus 1, and this decomposes like this."},{"Start":"01:37.445 ","End":"01:41.690","Text":"Now, I\u0027m going to forget about the 1/2 for the moment,"},{"Start":"01:41.690 ","End":"01:44.665","Text":"but we have to remember to come back to it."},{"Start":"01:44.665 ","End":"01:49.100","Text":"I\u0027ll just highlight it so I don\u0027t forget that I\u0027m sort of omitting it for now,"},{"Start":"01:49.100 ","End":"01:50.780","Text":"but then I have to restore it."},{"Start":"01:50.780 ","End":"01:54.200","Text":"We\u0027ll just take this part and write it as something"},{"Start":"01:54.200 ","End":"01:58.770","Text":"over x plus 1/2 and something else over x minus 2."},{"Start":"01:58.770 ","End":"02:01.530","Text":"The way we do this is, first of all,"},{"Start":"02:01.530 ","End":"02:06.800","Text":"we get rid of the denominators"},{"Start":"02:06.800 ","End":"02:12.990","Text":"of the fraction by multiplying by all of this denominator."},{"Start":"02:13.880 ","End":"02:19.400","Text":"On the left, we just have the 8x minus 1 because we multiply it by this times this."},{"Start":"02:19.400 ","End":"02:22.145","Text":"On the right here,"},{"Start":"02:22.145 ","End":"02:27.210","Text":"the x plus 1/2 cancels and we\u0027re left with x minus 2 multiplied by A,"},{"Start":"02:27.210 ","End":"02:29.930","Text":"and B gets multiplied with the other one,"},{"Start":"02:29.930 ","End":"02:31.055","Text":"the x plus 1/2,"},{"Start":"02:31.055 ","End":"02:33.350","Text":"because the x minus 2 cancels."},{"Start":"02:33.350 ","End":"02:34.970","Text":"We get this equation,"},{"Start":"02:34.970 ","End":"02:36.740","Text":"which is actually an identity,"},{"Start":"02:36.740 ","End":"02:38.420","Text":"meaning it\u0027s true for all x,"},{"Start":"02:38.420 ","End":"02:41.000","Text":"so we can substitute wherever x is convenient."},{"Start":"02:41.000 ","End":"02:44.260","Text":"For example, 2 would be a good value to substitute,"},{"Start":"02:44.260 ","End":"02:50.610","Text":"because, or minus 1/2,"},{"Start":"02:50.610 ","End":"02:52.750","Text":"we\u0027ll do the 2 later."},{"Start":"02:52.940 ","End":"02:56.640","Text":"If we put minus 1/2,"},{"Start":"02:56.640 ","End":"02:59.325","Text":"then this thing comes out 0 here,"},{"Start":"02:59.325 ","End":"03:01.050","Text":"and here we have minus 1/2,"},{"Start":"03:01.050 ","End":"03:03.600","Text":"minus 2 is minus 2 1/2,"},{"Start":"03:03.600 ","End":"03:06.510","Text":"and on the other side,"},{"Start":"03:06.510 ","End":"03:08.250","Text":"minus 1/2 times 8 is minus 4,"},{"Start":"03:08.250 ","End":"03:09.690","Text":"minus 1 is minus 5,"},{"Start":"03:09.690 ","End":"03:13.410","Text":"so we get that minus 2 1/2A is minus 5,"},{"Start":"03:13.410 ","End":"03:15.797","Text":"so A is minus 5 over 2 1/2,"},{"Start":"03:15.797 ","End":"03:17.310","Text":"in short, A is 2."},{"Start":"03:17.310 ","End":"03:19.095","Text":"Now we\u0027ll get to the other one,"},{"Start":"03:19.095 ","End":"03:27.845","Text":"2 is the other good value to substitute because the coefficient of A is 0 here."},{"Start":"03:27.845 ","End":"03:31.410","Text":"We\u0027re left with B times 2 plus 1/2."},{"Start":"03:31.570 ","End":"03:36.170","Text":"Here, 8 times 2 minus 1 is 15."},{"Start":"03:36.170 ","End":"03:41.325","Text":"Just, 15 over 2 1/2 is what B is, and that\u0027s 6."},{"Start":"03:41.325 ","End":"03:43.310","Text":"Now that we have A and B,"},{"Start":"03:43.310 ","End":"03:46.680","Text":"we substitute them back in here."},{"Start":"03:49.670 ","End":"03:53.175","Text":"The 1/2 that was up here,"},{"Start":"03:53.175 ","End":"03:55.750","Text":"I now restored it."},{"Start":"03:56.120 ","End":"03:59.125","Text":"Remember to do that."},{"Start":"03:59.125 ","End":"04:02.180","Text":"We could leave the answer like this,"},{"Start":"04:02.180 ","End":"04:04.205","Text":"but it could be simplified a bit."},{"Start":"04:04.205 ","End":"04:07.580","Text":"If I multiply 1/2 by each of them,"},{"Start":"04:07.580 ","End":"04:10.220","Text":"if I multiply 1/2 by this,"},{"Start":"04:10.220 ","End":"04:12.100","Text":"the 2 and the 2."},{"Start":"04:12.100 ","End":"04:14.940","Text":"Sorry, I don\u0027t want to cancel,"},{"Start":"04:14.940 ","End":"04:18.475","Text":"I want to multiply 2 by the denominator,"},{"Start":"04:18.475 ","End":"04:22.970","Text":"so that I can get rid of this decimal."},{"Start":"04:22.970 ","End":"04:26.930","Text":"2 times x plus 1/2 is 2x plus 1."},{"Start":"04:26.930 ","End":"04:28.925","Text":"When I multiply the other one,"},{"Start":"04:28.925 ","End":"04:32.950","Text":"then I use the cancellation of 2 with 6 as 3,"},{"Start":"04:32.950 ","End":"04:40.230","Text":"so this is a little bit neater than this, and we\u0027re done."}],"ID":5274},{"Watched":false,"Name":"Exercise 5","Duration":"2m ","ChapterTopicVideoID":5269,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5269.jpeg","UploadDate":"2016-03-07T09:07:20.9170000","DurationForVideoObject":"PT2M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this exercise, we have to determine"},{"Start":"00:02.310 ","End":"00:06.480","Text":"the partial fraction decomposition of the following expression."},{"Start":"00:06.480 ","End":"00:10.620","Text":"It\u0027s a rational expression, polynomial over polynomial."},{"Start":"00:10.620 ","End":"00:14.238","Text":"It\u0027s already been factorized,"},{"Start":"00:14.238 ","End":"00:17.970","Text":"so all we have to do is determine the general form."},{"Start":"00:17.970 ","End":"00:21.090","Text":"Note that there is an (x-1)^2 here."},{"Start":"00:21.090 ","End":"00:23.010","Text":"Whenever we have something to a power,"},{"Start":"00:23.010 ","End":"00:28.500","Text":"we have to represent all powers from 1 up to 2 in this case."},{"Start":"00:28.500 ","End":"00:35.050","Text":"They\u0027ll be an x-1 and an (x-1)^2 with constants above each one."},{"Start":"00:35.050 ","End":"00:38.600","Text":"The common denominator will be (x-1)^2."},{"Start":"00:38.600 ","End":"00:42.200","Text":"Multiply both sides, and we get here,"},{"Start":"00:42.200 ","End":"00:43.910","Text":"x+4, here just B,"},{"Start":"00:43.910 ","End":"00:47.230","Text":"but here we still have an x-1."},{"Start":"00:47.230 ","End":"00:49.380","Text":"We have to find A and B."},{"Start":"00:49.380 ","End":"00:52.190","Text":"Now, this is not just an equation, it\u0027s an identity."},{"Start":"00:52.190 ","End":"00:55.055","Text":"Any value of x should make this work."},{"Start":"00:55.055 ","End":"00:58.550","Text":"For example, if I put x=1,"},{"Start":"00:58.550 ","End":"01:02.210","Text":"then I get that,"},{"Start":"01:02.210 ","End":"01:06.210","Text":"1-1 is 0 here, and B is 1+4 is 5."},{"Start":"01:10.390 ","End":"01:20.108","Text":"Once I know that B is 5,"},{"Start":"01:20.108 ","End":"01:22.730","Text":"I can put another value of x."},{"Start":"01:22.730 ","End":"01:26.855","Text":"It doesn\u0027t really matter, say x=2."},{"Start":"01:26.855 ","End":"01:32.418","Text":"Here we have x+4 is 6,"},{"Start":"01:32.418 ","End":"01:35.070","Text":"B we know is 5, so 6=5+A."},{"Start":"01:38.000 ","End":"01:42.900","Text":"This is 1 because that\u0027s the x-1 and x is 2 here."},{"Start":"01:42.900 ","End":"01:51.330","Text":"This just gives us that A is (6-5)/1, and it\u0027s 1."},{"Start":"01:51.330 ","End":"01:52.650","Text":"Now that we have A and B,"},{"Start":"01:52.650 ","End":"01:58.320","Text":"we just plug them in here and this gives us our partial fraction decomposition,"},{"Start":"01:58.320 ","End":"02:00.580","Text":"and we are done."}],"ID":5275},{"Watched":false,"Name":"Exercise 6","Duration":"4m 24s","ChapterTopicVideoID":5270,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5270.jpeg","UploadDate":"2016-03-07T09:08:04.0400000","DurationForVideoObject":"PT4M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"In this exercise, we have to determine"},{"Start":"00:02.520 ","End":"00:05.730","Text":"the partial fraction decomposition of"},{"Start":"00:05.730 ","End":"00:10.050","Text":"this rational expression polynomial over polynomial."},{"Start":"00:10.050 ","End":"00:14.505","Text":"The first thing we want to do is factorize the denominator."},{"Start":"00:14.505 ","End":"00:17.460","Text":"What we do is we when have a quadratic,"},{"Start":"00:17.460 ","End":"00:20.370","Text":"is we solve the equation where the quadratic is 0,"},{"Start":"00:20.370 ","End":"00:22.570","Text":"and that\u0027s how we find the roots,"},{"Start":"00:22.570 ","End":"00:24.880","Text":"and then we use that to factorize."},{"Start":"00:24.880 ","End":"00:32.150","Text":"In other words, we start off by letting x^2+8x+16=0 and find both routes."},{"Start":"00:32.150 ","End":"00:33.230","Text":"Now as it turns out,"},{"Start":"00:33.230 ","End":"00:37.830","Text":"they both come out the same -4 and -4."},{"Start":"00:37.830 ","End":"00:40.400","Text":"If you\u0027re wondering how this happened,"},{"Start":"00:40.400 ","End":"00:43.880","Text":"it\u0027s because if you did it with the formula method,"},{"Start":"00:43.880 ","End":"00:49.100","Text":"you get -8 + or - the square root of 0 over 2."},{"Start":"00:49.100 ","End":"00:54.900","Text":"The b^2-4ac comes out 64-4 times 16,"},{"Start":"00:54.900 ","End":"00:58.520","Text":"and so both + and -0 come out the same."},{"Start":"00:58.520 ","End":"01:01.070","Text":"It comes out -8 over 2 in both cases,"},{"Start":"01:01.070 ","End":"01:02.920","Text":"which is -4,"},{"Start":"01:02.920 ","End":"01:07.500","Text":"but we write it twice because it\u0027s a + and a -."},{"Start":"01:08.230 ","End":"01:12.740","Text":"That means that what we get when we"},{"Start":"01:12.740 ","End":"01:20.900","Text":"factorize is x - -4 times x - -4."},{"Start":"01:20.900 ","End":"01:23.510","Text":"It\u0027s always x - 1 root x - the other route,"},{"Start":"01:23.510 ","End":"01:24.935","Text":"but they\u0027re both the same,"},{"Start":"01:24.935 ","End":"01:28.660","Text":"and so this comes out (x+4)^2."},{"Start":"01:28.660 ","End":"01:32.045","Text":"I\u0027ll just mention that there is another way you could have done it if you had"},{"Start":"01:32.045 ","End":"01:35.810","Text":"spotted that 16 is a perfect square, and so is x."},{"Start":"01:35.810 ","End":"01:39.920","Text":"You might have tried the special binomial expansion."},{"Start":"01:39.920 ","End":"01:48.270","Text":"The one that says that (a+b)^2 is a^2+2ab+b^2,"},{"Start":"01:48.270 ","End":"01:53.625","Text":"and then if a is x and b is 4,"},{"Start":"01:53.625 ","End":"01:58.330","Text":"then we get x^2 is x^2,"},{"Start":"01:58.330 ","End":"02:03.980","Text":"4^2 is 16, and the middle term twice x times 4 is 8x."},{"Start":"02:03.980 ","End":"02:05.600","Text":"So we could have done it that way,"},{"Start":"02:05.600 ","End":"02:10.770","Text":"and let us say a is x and b is 4,"},{"Start":"02:10.770 ","End":"02:12.565","Text":"and then we\u0027d get (x+4)^2."},{"Start":"02:12.565 ","End":"02:15.980","Text":"So either way, we reach this point."},{"Start":"02:15.980 ","End":"02:18.380","Text":"Now we can continue from here,"},{"Start":"02:18.380 ","End":"02:23.980","Text":"and remember when we have a square of a linear term,"},{"Start":"02:23.980 ","End":"02:30.495","Text":"we have to include both x+4 and (x+4)^2."},{"Start":"02:30.495 ","End":"02:32.395","Text":"So we put something,"},{"Start":"02:32.395 ","End":"02:39.210","Text":"call it a over x+4 and something else, b over (x+4)^2."},{"Start":"02:39.210 ","End":"02:43.960","Text":"As usual, we get rid of the fractions by multiplying by the common denominator,"},{"Start":"02:43.960 ","End":"02:46.300","Text":"which will be (x+4)^2."},{"Start":"02:46.300 ","End":"02:53.250","Text":"That will give us here just x+6 and here just the b here multiply by (x+4)^2,"},{"Start":"02:53.250 ","End":"02:55.230","Text":"1 of the x+4 cancels,"},{"Start":"02:55.230 ","End":"02:56.715","Text":"and this is what we get,"},{"Start":"02:56.715 ","End":"03:06.865","Text":"and what we want to do is plug in different values to try and get a and b."},{"Start":"03:06.865 ","End":"03:10.895","Text":"The obvious thing to substitute would be - 4,"},{"Start":"03:10.895 ","End":"03:12.860","Text":"because then this comes out zeros."},{"Start":"03:12.860 ","End":"03:14.870","Text":"So if I let x=-4 here,"},{"Start":"03:14.870 ","End":"03:19.159","Text":"I\u0027ve got 6- -4 is 10, -4+4 is 0."},{"Start":"03:19.159 ","End":"03:23.664","Text":"So you get straightaway, 10=b or b=10."},{"Start":"03:23.664 ","End":"03:26.340","Text":"Then you want to try another value,"},{"Start":"03:26.340 ","End":"03:29.160","Text":"it doesn\u0027t really matter which."},{"Start":"03:29.160 ","End":"03:33.195","Text":"We could like -3 is just a sample."},{"Start":"03:33.195 ","End":"03:40.815","Text":"So 6- -3 is 9, -3+4 is 1."},{"Start":"03:40.815 ","End":"03:43.820","Text":"So there was a use for -3 because it made"},{"Start":"03:43.820 ","End":"03:47.540","Text":"the coefficient of a come out 1 which is convenient,"},{"Start":"03:47.540 ","End":"03:54.435","Text":"and I have got 9=a+b."},{"Start":"03:54.435 ","End":"03:56.970","Text":"But we know that b is 10,"},{"Start":"03:56.970 ","End":"04:06.630","Text":"so what we have is 9=a+10."},{"Start":"04:06.630 ","End":"04:07.905","Text":"If I rewrite this,"},{"Start":"04:07.905 ","End":"04:10.890","Text":"and so clearly a is -1."},{"Start":"04:10.890 ","End":"04:13.680","Text":"Now we have b and we have a,"},{"Start":"04:13.680 ","End":"04:21.300","Text":"so I just stick them here and here and what\u0027s left is what\u0027s written here,"},{"Start":"04:21.300 ","End":"04:23.830","Text":"and that\u0027s the answer."}],"ID":5276},{"Watched":false,"Name":"Exercise 7","Duration":"3m 10s","ChapterTopicVideoID":5271,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5271.jpeg","UploadDate":"2016-03-07T09:08:34.3600000","DurationForVideoObject":"PT3M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.745","Text":"In this exercise, we have to"},{"Start":"00:02.745 ","End":"00:06.450","Text":"determine the partial fraction decomposition of the following."},{"Start":"00:06.450 ","End":"00:08.460","Text":"It\u0027s a rational expression."},{"Start":"00:08.460 ","End":"00:11.580","Text":"It\u0027s a polynomial over polynomial."},{"Start":"00:11.580 ","End":"00:16.155","Text":"The degree in the numerator is less than that in the denominator."},{"Start":"00:16.155 ","End":"00:20.128","Text":"The first thing we want to do is factorize the denominator."},{"Start":"00:20.128 ","End":"00:22.545","Text":"Clearly, x can be taken out."},{"Start":"00:22.545 ","End":"00:25.755","Text":"If we do that, then we\u0027re left with x^2 minus 1."},{"Start":"00:25.755 ","End":"00:27.903","Text":"This is the difference of squares,"},{"Start":"00:27.903 ","End":"00:32.430","Text":"so the x^2 minus 1 can be written as x minus 1 x plus 1."},{"Start":"00:32.430 ","End":"00:35.520","Text":"Now the denominator is fully factored."},{"Start":"00:35.520 ","End":"00:39.650","Text":"Notice that there\u0027s three different linear factors,"},{"Start":"00:39.650 ","End":"00:42.110","Text":"and that\u0027s the simplest case."},{"Start":"00:42.110 ","End":"00:46.040","Text":"We simply have a letter A,"},{"Start":"00:46.040 ","End":"00:47.690","Text":"B, and C. Well, I\u0027ll just show you."},{"Start":"00:47.690 ","End":"00:51.230","Text":"A over the x factor,"},{"Start":"00:51.230 ","End":"00:54.110","Text":"B over the x minus 1 factor,"},{"Start":"00:54.110 ","End":"00:56.675","Text":"and C over the x plus 1."},{"Start":"00:56.675 ","End":"01:04.955","Text":"We get rid of fractions by multiplying by the common denominator, which is this."},{"Start":"01:04.955 ","End":"01:07.445","Text":"What we get then is,"},{"Start":"01:07.445 ","End":"01:11.330","Text":"A is multiplied by the missing factors,"},{"Start":"01:11.330 ","End":"01:13.040","Text":"which is everything without the x,"},{"Start":"01:13.040 ","End":"01:14.930","Text":"x minus 1 x plus 1,"},{"Start":"01:14.930 ","End":"01:19.850","Text":"B goes with the missing ones of this and this,"},{"Start":"01:19.850 ","End":"01:24.420","Text":"and C goes with this one and this one."},{"Start":"01:24.420 ","End":"01:25.880","Text":"At this point,"},{"Start":"01:25.880 ","End":"01:30.550","Text":"we can already start substituting values."},{"Start":"01:30.550 ","End":"01:35.090","Text":"What we\u0027ll choose is things that make one of these 0."},{"Start":"01:35.090 ","End":"01:38.280","Text":"We have x, we have x minus 1, and x plus 1."},{"Start":"01:38.280 ","End":"01:45.090","Text":"We start with x=0, that will make this term 0 and this one 0."},{"Start":"01:45.090 ","End":"01:48.375","Text":"So what we\u0027re left with, really,"},{"Start":"01:48.375 ","End":"01:55.115","Text":"is, this one and this one are 0, this one is left."},{"Start":"01:55.115 ","End":"01:59.075","Text":"If x is 0, we get minus 1 times plus 1 is minus 1."},{"Start":"01:59.075 ","End":"02:02.420","Text":"So basically, we just get minus 1 is minus A."},{"Start":"02:02.420 ","End":"02:06.050","Text":"These two are 0 and can be ignored, and A=1."},{"Start":"02:06.050 ","End":"02:09.925","Text":"Next, we should let x either be 1 or minus 1."},{"Start":"02:09.925 ","End":"02:12.330","Text":"I\u0027ll go with the 1 first."},{"Start":"02:12.330 ","End":"02:16.670","Text":"That 1 will make this 0 and this 0,"},{"Start":"02:16.670 ","End":"02:18.020","Text":"the x minus 1 terms."},{"Start":"02:18.020 ","End":"02:20.165","Text":"All we\u0027re left with is the B."},{"Start":"02:20.165 ","End":"02:23.160","Text":"If x is 1, 1 times 1 plus 1 is 2,"},{"Start":"02:23.160 ","End":"02:24.757","Text":"so we\u0027ve got 2B."},{"Start":"02:24.757 ","End":"02:27.390","Text":"On the left we have,"},{"Start":"02:27.390 ","End":"02:30.630","Text":"if we put in 1,"},{"Start":"02:30.630 ","End":"02:32.910","Text":"we have 1 plus 1 minus 1 is 1,"},{"Start":"02:32.910 ","End":"02:35.175","Text":"so 2B=1B is a half."},{"Start":"02:35.175 ","End":"02:37.910","Text":"It\u0027s just very simple algebra."},{"Start":"02:37.910 ","End":"02:40.485","Text":"Then we\u0027re going to let x equal minus 1."},{"Start":"02:40.485 ","End":"02:44.685","Text":"If we do that, it\u0027s fairly straightforward."},{"Start":"02:44.685 ","End":"02:50.820","Text":"Just substitute the minus 1 here and then these first two"},{"Start":"02:50.820 ","End":"02:56.720","Text":"become 0 and we just can conclude what C is from here, is minus a half."},{"Start":"02:56.720 ","End":"02:59.460","Text":"The last thing we do is to take A, B,"},{"Start":"02:59.460 ","End":"03:02.240","Text":"and C, and put them here instead of A, B,"},{"Start":"03:02.240 ","End":"03:05.330","Text":"and C. This gives us our final answer,"},{"Start":"03:05.330 ","End":"03:10.500","Text":"which is the partial fraction decomposition. We\u0027re done."}],"ID":5277},{"Watched":false,"Name":"Exercise 8","Duration":"6m 15s","ChapterTopicVideoID":5272,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5272.jpeg","UploadDate":"2016-03-07T09:09:40.4730000","DurationForVideoObject":"PT6M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this exercise, we have to determine"},{"Start":"00:02.310 ","End":"00:08.070","Text":"the partial fraction decomposition of the following rational expression."},{"Start":"00:08.070 ","End":"00:14.085","Text":"First thing to do is to factorize the denominator."},{"Start":"00:14.085 ","End":"00:16.260","Text":"We should also check that the degree in"},{"Start":"00:16.260 ","End":"00:20.140","Text":"the numerator is less than the denominator, but that\u0027s clear."},{"Start":"00:21.230 ","End":"00:29.019","Text":"As I said, we\u0027re going to factorize this degree for a polynomial."},{"Start":"00:29.450 ","End":"00:34.845","Text":"We\u0027d like to get it as is of the following form,"},{"Start":"00:34.845 ","End":"00:39.810","Text":"where we find all 4 roots of a fourth degree polynomial and we get x minus 1 root,"},{"Start":"00:39.810 ","End":"00:43.160","Text":"the other, the third, the fourth, and so on."},{"Start":"00:43.160 ","End":"00:45.440","Text":"This is a fourth degree,"},{"Start":"00:45.440 ","End":"00:47.855","Text":"but there\u0027s no x^3 and x terms."},{"Start":"00:47.855 ","End":"00:51.695","Text":"In this case, what we\u0027d usually do is a substitution."},{"Start":"00:51.695 ","End":"00:54.590","Text":"We take this polynomial and we substitute,"},{"Start":"00:54.590 ","End":"00:55.955","Text":"instead of x^2,"},{"Start":"00:55.955 ","End":"00:59.690","Text":"some letter such as t. We\u0027ll choose that 1."},{"Start":"00:59.690 ","End":"01:02.615","Text":"Then x^4 would be t^2,"},{"Start":"01:02.615 ","End":"01:11.400","Text":"x^2 is just t and so we get a quadratic equation in t. Now in this chapter,"},{"Start":"01:11.400 ","End":"01:13.520","Text":"I don\u0027t solve quadratic equations for you."},{"Start":"01:13.520 ","End":"01:15.035","Text":"You know how to do that."},{"Start":"01:15.035 ","End":"01:17.075","Text":"I\u0027ll just tell you the answers."},{"Start":"01:17.075 ","End":"01:19.010","Text":"That we get 2 t\u0027s,"},{"Start":"01:19.010 ","End":"01:22.900","Text":"we\u0027ve got the first 1, 9 and the other 1, 4."},{"Start":"01:22.900 ","End":"01:25.195","Text":"Call them t_1 and t_2."},{"Start":"01:25.195 ","End":"01:27.380","Text":"However, we\u0027re not looking for t,"},{"Start":"01:27.380 ","End":"01:28.820","Text":"we\u0027re looking for x,"},{"Start":"01:28.820 ","End":"01:34.670","Text":"and so what that tells us is that x^2 could be"},{"Start":"01:34.670 ","End":"01:41.710","Text":"9 or x^2 could be 4 because t is x^2."},{"Start":"01:42.380 ","End":"01:46.825","Text":"Now, if x^2=9,"},{"Start":"01:46.825 ","End":"01:50.870","Text":"that gives us that x is plus or minus 3."},{"Start":"01:50.870 ","End":"01:54.275","Text":"If x^2=4, then x is plus or minus 2,"},{"Start":"01:54.275 ","End":"01:56.300","Text":"so we have 4 possibilities,"},{"Start":"01:56.300 ","End":"01:59.155","Text":"-3, 3, -2, 2."},{"Start":"01:59.155 ","End":"02:01.370","Text":"These are 4 different solutions."},{"Start":"02:01.370 ","End":"02:02.390","Text":"Which we call them next x_1,"},{"Start":"02:02.390 ","End":"02:04.640","Text":"x_2, x_3, and x_4."},{"Start":"02:04.640 ","End":"02:08.150","Text":"That means that according to what we said above,"},{"Start":"02:08.150 ","End":"02:12.530","Text":"that we can factorize this as the following product,"},{"Start":"02:12.530 ","End":"02:15.035","Text":"x minus minus 3,"},{"Start":"02:15.035 ","End":"02:17.525","Text":"x minus 3, and so on."},{"Start":"02:17.525 ","End":"02:25.295","Text":"Now that we have this factorized and the factors are linear and all different,"},{"Start":"02:25.295 ","End":"02:29.210","Text":"the way we go about it now is to say what"},{"Start":"02:29.210 ","End":"02:34.085","Text":"we generally expect the partial decomposition to be."},{"Start":"02:34.085 ","End":"02:42.130","Text":"The original thing, I\u0027ve just rewritten first is 10x over the factorized denominator."},{"Start":"02:42.130 ","End":"02:45.320","Text":"Because all the factors are linear, like I said,"},{"Start":"02:45.320 ","End":"02:48.365","Text":"we have just some constant over each of them."},{"Start":"02:48.365 ","End":"02:51.755","Text":"We take each of these factors and put a constant over each 1."},{"Start":"02:51.755 ","End":"02:55.040","Text":"This is going to be the general form and we have to find out what are A,"},{"Start":"02:55.040 ","End":"02:56.925","Text":"B, C,"},{"Start":"02:56.925 ","End":"03:04.890","Text":"and D. We get rid of the denominator,"},{"Start":"03:04.890 ","End":"03:10.160","Text":"and we multiply everything by this product of 4 things."},{"Start":"03:10.160 ","End":"03:12.870","Text":"Each constant A, B,"},{"Start":"03:12.870 ","End":"03:15.600","Text":"C, or D gets multiplied by the 3 missing ones."},{"Start":"03:15.600 ","End":"03:17.205","Text":"If I have x plus 3,"},{"Start":"03:17.205 ","End":"03:19.860","Text":"then I multiply it by this, this, and this."},{"Start":"03:19.860 ","End":"03:22.380","Text":"B is over x minus 3,"},{"Start":"03:22.380 ","End":"03:24.420","Text":"it\u0027s multiplied by this,"},{"Start":"03:24.420 ","End":"03:27.465","Text":"and so on, then we get the following expression."},{"Start":"03:27.465 ","End":"03:33.365","Text":"Just a moment to check that you\u0027ll see that\u0027s what it comes out to."},{"Start":"03:33.365 ","End":"03:36.935","Text":"Then we start the process of substitution."},{"Start":"03:36.935 ","End":"03:40.865","Text":"Each time we substitute 1 of these values,"},{"Start":"03:40.865 ","End":"03:43.040","Text":"that makes this 0."},{"Start":"03:43.040 ","End":"03:46.175","Text":"If we substitute minus 3,"},{"Start":"03:46.175 ","End":"03:53.010","Text":"that\u0027s going to make the x plus 3 term 0."},{"Start":"03:53.010 ","End":"03:54.945","Text":"This is going to be 0,"},{"Start":"03:54.945 ","End":"03:57.855","Text":"wherever there\u0027s an x plus 3."},{"Start":"03:57.855 ","End":"03:59.490","Text":"That\u0027s going to give us A,"},{"Start":"03:59.490 ","End":"04:02.835","Text":"because we get 10 times minus 3 is 30,"},{"Start":"04:02.835 ","End":"04:12.460","Text":"A and x"},{"Start":"04:12.460 ","End":"04:13.650","Text":"is minus 3,"},{"Start":"04:13.650 ","End":"04:15.555","Text":"so that\u0027s minus 6."},{"Start":"04:15.555 ","End":"04:18.870","Text":"Then this is going to be minus 1,"},{"Start":"04:18.870 ","End":"04:21.450","Text":"and this is going to be minus 5,"},{"Start":"04:21.450 ","End":"04:30.190","Text":"so minus 6 times minus 1 times minus 5 is minus 30."},{"Start":"04:30.190 ","End":"04:32.060","Text":"That\u0027s this minus 30."},{"Start":"04:32.060 ","End":"04:34.220","Text":"Then the rest of these are 0."},{"Start":"04:34.220 ","End":"04:38.442","Text":"We get minus 30 equals minus 30A, so A is 1."},{"Start":"04:38.442 ","End":"04:45.195","Text":"Continuing, next time we let x=3,"},{"Start":"04:45.195 ","End":"04:49.410","Text":"and then all the terms with x minus 3 will be 0."},{"Start":"04:49.410 ","End":"04:51.140","Text":"It\u0027ll get 0 here,"},{"Start":"04:51.140 ","End":"04:53.280","Text":"and we\u0027ll get B."},{"Start":"04:54.130 ","End":"04:57.600","Text":"B comes out to be,"},{"Start":"04:57.600 ","End":"05:00.080","Text":"let\u0027s see, 10x is 30,"},{"Start":"05:00.080 ","End":"05:03.980","Text":"and the coefficient of B is going to be 3 plus 3,"},{"Start":"05:03.980 ","End":"05:06.690","Text":"3 plus 2, 3 minus 2,"},{"Start":"05:06.690 ","End":"05:09.490","Text":"6 times 5 times 1 is 30."},{"Start":"05:09.490 ","End":"05:14.700","Text":"Next, we\u0027ll be substituting x=-2."},{"Start":"05:14.700 ","End":"05:23.055","Text":"Then everything could be 0 except where there\u0027s no x plus 2 term."},{"Start":"05:23.055 ","End":"05:25.260","Text":"Wherever there\u0027s x plus 2 it\u0027s 0."},{"Start":"05:25.260 ","End":"05:27.150","Text":"This, this, and this is 0."},{"Start":"05:27.150 ","End":"05:29.265","Text":"Only the C remains."},{"Start":"05:29.265 ","End":"05:31.335","Text":"If you let x=-2,"},{"Start":"05:31.335 ","End":"05:37.170","Text":"minus 2 plus 3 is 1 times minus 5 times minus 4,"},{"Start":"05:37.170 ","End":"05:39.090","Text":"it comes out minus 20."},{"Start":"05:39.090 ","End":"05:42.680","Text":"The last 1. I\u0027ll just gloss over that."},{"Start":"05:42.680 ","End":"05:44.165","Text":"If you let x=2,"},{"Start":"05:44.165 ","End":"05:47.240","Text":"only this 1 is non-zero and this is the equation we get,"},{"Start":"05:47.240 ","End":"05:49.135","Text":"and this is the answer for D,"},{"Start":"05:49.135 ","End":"05:50.860","Text":"similar to these 3."},{"Start":"05:50.860 ","End":"05:55.070","Text":"Finally, we get these 4 constants, A, B, C,"},{"Start":"05:55.070 ","End":"05:58.910","Text":"and D, and we substitute them in the,"},{"Start":"05:58.910 ","End":"06:00.770","Text":"I\u0027ll show you where,"},{"Start":"06:00.770 ","End":"06:03.560","Text":"we substitute them here, A, B, C,"},{"Start":"06:03.560 ","End":"06:08.620","Text":"and D. Then we get 1,"},{"Start":"06:08.620 ","End":"06:11.376","Text":"1 minus 1, minus 1,"},{"Start":"06:11.376 ","End":"06:15.040","Text":"and this is the answer."}],"ID":5278},{"Watched":false,"Name":"Exercise 9","Duration":"3m 47s","ChapterTopicVideoID":5273,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5273.jpeg","UploadDate":"2016-03-07T09:10:19.9770000","DurationForVideoObject":"PT3M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.510","Text":"Here we have to do a partial fraction decomposition of this rational expression."},{"Start":"00:06.510 ","End":"00:11.300","Text":"It\u0027s a polynomial on the top and a polynomial on the bottom."},{"Start":"00:11.300 ","End":"00:16.365","Text":"Clearly the degree on top is less than that of the bottom, so we\u0027re okay."},{"Start":"00:16.365 ","End":"00:20.610","Text":"The denominator, the bottom,"},{"Start":"00:20.610 ","End":"00:23.295","Text":"whatever is already factorized."},{"Start":"00:23.295 ","End":"00:25.170","Text":"That saves us that."},{"Start":"00:25.170 ","End":"00:27.450","Text":"What we see is we have x plus 2 1,"},{"Start":"00:27.450 ","End":"00:29.640","Text":"so we have x minus 2 squared."},{"Start":"00:29.640 ","End":"00:37.155","Text":"What we need are representatives of x plus 2 here."},{"Start":"00:37.155 ","End":"00:38.700","Text":"But for the x minus 2^2,"},{"Start":"00:38.700 ","End":"00:42.000","Text":"we need to take both x minus 2 and x minus 2^2."},{"Start":"00:42.000 ","End":"00:43.845","Text":"That\u0027s the way it works."},{"Start":"00:43.845 ","End":"00:46.020","Text":"We just put constants A, B,"},{"Start":"00:46.020 ","End":"00:47.780","Text":"and C over each of them as usual."},{"Start":"00:47.780 ","End":"00:51.290","Text":"Then we multiply by the common denominator,"},{"Start":"00:51.290 ","End":"00:53.550","Text":"which is all of this."},{"Start":"00:54.260 ","End":"00:58.530","Text":"Here, we just get 8x, here A goes with"},{"Start":"00:58.530 ","End":"01:01.650","Text":"the 2 missing factors as an x minus"},{"Start":"01:01.650 ","End":"01:05.915","Text":"2 and an x plus 2 missing here I\u0027m just missing the x plus 2."},{"Start":"01:05.915 ","End":"01:09.649","Text":"Here I\u0027m missing x minus 2^2."},{"Start":"01:09.649 ","End":"01:14.270","Text":"We now start substituting different values in order to find a,"},{"Start":"01:14.270 ","End":"01:19.660","Text":"b, and c. If we let x equals 2,"},{"Start":"01:19.660 ","End":"01:25.470","Text":"then we\u0027re going to get that x minus 2 is 0 and x minus"},{"Start":"01:25.470 ","End":"01:30.780","Text":"2^2 is also 0. x equals 2 will help us find B."},{"Start":"01:30.780 ","End":"01:34.175","Text":"What we get is 8 times 2 is 16."},{"Start":"01:34.175 ","End":"01:39.135","Text":"I will say this is 0 and this is 0 because of the x minus 2."},{"Start":"01:39.135 ","End":"01:44.490","Text":"All we\u0027re left with is B with x plus 2,"},{"Start":"01:44.490 ","End":"01:46.380","Text":"which is 2 plus 2 which is 4."},{"Start":"01:46.380 ","End":"01:49.950","Text":"4B is 16 and if 4B is 16,"},{"Start":"01:49.950 ","End":"01:51.905","Text":"then B is 4."},{"Start":"01:51.905 ","End":"01:56.780","Text":"Next thing you want to do is let x equal minus 2,"},{"Start":"01:56.780 ","End":"02:02.340","Text":"and that will make the x plus 2 factor 0."},{"Start":"02:02.600 ","End":"02:05.710","Text":"If we do put it as minus 2,"},{"Start":"02:05.710 ","End":"02:08.765","Text":"this becomes 0 and this becomes 0."},{"Start":"02:08.765 ","End":"02:11.275","Text":"We\u0027re just left with the C term,"},{"Start":"02:11.275 ","End":"02:14.070","Text":"minus 2, minus 2 is minus 4,"},{"Start":"02:14.070 ","End":"02:15.795","Text":"but squared is plus 16."},{"Start":"02:15.795 ","End":"02:18.359","Text":"Here 8 times minus 2 is minus 16."},{"Start":"02:18.359 ","End":"02:20.880","Text":"These 2 are 0 divided by 16,"},{"Start":"02:20.880 ","End":"02:23.410","Text":"C is minus 1."},{"Start":"02:23.510 ","End":"02:28.625","Text":"That exhausts all possibilities for x that will make something 0."},{"Start":"02:28.625 ","End":"02:32.595","Text":"We try any other x, for example,"},{"Start":"02:32.595 ","End":"02:38.735","Text":"x equals 0, where 1 reason that is that it\u0027s easy to compute to substitute."},{"Start":"02:38.735 ","End":"02:42.830","Text":"On the left we have 0 minus 2,"},{"Start":"02:42.830 ","End":"02:46.100","Text":"0 plus 2 is minus 4."},{"Start":"02:46.100 ","End":"02:53.190","Text":"Then here we have 0 plus 2 is 2,"},{"Start":"02:53.190 ","End":"02:55.365","Text":"so it\u0027s B times 2."},{"Start":"02:55.365 ","End":"03:05.500","Text":"Here, 0 minus 2 squared is minus 2 squared is plus 4C."},{"Start":"03:05.500 ","End":"03:08.840","Text":"Now remember that we already have B and C. There,"},{"Start":"03:08.840 ","End":"03:10.820","Text":"they\u0027re written in small writing."},{"Start":"03:10.820 ","End":"03:12.350","Text":"B we found was 4,"},{"Start":"03:12.350 ","End":"03:14.165","Text":"so make a note of that here."},{"Start":"03:14.165 ","End":"03:15.650","Text":"C is minus 1,"},{"Start":"03:15.650 ","End":"03:17.810","Text":"make a note of that here."},{"Start":"03:17.810 ","End":"03:20.000","Text":"Then if we do the computation,"},{"Start":"03:20.000 ","End":"03:22.340","Text":"we get 4 times 2 is 8,"},{"Start":"03:22.340 ","End":"03:25.535","Text":"minus 4 is 4."},{"Start":"03:25.535 ","End":"03:27.620","Text":"If you bring this to the other side,"},{"Start":"03:27.620 ","End":"03:28.970","Text":"we get 4A is 4,"},{"Start":"03:28.970 ","End":"03:30.350","Text":"so A is 1."},{"Start":"03:30.350 ","End":"03:31.730","Text":"We now have A, B,"},{"Start":"03:31.730 ","End":"03:36.140","Text":"and C. All that remains to do is to substitute them here, here and here,"},{"Start":"03:36.140 ","End":"03:40.895","Text":"which I\u0027ll do down here and we get 1,"},{"Start":"03:40.895 ","End":"03:43.256","Text":"4 minus 1, 1,"},{"Start":"03:43.256 ","End":"03:44.865","Text":"4 minus 1,"},{"Start":"03:44.865 ","End":"03:47.500","Text":"and that\u0027s the answer."}],"ID":5279},{"Watched":false,"Name":"Exercise 10","Duration":"3m 20s","ChapterTopicVideoID":5274,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5274.jpeg","UploadDate":"2016-03-07T09:10:55.7900000","DurationForVideoObject":"PT3M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this exercise, we have this expression,"},{"Start":"00:02.940 ","End":"00:08.520","Text":"a rational expression and we have to decompose it as a partial fraction."},{"Start":"00:08.520 ","End":"00:14.370","Text":"The first thing we do is to factorize the denominator,"},{"Start":"00:14.370 ","End":"00:18.930","Text":"of course we also check that the degree on the top is less than the degree on the bottom,"},{"Start":"00:18.930 ","End":"00:20.745","Text":"which it certainly is."},{"Start":"00:20.745 ","End":"00:24.070","Text":"We start to factorize."},{"Start":"00:24.170 ","End":"00:27.360","Text":"First of all, we take out the x^2 and then we\u0027re"},{"Start":"00:27.360 ","End":"00:30.480","Text":"left with x plus 1 and that\u0027s as far as we can go."},{"Start":"00:30.480 ","End":"00:32.310","Text":"There\u0027s kind of 3 factors,"},{"Start":"00:32.310 ","End":"00:34.620","Text":"x, x and x plus 1."},{"Start":"00:34.620 ","End":"00:36.255","Text":"Because of the x^2,"},{"Start":"00:36.255 ","End":"00:38.100","Text":"which is a double factor,"},{"Start":"00:38.100 ","End":"00:46.295","Text":"we need to write the partial fraction as something over x and something over x^2."},{"Start":"00:46.295 ","End":"00:48.710","Text":"We need to represent both of them."},{"Start":"00:48.710 ","End":"00:53.480","Text":"The x plus 1 on its own just has a constant of its own."},{"Start":"00:53.480 ","End":"00:57.140","Text":"This is what it will look like and as usual,"},{"Start":"00:57.140 ","End":"01:03.380","Text":"we get rid of the denominators by multiplying by x^2 times x plus 1."},{"Start":"01:03.380 ","End":"01:07.700","Text":"That gives us here just the 5 minus x that was on the top."},{"Start":"01:07.700 ","End":"01:12.450","Text":"Here with A we\u0027re missing an x and an x plus 1, that\u0027s there,"},{"Start":"01:12.450 ","End":"01:14.985","Text":"the x^2 is missing an x plus 1,"},{"Start":"01:14.985 ","End":"01:16.740","Text":"and C is missing the x^2,"},{"Start":"01:16.740 ","End":"01:21.110","Text":"so this is the algebraic expression we get."},{"Start":"01:21.110 ","End":"01:23.990","Text":"This is actually not an equation,"},{"Start":"01:23.990 ","End":"01:27.180","Text":"but then identity meaning that it\u0027s true for all x,"},{"Start":"01:27.180 ","End":"01:30.080","Text":"so we can substitute whatever is convenient for us."},{"Start":"01:30.080 ","End":"01:32.220","Text":"Certainly 2 convenient values,"},{"Start":"01:32.220 ","End":"01:35.885","Text":"if we put in 0, that will make this and this 0."},{"Start":"01:35.885 ","End":"01:37.705","Text":"If we put minus 1,"},{"Start":"01:37.705 ","End":"01:39.840","Text":"this and this will be 0,"},{"Start":"01:39.840 ","End":"01:42.500","Text":"let\u0027s first of all put the 0 in."},{"Start":"01:42.500 ","End":"01:45.200","Text":"Then this is 0 and the last one is 0,"},{"Start":"01:45.200 ","End":"01:49.350","Text":"so we\u0027re just left with B times 1 is 5 minus 0,"},{"Start":"01:49.350 ","End":"01:53.865","Text":"which is 5, so 5=B or B=5."},{"Start":"01:53.865 ","End":"01:55.710","Text":"Then we\u0027ll go with the A for the other one,"},{"Start":"01:55.710 ","End":"01:57.330","Text":"the minus 1,"},{"Start":"01:57.330 ","End":"02:00.765","Text":"and that makes this and this 0."},{"Start":"02:00.765 ","End":"02:03.045","Text":"The minus 1^2 is 1,"},{"Start":"02:03.045 ","End":"02:05.550","Text":"and 5 minus minus 1 is 6."},{"Start":"02:05.550 ","End":"02:09.140","Text":"We straightaway get C=6."},{"Start":"02:09.140 ","End":"02:12.530","Text":"Then we just have to choose any other value of x, really,"},{"Start":"02:12.530 ","End":"02:15.065","Text":"whatever is convenient for computation."},{"Start":"02:15.065 ","End":"02:17.665","Text":"For example, x=1."},{"Start":"02:17.665 ","End":"02:20.850","Text":"Then on the left we get 5 minus 1 is 4,"},{"Start":"02:20.850 ","End":"02:22.980","Text":"1 times 2 is 2,"},{"Start":"02:22.980 ","End":"02:28.190","Text":"1 plus 1 is 2, 1^2 is 1."},{"Start":"02:28.190 ","End":"02:31.470","Text":"We have 3 variables here,"},{"Start":"02:31.470 ","End":"02:33.920","Text":"A, B, and C. However,"},{"Start":"02:33.920 ","End":"02:36.350","Text":"we already found here that B is 5,"},{"Start":"02:36.350 ","End":"02:38.420","Text":"so I\u0027ll just write that above B,"},{"Start":"02:38.420 ","End":"02:39.590","Text":"and C is 6,"},{"Start":"02:39.590 ","End":"02:41.090","Text":"so I can write that."},{"Start":"02:41.090 ","End":"02:43.220","Text":"Really we only have one unknown left,"},{"Start":"02:43.220 ","End":"02:46.745","Text":"that\u0027s A, we get the 2A."},{"Start":"02:46.745 ","End":"02:49.100","Text":"I believe that on the right is 4 minus 10 minus 6,"},{"Start":"02:49.100 ","End":"02:58.170","Text":"and that comes out to minus 12,"},{"Start":"02:58.170 ","End":"03:00.000","Text":"so 2A is minus 12,"},{"Start":"03:00.000 ","End":"03:01.950","Text":"A is minus 6."},{"Start":"03:01.950 ","End":"03:07.370","Text":"The last step is just to put these values in order."},{"Start":"03:07.370 ","End":"03:09.560","Text":"They would be minus 6, 5,"},{"Start":"03:09.560 ","End":"03:14.595","Text":"6 here, and here we are."},{"Start":"03:14.595 ","End":"03:16.880","Text":"The A is minus 6, B,"},{"Start":"03:16.880 ","End":"03:20.140","Text":"5, C is 6 This is the answer."}],"ID":5280},{"Watched":false,"Name":"Exercise 11","Duration":"3m 54s","ChapterTopicVideoID":5275,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5275.jpeg","UploadDate":"2016-03-07T09:11:36.1970000","DurationForVideoObject":"PT3M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In this exercise, we have this expression,"},{"Start":"00:02.490 ","End":"00:07.620","Text":"it\u0027s a rational expression and we want to decompose it into partial fractions."},{"Start":"00:07.620 ","End":"00:13.545","Text":"Note that the degree on the top is less than the degree on the bottom, so that\u0027s okay."},{"Start":"00:13.545 ","End":"00:16.890","Text":"Then we want to factorize the denominator."},{"Start":"00:16.890 ","End":"00:18.420","Text":"First thing we can do is,"},{"Start":"00:18.420 ","End":"00:21.194","Text":"we can take x outside the brackets,"},{"Start":"00:21.194 ","End":"00:25.065","Text":"and then we\u0027re left with x^2 plus 6x plus 9."},{"Start":"00:25.065 ","End":"00:29.145","Text":"You can probably recognize this as a perfect square,"},{"Start":"00:29.145 ","End":"00:30.900","Text":"one of those special products;"},{"Start":"00:30.900 ","End":"00:34.920","Text":"a plus b^2 is a^2 plus 2ab plus b^2."},{"Start":"00:34.920 ","End":"00:37.800","Text":"Any event, it\u0027s x plus 3^2,"},{"Start":"00:37.800 ","End":"00:41.705","Text":"and you can always check by multiplying this out."},{"Start":"00:41.705 ","End":"00:45.520","Text":"You could also solve the quadratic and get two solutions,"},{"Start":"00:45.520 ","End":"00:46.930","Text":"both of them minus 3."},{"Start":"00:46.930 ","End":"00:50.750","Text":"That will also give you this factorization."},{"Start":"00:50.750 ","End":"00:55.180","Text":"Now that we have the factorization,"},{"Start":"00:55.180 ","End":"00:57.790","Text":"we see we have x once and x plus 3 twice,"},{"Start":"00:57.790 ","End":"00:59.680","Text":"so we need a constant for x,"},{"Start":"00:59.680 ","End":"01:00.790","Text":"one for x plus 3,"},{"Start":"01:00.790 ","End":"01:03.095","Text":"and one for x plus 3^2."},{"Start":"01:03.095 ","End":"01:04.685","Text":"We\u0027ll call them A, B,"},{"Start":"01:04.685 ","End":"01:07.225","Text":"and C as follows."},{"Start":"01:07.225 ","End":"01:10.030","Text":"It\u0027s important not to forget that when you have something squared,"},{"Start":"01:10.030 ","End":"01:16.120","Text":"you have to have it represented both as the linear and as a square,"},{"Start":"01:16.120 ","End":"01:20.650","Text":"x plus 3 and x plus 3^2 and multiply by this,"},{"Start":"01:20.650 ","End":"01:22.510","Text":"which is our common denominator."},{"Start":"01:22.510 ","End":"01:24.660","Text":"After we multiply out,"},{"Start":"01:24.660 ","End":"01:26.390","Text":"this is what we get."},{"Start":"01:26.390 ","End":"01:28.440","Text":"For each of these constants,"},{"Start":"01:28.440 ","End":"01:31.430","Text":"we multiply by the factors that are missing on the denominator."},{"Start":"01:31.430 ","End":"01:33.350","Text":"Here we\u0027re missing x plus 3^2,"},{"Start":"01:33.350 ","End":"01:35.605","Text":"here we\u0027re missing an x,"},{"Start":"01:35.605 ","End":"01:39.830","Text":"and another x plus 3 and here we\u0027re just missing the x."},{"Start":"01:39.830 ","End":"01:45.940","Text":"Now we want to substitute values that will make part of this 0."},{"Start":"01:45.940 ","End":"01:49.050","Text":"What we\u0027ll do is, well, there\u0027s two possibilities."},{"Start":"01:49.050 ","End":"01:53.250","Text":"We can add x is 0 or x=minus 3."},{"Start":"01:53.250 ","End":"01:59.250","Text":"We\u0027ll go with the 0 first and that will make this 0 and this 0,"},{"Start":"01:59.250 ","End":"02:04.140","Text":"so here and here we have 0 and 0 plus 3^2 is 9."},{"Start":"02:04.140 ","End":"02:06.870","Text":"9 times 0 plus 36 is 36."},{"Start":"02:06.870 ","End":"02:09.735","Text":"So 9A is 36 and A is 4."},{"Start":"02:09.735 ","End":"02:12.240","Text":"Then we want to try the minus 3,"},{"Start":"02:12.240 ","End":"02:15.080","Text":"that would make this 0 and this 0."},{"Start":"02:15.080 ","End":"02:17.645","Text":"We have C times minus 3."},{"Start":"02:17.645 ","End":"02:23.160","Text":"On this side, 9 times minus 3 plus 36."},{"Start":"02:23.160 ","End":"02:26.205","Text":"Minus 27 plus 36 is 9."},{"Start":"02:26.205 ","End":"02:29.760","Text":"Minus 3C is 9. So C is 9 over minus 3,"},{"Start":"02:29.760 ","End":"02:32.170","Text":"which is minus 3."},{"Start":"02:32.170 ","End":"02:34.460","Text":"That only gives us two of them,"},{"Start":"02:34.460 ","End":"02:39.335","Text":"A and C. We also need B to substitute any value that\u0027s convenient."},{"Start":"02:39.335 ","End":"02:43.260","Text":"For example, x equals 1 is easy to compute with."},{"Start":"02:43.440 ","End":"02:48.720","Text":"If x is 1, 1 plus 3^2 is 4^2 is 16."},{"Start":"02:48.720 ","End":"02:53.010","Text":"1 times 1 plus 3 is 4 and 1 is just 1."},{"Start":"02:53.010 ","End":"02:57.945","Text":"On this side, 9 times 1 plus 36 is 45."},{"Start":"02:57.945 ","End":"03:00.705","Text":"Because we already found A and C,"},{"Start":"03:00.705 ","End":"03:02.130","Text":"I can write that here,"},{"Start":"03:02.130 ","End":"03:04.950","Text":"that A is 4 from here,"},{"Start":"03:04.950 ","End":"03:06.420","Text":"and C is minus 3,"},{"Start":"03:06.420 ","End":"03:07.795","Text":"I can put that there."},{"Start":"03:07.795 ","End":"03:09.745","Text":"Now if you do the arithmetic,"},{"Start":"03:09.745 ","End":"03:12.554","Text":"we get the 4B is what,"},{"Start":"03:12.554 ","End":"03:18.110","Text":"45 minus 64 plus 3,"},{"Start":"03:18.110 ","End":"03:23.020","Text":"and that comes out to be minus 4."},{"Start":"03:24.180 ","End":"03:30.375","Text":"I didn\u0027t say that right. If I do 45 plus 3 is 48,"},{"Start":"03:30.375 ","End":"03:33.825","Text":"and then minus 64, it\u0027s minus 16."},{"Start":"03:33.825 ","End":"03:36.480","Text":"4B is minus 16,"},{"Start":"03:36.480 ","End":"03:37.950","Text":"so B is minus 4."},{"Start":"03:37.950 ","End":"03:40.360","Text":"That\u0027s how I should have said it."},{"Start":"03:40.360 ","End":"03:43.280","Text":"Now we just plug them in over here."},{"Start":"03:43.280 ","End":"03:45.470","Text":"A is 4,"},{"Start":"03:45.470 ","End":"03:46.880","Text":"next is B,"},{"Start":"03:46.880 ","End":"03:52.250","Text":"which is minus 4 here and then C is minus 3 and that\u0027s here,"},{"Start":"03:52.250 ","End":"03:54.750","Text":"and so we\u0027re done."}],"ID":5281},{"Watched":false,"Name":"Exercise 12","Duration":"5m 24s","ChapterTopicVideoID":5276,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5276.jpeg","UploadDate":"2016-03-07T09:12:35.3370000","DurationForVideoObject":"PT5M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.260","Text":"In this exercise,"},{"Start":"00:01.260 ","End":"00:06.480","Text":"we have to figure out the partial fraction decomposition of this,"},{"Start":"00:06.480 ","End":"00:09.015","Text":"which is a rational expression."},{"Start":"00:09.015 ","End":"00:15.780","Text":"What we have to do is factorize the denominator."},{"Start":"00:15.780 ","End":"00:19.895","Text":"We can see that these are both perfect squares."},{"Start":"00:19.895 ","End":"00:22.600","Text":"That\u0027s that special binomial expansion."},{"Start":"00:22.600 ","End":"00:27.240","Text":"(a - b)^2 is a^2 - 2ab + b^2."},{"Start":"00:27.240 ","End":"00:28.500","Text":"Anyway, if we apply that,"},{"Start":"00:28.500 ","End":"00:30.375","Text":"which you should be very familiar with,"},{"Start":"00:30.375 ","End":"00:36.945","Text":"this one comes out to be x - 1^2 and this 1 comes out to be x - 2^2."},{"Start":"00:36.945 ","End":"00:39.362","Text":"You can multiply it out and verify."},{"Start":"00:39.362 ","End":"00:41.735","Text":"I\u0027m not going to spend time on that."},{"Start":"00:41.735 ","End":"00:44.555","Text":"Now, we have 2 factors,"},{"Start":"00:44.555 ","End":"00:45.620","Text":"each of them squared,"},{"Start":"00:45.620 ","End":"00:49.640","Text":"so we\u0027re going to need 4 pieces in the partial fraction."},{"Start":"00:49.640 ","End":"00:54.935","Text":"We\u0027re going to need x - 1^2, x - 2^2."},{"Start":"00:54.935 ","End":"00:56.860","Text":"This is how it\u0027s going to look."},{"Start":"00:56.860 ","End":"01:00.120","Text":"You need 4 constants."},{"Start":"01:00.120 ","End":"01:04.580","Text":"The next step is to get rid of the denominator."},{"Start":"01:04.580 ","End":"01:08.480","Text":"If we multiply both sides by all of this,"},{"Start":"01:08.480 ","End":"01:11.240","Text":"then each constant here will get multiplied"},{"Start":"01:11.240 ","End":"01:14.765","Text":"by wherever factors are missing in the denominator."},{"Start":"01:14.765 ","End":"01:21.728","Text":"We get A times an x- 1 and both x-2s."},{"Start":"01:21.728 ","End":"01:23.450","Text":"Then in B we have these."},{"Start":"01:23.450 ","End":"01:26.700","Text":"We need the x - 2^2 and so on."},{"Start":"01:26.980 ","End":"01:30.710","Text":"Now we don\u0027t have to simplify or anything,"},{"Start":"01:30.710 ","End":"01:33.725","Text":"we just have to substitute values."},{"Start":"01:33.725 ","End":"01:37.445","Text":"The best thing to do is substitute something that makes something"},{"Start":"01:37.445 ","End":"01:41.735","Text":"0,1 and 2 are going to be perfect for that."},{"Start":"01:41.735 ","End":"01:44.390","Text":"If we substitute x = 1,"},{"Start":"01:44.390 ","End":"01:52.460","Text":"then what we get here is the 1 makes this 0 and it makes this 0 and this 0."},{"Start":"01:52.460 ","End":"01:55.910","Text":"We\u0027re just left with 1 - 2^2 is 1."},{"Start":"01:55.910 ","End":"01:57.470","Text":"B times 1 is 1,"},{"Start":"01:57.470 ","End":"01:58.940","Text":"that gives B as 1."},{"Start":"01:58.940 ","End":"02:01.640","Text":"Next, you want to try x=2."},{"Start":"02:01.640 ","End":"02:05.885","Text":"That will make everything 0 except for the last 1,"},{"Start":"02:05.885 ","End":"02:08.340","Text":"2 - 1^2 is 1."},{"Start":"02:08.340 ","End":"02:09.750","Text":"We get 1D = 1,"},{"Start":"02:09.750 ","End":"02:11.355","Text":"and then D is 1."},{"Start":"02:11.355 ","End":"02:14.467","Text":"We can\u0027t make anything else 0,"},{"Start":"02:14.467 ","End":"02:19.340","Text":"so we just basically substitute any values that are convenient."},{"Start":"02:19.340 ","End":"02:23.300","Text":"Zero is often good because it\u0027s easy to compute with."},{"Start":"02:23.300 ","End":"02:25.745","Text":"We put in x = 0."},{"Start":"02:25.745 ","End":"02:29.000","Text":"Here you get - 1,"},{"Start":"02:29.000 ","End":"02:32.893","Text":"and - 2^2 is -1 times 4 is -1."},{"Start":"02:32.893 ","End":"02:39.060","Text":"Compute each of these bits and this is what we get."},{"Start":"02:39.060 ","End":"02:43.080","Text":"However, we do have 2 of the variables,"},{"Start":"02:43.080 ","End":"02:45.945","Text":"2 constant variables so whatever."},{"Start":"02:45.945 ","End":"02:50.130","Text":"We have B and D, they are 1 and 1."},{"Start":"02:50.130 ","End":"02:53.400","Text":"We only have 2 unknowns here,"},{"Start":"02:53.400 ","End":"02:56.160","Text":"That\u0027s A and C after we substitute."},{"Start":"02:56.160 ","End":"02:58.430","Text":"Then we pick another value."},{"Start":"02:58.430 ","End":"03:03.200","Text":"For example, 3 could be anything."},{"Start":"03:03.200 ","End":"03:06.230","Text":"If we put in 3, you get 3-1 is 2,"},{"Start":"03:06.230 ","End":"03:10.650","Text":"3-2^2 is 1,"},{"Start":"03:10.650 ","End":"03:12.245","Text":"and you get 2, and so on."},{"Start":"03:12.245 ","End":"03:17.180","Text":"Substitute for all of them and you get the coefficients of A,"},{"Start":"03:17.180 ","End":"03:20.720","Text":"B, C, and D as 2,1,4, and 4."},{"Start":"03:20.720 ","End":"03:24.420","Text":"Then we again plug in B and D,"},{"Start":"03:24.420 ","End":"03:26.660","Text":"we already found before."},{"Start":"03:26.660 ","End":"03:33.620","Text":"We get another equation also in A and C. Now we have 2 equations in 2 unknowns,"},{"Start":"03:33.620 ","End":"03:39.875","Text":"A and C. Just put a curly brace around them."},{"Start":"03:39.875 ","End":"03:42.800","Text":"Everything\u0027s divisible by 2,"},{"Start":"03:42.800 ","End":"03:44.740","Text":"so we simplify a bit."},{"Start":"03:44.740 ","End":"03:47.220","Text":"Here if we divide by - 2,"},{"Start":"03:47.220 ","End":"03:48.965","Text":"we get 2 equals to A plus C,"},{"Start":"03:48.965 ","End":"03:57.570","Text":"and here we divide just by 2 and we get -2 equals A plus 2C."},{"Start":"03:57.570 ","End":"04:01.165","Text":"This is convenient, this way for this to be positive."},{"Start":"04:01.165 ","End":"04:05.370","Text":"What we can do is, for example,"},{"Start":"04:05.370 ","End":"04:10.425","Text":"say from here that C is equal to"},{"Start":"04:10.425 ","End":"04:18.345","Text":"2- 2A and then substitute that in the second one,"},{"Start":"04:18.345 ","End":"04:23.175","Text":"this C here plug-in, what is here."},{"Start":"04:23.175 ","End":"04:29.535","Text":"We get -2 equals A plus twice 2-2A."},{"Start":"04:29.535 ","End":"04:30.990","Text":"If we solve that,"},{"Start":"04:30.990 ","End":"04:33.120","Text":"we get A = 2."},{"Start":"04:33.120 ","End":"04:35.560","Text":"Then once we get that A =2,"},{"Start":"04:35.560 ","End":"04:40.565","Text":"then we substitute that in here and we get C is 2 - 4 is - 2."},{"Start":"04:40.565 ","End":"04:44.855","Text":"I\u0027m not going to go into every single detail here."},{"Start":"04:44.855 ","End":"04:46.190","Text":"Now we have B,"},{"Start":"04:46.190 ","End":"04:47.660","Text":"we have D, we have A,"},{"Start":"04:47.660 ","End":"04:54.890","Text":"and we have C and all that\u0027s left is to look back here and say that we put A, B, C,"},{"Start":"04:54.890 ","End":"05:00.750","Text":"and D, as according to see if I can fit it all in, yeah,"},{"Start":"05:00.750 ","End":"05:09.360","Text":"just put in respectively 2,1,- 2 and 1 in order."},{"Start":"05:09.360 ","End":"05:16.350","Text":"Then we get this."},{"Start":"05:16.350 ","End":"05:20.235","Text":"Like I said, the 2,1, - 2,1."},{"Start":"05:20.235 ","End":"05:21.450","Text":"That\u0027s the right 1."},{"Start":"05:21.450 ","End":"05:24.460","Text":"This is the answer. We\u0027re done."}],"ID":5282},{"Watched":false,"Name":"Exercise 13","Duration":"3m 38s","ChapterTopicVideoID":5277,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5277.jpeg","UploadDate":"2016-03-07T09:13:11.9770000","DurationForVideoObject":"PT3M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we have to take this expression,"},{"Start":"00:04.080 ","End":"00:09.525","Text":"this is the rational expression and decompose it into partial fractions."},{"Start":"00:09.525 ","End":"00:19.770","Text":"This is probably the first time we\u0027ve done a cubic term, x-1^3."},{"Start":"00:19.770 ","End":"00:23.515","Text":"I\u0027ll emphasize what we do here."},{"Start":"00:23.515 ","End":"00:30.020","Text":"What we have to do is to take representatives of x minus 1 and all powers from 1 up to 3."},{"Start":"00:30.020 ","End":"00:31.895","Text":"In other words, we need an x minus 1,"},{"Start":"00:31.895 ","End":"00:36.025","Text":"an x minus 1 squared and an x minus 1 cubed."},{"Start":"00:36.025 ","End":"00:37.320","Text":"Something like this,"},{"Start":"00:37.320 ","End":"00:38.344","Text":"we put a constant,"},{"Start":"00:38.344 ","End":"00:40.100","Text":"a over x minus 1,"},{"Start":"00:40.100 ","End":"00:41.840","Text":"b over x squared,"},{"Start":"00:41.840 ","End":"00:44.555","Text":"and c over the cubed of this thing."},{"Start":"00:44.555 ","End":"00:48.380","Text":"There are some difficulties, well, which we\u0027ll see."},{"Start":"00:48.380 ","End":"00:52.670","Text":"As usual, we multiply by this denominator"},{"Start":"00:52.670 ","End":"00:56.460","Text":"and this will be good for the common denominator for all,"},{"Start":"00:56.460 ","End":"00:58.950","Text":"and what we get here is x plus 4,"},{"Start":"00:58.950 ","End":"01:00.285","Text":"here we get the c,"},{"Start":"01:00.285 ","End":"01:06.600","Text":"and here we just have a missing x minus 1 squared here we have a missing x minus 1."},{"Start":"01:07.190 ","End":"01:13.355","Text":"The thing is that we usually used to get by with finding most of the constants,"},{"Start":"01:13.355 ","End":"01:19.040","Text":"by plugging in something that makes everything but 1, 0 here,"},{"Start":"01:19.040 ","End":"01:23.845","Text":"the only thing we can substitute to make something 0 is x equals 1,"},{"Start":"01:23.845 ","End":"01:26.405","Text":"and we certainly will do that."},{"Start":"01:26.405 ","End":"01:30.875","Text":"But that just gives us the value of C. Let\u0027s just go over this step."},{"Start":"01:30.875 ","End":"01:33.275","Text":"If x is 1, 1 plus 4 is 5,"},{"Start":"01:33.275 ","End":"01:36.950","Text":"this one is 0 because 1 minus 1 is 0, this one is 0,"},{"Start":"01:36.950 ","End":"01:39.320","Text":"and we get C. So C is 5,"},{"Start":"01:39.320 ","End":"01:42.470","Text":"but the other two substitutions,"},{"Start":"01:42.470 ","End":"01:45.155","Text":"we just substitute whatever value is convenient."},{"Start":"01:45.155 ","End":"01:47.960","Text":"For example, x equals 2."},{"Start":"01:47.960 ","End":"01:52.740","Text":"It will give us 2 minus 1 is 1,"},{"Start":"01:52.740 ","End":"01:58.195","Text":"and then we just get A plus B plus C is 6."},{"Start":"01:58.195 ","End":"02:03.965","Text":"But since we have from here that C is 5, basically,"},{"Start":"02:03.965 ","End":"02:10.110","Text":"we take 6 minus 5 is 1 so 1 equals A plus B,"},{"Start":"02:10.110 ","End":"02:12.660","Text":"and that\u0027s how got this."},{"Start":"02:12.660 ","End":"02:15.375","Text":"Then we try another value,"},{"Start":"02:15.375 ","End":"02:18.875","Text":"0 is often good because it\u0027s easy to compute with,"},{"Start":"02:18.875 ","End":"02:21.410","Text":"and if we put 0 in here,"},{"Start":"02:21.410 ","End":"02:25.295","Text":"we get minus 1 squared is 1."},{"Start":"02:25.295 ","End":"02:27.755","Text":"And here we get a minus 1."},{"Start":"02:27.755 ","End":"02:32.500","Text":"So we\u0027ve got A minus B plus C is 4,"},{"Start":"02:32.500 ","End":"02:39.230","Text":"and once again we have that C is 5,"},{"Start":"02:39.230 ","End":"02:40.910","Text":"same as above it from here."},{"Start":"02:40.910 ","End":"02:45.290","Text":"So this time if I subtract the 5 from both sides,"},{"Start":"02:45.290 ","End":"02:47.240","Text":"4 minus 5 is minus 1,"},{"Start":"02:47.240 ","End":"02:49.775","Text":"and I\u0027m left here with A minus B."},{"Start":"02:49.775 ","End":"02:53.510","Text":"We have two equations and two unknowns,"},{"Start":"02:53.510 ","End":"02:55.280","Text":"we won\u0027t spend a lot of time, but look,"},{"Start":"02:55.280 ","End":"02:57.275","Text":"if you add the two equations together,"},{"Start":"02:57.275 ","End":"03:01.820","Text":"you get 0 equals 2A so A is 0."},{"Start":"03:01.820 ","End":"03:04.240","Text":"Once you have A is 0,"},{"Start":"03:04.240 ","End":"03:06.870","Text":"then if you put A equal 0 here,"},{"Start":"03:06.870 ","End":"03:10.170","Text":"you get 1 equals B and C we already know is 5,"},{"Start":"03:10.170 ","End":"03:12.195","Text":"so we have our A,"},{"Start":"03:12.195 ","End":"03:13.545","Text":"B, and C,"},{"Start":"03:13.545 ","End":"03:17.720","Text":"and all that remains to do is to substitute the A, the B,"},{"Start":"03:17.720 ","End":"03:19.460","Text":"and the c in here,"},{"Start":"03:19.460 ","End":"03:25.400","Text":"and that gives us that the answer is this."},{"Start":"03:25.400 ","End":"03:27.680","Text":"We could leave this as the answer,"},{"Start":"03:27.680 ","End":"03:29.719","Text":"but really when you have a 0,"},{"Start":"03:29.719 ","End":"03:31.520","Text":"we don\u0027t want to write the 0 in,"},{"Start":"03:31.520 ","End":"03:34.685","Text":"so you would write the answer just as this plus this,"},{"Start":"03:34.685 ","End":"03:38.100","Text":"and we are done."}],"ID":5283},{"Watched":false,"Name":"Exercise 14","Duration":"3m 23s","ChapterTopicVideoID":5278,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5278.jpeg","UploadDate":"2016-03-07T09:13:45.6030000","DurationForVideoObject":"PT3M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.525","Text":"Here we have to decompose this rational expression into partial fractions."},{"Start":"00:06.525 ","End":"00:10.110","Text":"Note that the denominator has been factorized"},{"Start":"00:10.110 ","End":"00:13.980","Text":"and also note that the degree in the numerator,"},{"Start":"00:13.980 ","End":"00:15.120","Text":"which is 2,"},{"Start":"00:15.120 ","End":"00:17.070","Text":"is less than a degree in the denominator,"},{"Start":"00:17.070 ","End":"00:19.590","Text":"which is 3, so we can proceed."},{"Start":"00:19.590 ","End":"00:22.500","Text":"Because of the x minus 1^3,"},{"Start":"00:22.500 ","End":"00:24.600","Text":"we need to represent x minus 1,"},{"Start":"00:24.600 ","End":"00:28.140","Text":"x minus 1 squared and x minus 1^3."},{"Start":"00:28.140 ","End":"00:30.540","Text":"We put A over this, B over this,"},{"Start":"00:30.540 ","End":"00:33.630","Text":"C over this and as usual,"},{"Start":"00:33.630 ","End":"00:37.350","Text":"we first of all get rid of the denominators by multiplying by x"},{"Start":"00:37.350 ","End":"00:41.910","Text":"minus 1^3 and that leaves us with this A."},{"Start":"00:41.910 ","End":"00:44.690","Text":"It still has an x minus 1 squared missing and"},{"Start":"00:44.690 ","End":"00:49.760","Text":"its denominator B has a missing x minus 1 and C as is."},{"Start":"00:49.760 ","End":"00:51.800","Text":"Then we have to start finding A, B,"},{"Start":"00:51.800 ","End":"00:57.215","Text":"and C, for example by substituting values."},{"Start":"00:57.215 ","End":"01:02.150","Text":"One obvious thing to substitute is x=1 and if we do that,"},{"Start":"01:02.150 ","End":"01:06.165","Text":"then on the left we get 6 minus 4 plus 1 is 3."},{"Start":"01:06.165 ","End":"01:08.460","Text":"The 1 makes this and this disappear,"},{"Start":"01:08.460 ","End":"01:11.610","Text":"because these are 0 and all we\u0027re left with is C,"},{"Start":"01:11.610 ","End":"01:12.825","Text":"so C is 3."},{"Start":"01:12.825 ","End":"01:14.570","Text":"That\u0027s where we found 1 already,"},{"Start":"01:14.570 ","End":"01:18.980","Text":"but we can\u0027t make anything 0 by substituting anymore,"},{"Start":"01:18.980 ","End":"01:21.860","Text":"so we just choose other values wherever is convenient."},{"Start":"01:21.860 ","End":"01:24.520","Text":"For example, x equals 2,"},{"Start":"01:24.520 ","End":"01:27.710","Text":"it\u0027s easy to compute with because 2 minus 1 is 1,"},{"Start":"01:27.710 ","End":"01:30.620","Text":"so we have here just A plus B plus C,"},{"Start":"01:30.620 ","End":"01:34.520","Text":"as here it has 1 plus B times 1 plus C. On the left,"},{"Start":"01:34.520 ","End":"01:36.660","Text":"if we have x=2,"},{"Start":"01:36.660 ","End":"01:44.175","Text":"then we have 6 times 4 that is 24 minus 8 plus 1 in short 17."},{"Start":"01:44.175 ","End":"01:48.960","Text":"17 is A plus B plus C. However,"},{"Start":"01:48.960 ","End":"01:54.155","Text":"we know that C is 3 because it\u0027s written here."},{"Start":"01:54.155 ","End":"01:57.230","Text":"If we do 17 minus 3 is 14,"},{"Start":"01:57.230 ","End":"01:59.450","Text":"we get A plus B."},{"Start":"01:59.450 ","End":"02:01.760","Text":"We can also substitute x=0."},{"Start":"02:01.760 ","End":"02:05.585","Text":"That\u0027s often easy to compute with."},{"Start":"02:05.585 ","End":"02:11.370","Text":"If we put x equals 0, here we get just the 1 and here we get 1 squared is 1,"},{"Start":"02:11.370 ","End":"02:13.140","Text":"0 minus 1 is minus 1,"},{"Start":"02:13.140 ","End":"02:21.870","Text":"and we still have a C. This time we already have one equation with A plus B and here,"},{"Start":"02:21.870 ","End":"02:30.720","Text":"once again putting C equals 3 gives us that A minus B is 1."},{"Start":"02:30.720 ","End":"02:33.890","Text":"The 3 goes over to the other side and becomes minus 3."},{"Start":"02:33.890 ","End":"02:37.070","Text":"So we got minus 2 is a minus B."},{"Start":"02:37.070 ","End":"02:42.170","Text":"Now we have 2 equations and 2 unknowns, A and B."},{"Start":"02:42.170 ","End":"02:46.670","Text":"Now, I\u0027ll just give you the answer that A is 6 and B is 8."},{"Start":"02:46.670 ","End":"02:48.620","Text":"Though, if you want a quick way of seeing this,"},{"Start":"02:48.620 ","End":"02:50.165","Text":"if we add the two together,"},{"Start":"02:50.165 ","End":"02:53.610","Text":"we get 2A=12,"},{"Start":"02:53.610 ","End":"02:57.319","Text":"so A is 6 and if you subtract the 2 equations,"},{"Start":"02:57.319 ","End":"03:00.775","Text":"you get 14 minus minus 2 is 16."},{"Start":"03:00.775 ","End":"03:03.690","Text":"B minus minus B is 2B is 16,"},{"Start":"03:03.690 ","End":"03:06.000","Text":"so B is A, so we found already."},{"Start":"03:06.000 ","End":"03:11.515","Text":"We have everything and we just have to substitute this A, B, and C,"},{"Start":"03:11.515 ","End":"03:18.125","Text":"where it says so here and what that leaves us with is the following,"},{"Start":"03:18.125 ","End":"03:20.210","Text":"which is 6,"},{"Start":"03:20.210 ","End":"03:23.460","Text":"the 8, the 3, and this is our final answer."}],"ID":5284},{"Watched":false,"Name":"Exercise 15","Duration":"3m 48s","ChapterTopicVideoID":5279,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5279.jpeg","UploadDate":"2016-03-07T09:14:24.2900000","DurationForVideoObject":"PT3M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.790","Text":"In this exercise, we need to decompose this expression into partial fractions,"},{"Start":"00:05.790 ","End":"00:08.070","Text":"note that it\u0027s a rational expression and"},{"Start":"00:08.070 ","End":"00:11.070","Text":"the degree on top is less than the degree on the bottom,"},{"Start":"00:11.070 ","End":"00:15.090","Text":"because here it\u0027s 3 and here it\u0027s 2 so we can proceed."},{"Start":"00:15.090 ","End":"00:19.650","Text":"Now, when you have a linear factor x plus 2,"},{"Start":"00:19.650 ","End":"00:21.840","Text":"that just gets a constant,"},{"Start":"00:21.840 ","End":"00:24.345","Text":"but when you have a quadratic factor,"},{"Start":"00:24.345 ","End":"00:26.010","Text":"like x^2 plus 1,"},{"Start":"00:26.010 ","End":"00:30.570","Text":"which can\u0027t be reduced any further than you need an Ax plus B thing,"},{"Start":"00:30.570 ","End":"00:32.980","Text":"something x plus something."},{"Start":"00:33.710 ","End":"00:37.470","Text":"What we get is for the x^2 plus 1,"},{"Start":"00:37.470 ","End":"00:44.376","Text":"we have Ax plus B and for x plus 2, just a constant."},{"Start":"00:44.376 ","End":"00:48.850","Text":"I should have mentioned that the reason that we can\u0027t factorize x^2 plus 1 is that it has"},{"Start":"00:48.850 ","End":"00:54.190","Text":"no zeros x^2 plus 1 equals 0 would give us x^2 is minus 1,"},{"Start":"00:54.190 ","End":"00:56.670","Text":"which is not possible."},{"Start":"00:56.670 ","End":"01:00.980","Text":"That\u0027s why we need the Ax plus B and we"},{"Start":"01:00.980 ","End":"01:05.405","Text":"multiply out now by the common denominator, which is this."},{"Start":"01:05.405 ","End":"01:09.530","Text":"What we\u0027re left with is this times the missing factor,"},{"Start":"01:09.530 ","End":"01:12.665","Text":"which is x plus 2, and this times the x^2 plus 1."},{"Start":"01:12.665 ","End":"01:15.140","Text":"Now we want to try and find the constants A, B,"},{"Start":"01:15.140 ","End":"01:20.270","Text":"and C. We can substitute values of x this is not really inequality,"},{"Start":"01:20.270 ","End":"01:23.105","Text":"it\u0027s an identity for all x."},{"Start":"01:23.105 ","End":"01:26.330","Text":"Let\u0027s try x equals first of all,"},{"Start":"01:26.330 ","End":"01:31.365","Text":"we\u0027ll try the minus 2 that makes this thing 0."},{"Start":"01:31.365 ","End":"01:34.710","Text":"Minus 2 gives us on the left,"},{"Start":"01:34.710 ","End":"01:44.060","Text":"gives us, let\u0027s see minus 2^2 is 4,"},{"Start":"01:44.060 ","End":"01:45.690","Text":"4 times 8 is 8,"},{"Start":"01:45.690 ","End":"01:49.890","Text":"minus 4 is 5 that\u0027s correct."},{"Start":"01:49.890 ","End":"01:57.555","Text":"This is 0 here 2^2 plus 1 or minus 2^2 plus 1 is 5,"},{"Start":"01:57.555 ","End":"01:59.475","Text":"5 equals 5C,"},{"Start":"01:59.475 ","End":"02:01.155","Text":"and C is 1."},{"Start":"02:01.155 ","End":"02:06.605","Text":"That gives us 1 of the 3 constants."},{"Start":"02:06.605 ","End":"02:10.160","Text":"The next thing we can just let any value but zeros easy to"},{"Start":"02:10.160 ","End":"02:15.595","Text":"substitute so that gives us on the left 1 on the right,"},{"Start":"02:15.595 ","End":"02:19.605","Text":"0 plus 2 is 2 and this is 0."},{"Start":"02:19.605 ","End":"02:24.600","Text":"We get to B and here we get just C and we already"},{"Start":"02:24.600 ","End":"02:29.735","Text":"know that C is equal to 1 because we found that here."},{"Start":"02:29.735 ","End":"02:31.610","Text":"We get 1 equals 2B plus 1,"},{"Start":"02:31.610 ","End":"02:34.249","Text":"so 2B is 0, so B is 0."},{"Start":"02:34.249 ","End":"02:40.150","Text":"Next we can try substituting say 1 and then on the left we get, sorry,"},{"Start":"02:40.150 ","End":"02:42.565","Text":"2 plus 2 plus 1 is 5,"},{"Start":"02:42.565 ","End":"02:44.050","Text":"and then here,"},{"Start":"02:44.050 ","End":"02:48.095","Text":"A plus B times 1 plus 2 is 3."},{"Start":"02:48.095 ","End":"02:56.640","Text":"Here, C times 1 plus 1 is 2 and we already have that C is 1,"},{"Start":"02:56.640 ","End":"03:01.130","Text":"and B we know is 0."},{"Start":"03:01.130 ","End":"03:05.014","Text":"If we do that, we get 5 equals 3A"},{"Start":"03:05.014 ","End":"03:13.005","Text":"plus 2 and so 5 minus 2 is 3."},{"Start":"03:13.005 ","End":"03:15.410","Text":"A is 1 if you figure it out,"},{"Start":"03:15.410 ","End":"03:18.170","Text":"we do have all these 3 numbers, A, B,"},{"Start":"03:18.170 ","End":"03:22.620","Text":"and C and then we just put them here, here and here."},{"Start":"03:23.920 ","End":"03:28.280","Text":"What we get is this, of course,"},{"Start":"03:28.280 ","End":"03:30.965","Text":"I don\u0027t have to put the 0 in I mean,"},{"Start":"03:30.965 ","End":"03:33.710","Text":"Ax plus B is and the A is 1."},{"Start":"03:33.710 ","End":"03:36.245","Text":"You don\u0027t see that, but this is like,"},{"Start":"03:36.245 ","End":"03:41.345","Text":"if I did it literally it\u0027d be 1x plus 0 here but of course,"},{"Start":"03:41.345 ","End":"03:44.465","Text":"1x is just x and 0 is not needed."},{"Start":"03:44.465 ","End":"03:48.570","Text":"This is our answer and we are done."}],"ID":5285},{"Watched":false,"Name":"Exercise 16","Duration":"2m 51s","ChapterTopicVideoID":5280,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5280.jpeg","UploadDate":"2016-03-07T09:14:51.7070000","DurationForVideoObject":"PT2M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.060","Text":"Here we have yet another partial fraction decomposition of a rational expression."},{"Start":"00:06.060 ","End":"00:08.669","Text":"Note that the degree on top is 2,"},{"Start":"00:08.669 ","End":"00:13.005","Text":"the degree on the bottom is 3, so that\u0027s okay."},{"Start":"00:13.005 ","End":"00:17.160","Text":"Note also that x^2+1 can\u0027t be decomposed."},{"Start":"00:17.160 ","End":"00:18.645","Text":"It has no roots,"},{"Start":"00:18.645 ","End":"00:21.975","Text":"it can never be 0, it\u0027s always at least 1."},{"Start":"00:21.975 ","End":"00:27.544","Text":"For this we need a linear term and for this we need a constant term."},{"Start":"00:27.544 ","End":"00:30.720","Text":"The partial fraction decomposition looks something like this."},{"Start":"00:30.720 ","End":"00:32.910","Text":"For x^2+1 we get Ax+B,"},{"Start":"00:32.910 ","End":"00:39.350","Text":"and for x-3 we get a C. Multiply out by this denominator to get rid of it."},{"Start":"00:39.350 ","End":"00:41.090","Text":"What we get is this,"},{"Start":"00:41.090 ","End":"00:42.800","Text":"this times this,"},{"Start":"00:42.800 ","End":"00:45.150","Text":"and this times this."},{"Start":"00:45.290 ","End":"00:50.870","Text":"We can try substituting different values of x to try and compute A, B,"},{"Start":"00:50.870 ","End":"00:54.305","Text":"and C. If we let x=3,"},{"Start":"00:54.305 ","End":"00:59.585","Text":"for example, then this thing comes out to be 0."},{"Start":"00:59.585 ","End":"01:02.450","Text":"Here on the left we have 20. Why 20?"},{"Start":"01:02.450 ","End":"01:08.915","Text":"Because 3^2 is 9 times 2 is 18 plus 3 minus 1 is 20."},{"Start":"01:08.915 ","End":"01:10.580","Text":"Here, x minus 3 is 0,"},{"Start":"01:10.580 ","End":"01:13.910","Text":"so that\u0027s just 0, and 3^2 plus 1 is 10."},{"Start":"01:13.910 ","End":"01:15.780","Text":"So it\u0027s 10C."},{"Start":"01:15.780 ","End":"01:17.885","Text":"10C is 20, C is 2."},{"Start":"01:17.885 ","End":"01:23.540","Text":"Next we try something else like x=0 because it\u0027s easy to substitute."},{"Start":"01:23.540 ","End":"01:25.760","Text":"Here we get minus 1."},{"Start":"01:25.760 ","End":"01:28.170","Text":"Here we get 0A,"},{"Start":"01:28.170 ","End":"01:31.330","Text":"so A is not there,"},{"Start":"01:31.760 ","End":"01:34.237","Text":"also this x is not here,"},{"Start":"01:34.237 ","End":"01:35.810","Text":"we get minus 3B,"},{"Start":"01:35.810 ","End":"01:39.035","Text":"and then plus C times 1."},{"Start":"01:39.035 ","End":"01:40.400","Text":"But we already know C,"},{"Start":"01:40.400 ","End":"01:42.680","Text":"from here C is 2."},{"Start":"01:42.680 ","End":"01:45.920","Text":"So we have -1=-3B+2,"},{"Start":"01:45.920 ","End":"01:50.005","Text":"-3=-3B, so B is 1."},{"Start":"01:50.005 ","End":"01:53.165","Text":"Then we substitute, say x=1,"},{"Start":"01:53.165 ","End":"01:55.610","Text":"might be easy to substitute."},{"Start":"01:55.610 ","End":"01:58.696","Text":"We get 2=A+B,"},{"Start":"01:58.696 ","End":"02:03.115","Text":"1-3 is -2, 1+1 is 2."},{"Start":"02:03.115 ","End":"02:05.060","Text":"C and B, we already know."},{"Start":"02:05.060 ","End":"02:08.395","Text":"C is 2, B is 1."},{"Start":"02:08.395 ","End":"02:13.250","Text":"If we expand this and do the substitutions,"},{"Start":"02:13.250 ","End":"02:19.960","Text":"basically you get 2 is minus 2A minus 2 plus 4."},{"Start":"02:19.960 ","End":"02:21.980","Text":"The numbers on the left come out 0,"},{"Start":"02:21.980 ","End":"02:25.285","Text":"we get minus 2A is 0, so A is 0."},{"Start":"02:25.285 ","End":"02:30.500","Text":"Now that we have all of the A, B, and C,"},{"Start":"02:30.500 ","End":"02:35.630","Text":"we can put them here and what we get is"},{"Start":"02:35.630 ","End":"02:40.730","Text":"Ax plus B is just 0x plus 1,"},{"Start":"02:40.730 ","End":"02:42.123","Text":"so it\u0027s just 1,"},{"Start":"02:42.123 ","End":"02:43.775","Text":"and C is 2."},{"Start":"02:43.775 ","End":"02:51.960","Text":"So we get 1/(x^2+1) plus 2/x-3. That\u0027s the answer."}],"ID":5286},{"Watched":false,"Name":"Exercise 17","Duration":"3m 23s","ChapterTopicVideoID":5281,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5281.jpeg","UploadDate":"2016-03-07T09:15:27.6470000","DurationForVideoObject":"PT3M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.295","Text":"In this exercise, we have another partial fraction decomposition."},{"Start":"00:05.295 ","End":"00:07.860","Text":"We have a rational expression."},{"Start":"00:07.860 ","End":"00:10.470","Text":"This time it\u0027s going to be a bit different."},{"Start":"00:10.470 ","End":"00:16.080","Text":"Notice that in the denominator we have 2 quadratics and each of them is irreducible,"},{"Start":"00:16.080 ","End":"00:22.445","Text":"can\u0027t be factored because each of them can never be 0. x^2 can\u0027t be minus 1,"},{"Start":"00:22.445 ","End":"00:24.545","Text":"x^2 can\u0027t be minus 4."},{"Start":"00:24.545 ","End":"00:27.185","Text":"We have to leave these as is."},{"Start":"00:27.185 ","End":"00:31.490","Text":"For a quadratic factor in the denominator,"},{"Start":"00:31.490 ","End":"00:34.490","Text":"we need something x plus something, a linear term."},{"Start":"00:34.490 ","End":"00:41.420","Text":"In other words, the partial fraction form will be some linear Ax plus B over x^2 plus 1."},{"Start":"00:41.420 ","End":"00:45.340","Text":"On another linear Cx plus D over x^2 plus 4."},{"Start":"00:45.340 ","End":"00:48.500","Text":"The technique is going to be a bit different than usual."},{"Start":"00:48.500 ","End":"00:53.270","Text":"First, let\u0027s get rid of the denominator by multiplying by this."},{"Start":"00:53.270 ","End":"00:57.095","Text":"Then what we get is this,"},{"Start":"00:57.095 ","End":"00:59.610","Text":"which is an equation,"},{"Start":"00:59.610 ","End":"01:03.600","Text":"it\u0027s really an identity meaning for all x it\u0027s true."},{"Start":"01:03.850 ","End":"01:08.810","Text":"Rather than using the substitution method we\u0027ve been using,"},{"Start":"01:08.810 ","End":"01:11.979","Text":"we\u0027ll go about it a bit differently."},{"Start":"01:11.979 ","End":"01:13.720","Text":"We\u0027ll open up the brackets,"},{"Start":"01:13.720 ","End":"01:16.070","Text":"do some algebra on the right."},{"Start":"01:16.070 ","End":"01:21.280","Text":"This is routine, Ax times x^2 is ax^3 and so on."},{"Start":"01:21.280 ","End":"01:22.930","Text":"Each 1 of these with each 1 of these."},{"Start":"01:22.930 ","End":"01:25.645","Text":"In short, we expand this times this,"},{"Start":"01:25.645 ","End":"01:27.700","Text":"and we get these 4 terms,"},{"Start":"01:27.700 ","End":"01:30.235","Text":"this times this, we get the other 4 terms."},{"Start":"01:30.235 ","End":"01:33.370","Text":"Then we start collecting like terms."},{"Start":"01:33.370 ","End":"01:35.845","Text":"We collect the constant terms,"},{"Start":"01:35.845 ","End":"01:40.660","Text":"which we could also write as the x to the power of 0."},{"Start":"01:40.660 ","End":"01:42.880","Text":"I didn\u0027t have to write that."},{"Start":"01:42.880 ","End":"01:44.890","Text":"Then the x terms,"},{"Start":"01:44.890 ","End":"01:47.485","Text":"which are really x to the power of 1 terms,"},{"Start":"01:47.485 ","End":"01:50.140","Text":"and then x^2 terms, for example,"},{"Start":"01:50.140 ","End":"01:53.740","Text":"the x^2 we get from here and from here so it\u0027s B plus"},{"Start":"01:53.740 ","End":"01:57.380","Text":"D. The x^3 is from here and from here,"},{"Start":"01:57.380 ","End":"02:01.490","Text":"so it\u0027s A plus C and so on for all the terms."},{"Start":"02:01.490 ","End":"02:05.660","Text":"Then, because this is not a simple equality, it\u0027s an identity."},{"Start":"02:05.660 ","End":"02:08.150","Text":"These are the same polynomials."},{"Start":"02:08.150 ","End":"02:11.195","Text":"Then we know that at every level,"},{"Start":"02:11.195 ","End":"02:14.660","Text":"the constant term has to match."},{"Start":"02:14.660 ","End":"02:17.225","Text":"The constant term here is 3,"},{"Start":"02:17.225 ","End":"02:22.819","Text":"that\u0027s going to be 4B plus D. The term containing x here, which is 0,"},{"Start":"02:22.819 ","End":"02:28.690","Text":"has to be the 4A plus C. All the rest of them are going to be 0,"},{"Start":"02:28.690 ","End":"02:34.275","Text":"because there is no x^2 or x^3 on the left."},{"Start":"02:34.275 ","End":"02:38.045","Text":"There\u0027s only a constant term or x^0 term."},{"Start":"02:38.045 ","End":"02:42.175","Text":"Then the events, we get 4 equations in 4 unknowns."},{"Start":"02:42.175 ","End":"02:44.210","Text":"If we solve it,"},{"Start":"02:44.210 ","End":"02:45.860","Text":"and I\u0027m not going to spend the time solving it"},{"Start":"02:45.860 ","End":"02:48.110","Text":"because you know how to solve such a system."},{"Start":"02:48.110 ","End":"02:49.925","Text":"I\u0027ll just tell you the answers."},{"Start":"02:49.925 ","End":"02:51.570","Text":"B is 1, D is minus 1,"},{"Start":"02:51.570 ","End":"02:53.195","Text":"A is 0, C is 0."},{"Start":"02:53.195 ","End":"02:56.090","Text":"You can check by substituting them all so."},{"Start":"02:56.090 ","End":"02:58.220","Text":"Now that we have all these 4 constants,"},{"Start":"02:58.220 ","End":"03:03.945","Text":"we just put them in here in this decomposition."},{"Start":"03:03.945 ","End":"03:08.199","Text":"What we get is the following."},{"Start":"03:09.200 ","End":"03:14.850","Text":"This 1 is just 1 because A is 0,"},{"Start":"03:14.850 ","End":"03:19.960","Text":"C is also 0 and D is minus 1."},{"Start":"03:19.960 ","End":"03:23.010","Text":"So this is what we get and we are done."}],"ID":5287},{"Watched":false,"Name":"Exercise 18","Duration":"3m 2s","ChapterTopicVideoID":5282,"CourseChapterTopicPlaylistID":56154,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5282.jpeg","UploadDate":"2016-03-07T09:16:00.1500000","DurationForVideoObject":"PT3M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"Here we have yet another partial fraction decomposition"},{"Start":"00:03.810 ","End":"00:07.425","Text":"of a rational expression. This is the 1."},{"Start":"00:07.425 ","End":"00:12.659","Text":"We have a linear factor x and we have a quadratic factor, which is squared."},{"Start":"00:12.659 ","End":"00:15.420","Text":"When you have a squared quadratic factor,"},{"Start":"00:15.420 ","End":"00:17.670","Text":"you have to have it at all levels,"},{"Start":"00:17.670 ","End":"00:21.300","Text":"meaning at x squared plus 1 on its own and x squared plus 1 squared."},{"Start":"00:21.300 ","End":"00:24.780","Text":"Each of them has to have a linear expression on the top."},{"Start":"00:24.780 ","End":"00:27.480","Text":"But the x just have to have a constant expression."},{"Start":"00:27.480 ","End":"00:32.145","Text":"This is the general form of such an expression,"},{"Start":"00:32.145 ","End":"00:33.870","Text":"the x squared plus 1."},{"Start":"00:33.870 ","End":"00:35.490","Text":"First of all, it\u0027s squared,"},{"Start":"00:35.490 ","End":"00:37.620","Text":"so we have to take it each power."},{"Start":"00:37.620 ","End":"00:40.550","Text":"Because it\u0027s quadratic, a constant won\u0027t do,"},{"Start":"00:40.550 ","End":"00:43.800","Text":"a linear expression will do it."},{"Start":"00:44.900 ","End":"00:50.660","Text":"We multiply by the common denominator and get 1"},{"Start":"00:50.660 ","End":"00:58.350","Text":"equals A is missing the x squared plus 1 squared and Bx plus C,"},{"Start":"00:58.350 ","End":"01:02.505","Text":"we have to multiply by x and by x squared plus 1,"},{"Start":"01:02.505 ","End":"01:05.410","Text":"and the last one just by x."},{"Start":"01:05.500 ","End":"01:11.000","Text":"What we get is doing some algebra here."},{"Start":"01:11.000 ","End":"01:15.665","Text":"This thing squared comes out this from the A plus B squared,"},{"Start":"01:15.665 ","End":"01:19.310","Text":"the special binomial expansion."},{"Start":"01:19.310 ","End":"01:22.160","Text":"Here we\u0027re just multiplying,"},{"Start":"01:22.160 ","End":"01:24.050","Text":"put the x in here,"},{"Start":"01:24.050 ","End":"01:30.545","Text":"put the x in here and here we still have to multiply out this by this."},{"Start":"01:30.545 ","End":"01:35.165","Text":"Then here we have 2 terms times 2 terms and so on."},{"Start":"01:35.165 ","End":"01:41.510","Text":"In short after all the algebra and after collecting all the same powers together,"},{"Start":"01:41.510 ","End":"01:46.345","Text":"like constants, we have just Ax."},{"Start":"01:46.345 ","End":"01:52.040","Text":"With just x, we have E and C and that goes here."},{"Start":"01:52.040 ","End":"01:59.555","Text":"The x squareds are here and here and here and so on, collect things together."},{"Start":"01:59.555 ","End":"02:06.020","Text":"Now, we compare each power of x."},{"Start":"02:06.020 ","End":"02:09.180","Text":"The x^0 means constant term,"},{"Start":"02:09.180 ","End":"02:10.855","Text":"so A is 1,"},{"Start":"02:10.855 ","End":"02:15.275","Text":"and everything else is going to be 0,"},{"Start":"02:15.275 ","End":"02:19.640","Text":"C plus E has to be 0 because there are no x terms here."},{"Start":"02:19.640 ","End":"02:22.610","Text":"There\u0027s no x squared, there\u0027s no x cubed, there\u0027s no x^4."},{"Start":"02:22.610 ","End":"02:26.225","Text":"Each of these is 0, this is 0,"},{"Start":"02:26.225 ","End":"02:29.065","Text":"C is 0, they\u0027re all 0."},{"Start":"02:29.065 ","End":"02:33.710","Text":"We get 5 equations and 5 unknowns,"},{"Start":"02:33.710 ","End":"02:35.570","Text":"but it\u0027s not that bad because for example,"},{"Start":"02:35.570 ","End":"02:38.595","Text":"we already know that A is 1 and C is 0."},{"Start":"02:38.595 ","End":"02:43.988","Text":"In short, I\u0027m not going to spend time solving the system of equations."},{"Start":"02:43.988 ","End":"02:47.390","Text":"I\u0027ll just give you straightaway the answers."},{"Start":"02:47.390 ","End":"02:53.315","Text":"Then all that\u0027s left to do is to substitute these values in this expression here,"},{"Start":"02:53.315 ","End":"02:54.590","Text":"A, B, C, D,"},{"Start":"02:54.590 ","End":"02:58.370","Text":"and E. What we get is the following,"},{"Start":"02:58.370 ","End":"03:00.095","Text":"which is the answer."},{"Start":"03:00.095 ","End":"03:02.340","Text":"We are done."}],"ID":5288}],"Thumbnail":null,"ID":56154}]

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