In Exercises 15–32, multiply or divide as indicated. (x^3−25x)/4x^2 ⋅ (2x^2−2)/(x^2−6x+5) ÷ (x^2+5x)/(7x+7)

Verified step by step guidance

1. First, rewrite the division as multiplication by the reciprocal. The expression becomes: \( \frac{x^3 - 25x}{4x^2} \cdot \frac{2x^2 - 2}{x^2 - 6x + 5} \cdot \frac{7x + 7}{x^2 + 5x} \).
2. Factor each polynomial where possible. For example, factor \( x^3 - 25x \) as \( x(x^2 - 25) \) and further as \( x(x - 5)(x + 5) \).
3. Continue factoring: \( 2x^2 - 2 \) becomes \( 2(x^2 - 1) \) and further as \( 2(x - 1)(x + 1) \). Factor \( x^2 - 6x + 5 \) as \( (x - 5)(x - 1) \).
4. Factor \( x^2 + 5x \) as \( x(x + 5) \) and \( 7x + 7 \) as \( 7(x + 1) \).
5. Cancel out common factors in the numerator and the denominator across the entire expression.

Key Concepts

Polynomial Operations

Polynomial operations involve the addition, subtraction, multiplication, and division of polynomial expressions. In this question, we are required to multiply and divide polynomials, which requires understanding how to combine like terms and apply the distributive property. Recognizing the structure of polynomials is essential for simplifying expressions correctly.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This is crucial in simplifying expressions, especially when dividing polynomials, as it allows for cancellation of common factors. For example, recognizing that x^2 - 6x + 5 can be factored into (x - 1)(x - 5) simplifies the division process.

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Understanding how to manipulate these expressions, including multiplying and dividing them, is key to solving the problem. When dividing rational expressions, it is important to multiply by the reciprocal of the divisor, which can lead to further simplification and cancellation of terms.