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

Verified step by step guidance

1. Identify the operation: We need to divide two rational expressions: \( \frac{x^2 - 4}{x - 2} \) and \( \frac{x + 2}{4x - 8} \).
2. Rewrite the division as multiplication by the reciprocal: \( \frac{x^2 - 4}{x - 2} \times \frac{4x - 8}{x + 2} \).
3. Factor the numerators and denominators where possible: \( x^2 - 4 \) is a difference of squares, so it factors to \((x - 2)(x + 2)\). The expression \(4x - 8\) can be factored as \(4(x - 2)\).
4. Substitute the factored forms into the expression: \( \frac{(x - 2)(x + 2)}{x - 2} \times \frac{4(x - 2)}{x + 2} \).
5. Cancel out common factors in the numerator and the denominator: The \(x - 2\) and \(x + 2\) terms can be canceled where applicable.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the 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.