1Cademy - Simplifying $$ \frac{x^2-4x-32}{64-x^2} $$

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Opposites in a Rational Expression

Example

Simplifying $\frac{x^2-4x-32}{64-x^2}$

Simplify the rational expression $\frac{x^2 - 4x - 32}{64 - x^2}$ by factoring the numerator and denominator, recognizing opposite factors, and simplifying.

Step 1 — Factor the numerator and denominator.

Numerator: The trinomial $x^2 - 4x - 32$ requires two numbers whose product is $-32$ and whose sum is $-4$ . These numbers are $-8$ and $4$ , so $x^2 - 4x - 32 = (x - 8)(x + 4)$ .
Denominator: The binomial $64 - x^2$ is a difference of squares factored as $64 - x^2 = (8 - x)(8 + x)$ .

Substituting these factors into the fraction gives: $\frac{(x - 8)(x + 4)}{(8 - x)(8 + x)}$

Step 2 — Recognize opposite factors. The factors $(x - 8)$ in the numerator and $(8 - x)$ in the denominator are opposites since $8 - x = -(x - 8)$ . Their ratio simplifies to $-1$ : $\frac{x - 8}{8 - x} = -1$

Step 3 — Simplify. Replace the ratio of opposite factors with $-1$ and rewrite the remaining denominator factor as $(8 + x) = (x + 8)$ : $-1 · \frac{x + 4}{x + 8} = \frac{-(x + 4)}{x + 8}$