In Exercises 15–32, multiply or divide as indicated. (x²−4)/(x−2) ÷ (x+2)/(4x−8)

Verified step by step guidance

1. Rewrite the division problem as a multiplication problem by multiplying the first fraction by the reciprocal of the second fraction. This gives: \( \frac{x^2 - 4}{x - 2} \times \frac{4x - 8}{x + 2} \).
2. Factorize the numerator \(x^2 - 4\) in the first fraction using the difference of squares formula: \(x^2 - 4 = (x - 2)(x + 2)\).
3. Factorize the numerator \(4x - 8\) in the second fraction by factoring out the greatest common factor (GCF), which is 4: \(4x - 8 = 4(x - 2)\).
4. Substitute the factored forms into the expression: \( \frac{(x - 2)(x + 2)}{x - 2} \times \frac{4(x - 2)}{x + 2} \).
5. Simplify the expression by canceling out common factors in the numerator and denominator, such as \(x - 2\) and \(x + 2\), where applicable.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Understanding how to manipulate these expressions, including simplifying, multiplying, and dividing them, is crucial for solving problems involving them. In this question, we are dealing with rational expressions that require simplification before performing operations.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential for simplifying rational expressions, as it allows us to cancel common factors in the numerator and denominator. In the given expression, recognizing that both the numerator and denominator can be factored will facilitate easier computation.

Division of Fractions

Dividing fractions involves multiplying by the reciprocal of the divisor. In this case, to divide the first rational expression by the second, we will multiply the first expression by the reciprocal of the second. This concept is fundamental in algebra and is key to solving the problem correctly.