College Algebra

To solve the given mathematical expressions, we will follow a step-by-step approach to simplify each one. Let's start with the first expression: 1. \( y = 10 + 3x - 2.5x^2 + 6x^{1/2} \) This expression is already in its simplest form, as there are no like terms to combine. The terms consist of a constant, a linear term, a quadratic term, and a square root term. Since they are all different types of terms, we cannot simplify this expression any further. Now, let's move on to the second expression: 2. \( y = 2x(4x^2 + x - 3x^2) \) First, we will simplify the expression inside the parentheses by combining like terms: \( 4x^2 - 3x^2 = x^2 \) So the expression inside the parentheses becomes: \( x^2 + x \) Now, we distribute the \( 2x \) across the terms inside the parentheses: \( y = 2x \cdot x^2 + 2x \cdot x \) Simplify the terms: \( y = 2x^3 + 2x^2 \) This is the simplified form of the second expression. Finally, let's simplify the third expression: 3. \( y = (10x - 4x^2) - (2 + \frac{3}{x}) \) First, distribute the negative sign across the terms in the second set of parentheses: \( y = 10x - 4x^2 - 2 - \frac{3}{x} \) Now, we have an expression with a linear term, a quadratic term, a constant, and a term with \( x \) in the denominator. Since these are all different types of terms, we cannot combine them any further. Therefore, the simplified form of the third expression is: \( y = -4x^2 + 10x - 2 - \frac{3}{x} \) In summary, the simplified forms of the given expressions are: 1. \( y = 10 + 3x - 2.5x^2 + 6x^{1/2} \) (already simplified) 2. \( y = 2x^3 + 2x^2 \) 3. \( y = -4x^2 + 10x - 2 - \frac{3}{x} \)

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