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 as a trinomial, factoring the denominator as a difference of squares, recognizing opposite factors, and simplifying.

Step 1 — Factor the numerator and denominator. The numerator $x^2 - 4x - 32$ is a trinomial requiring two numbers whose product is $-32$ and whose sum is $-4$ : the pair $-8$ and $4$ works, since $(-8)(4) = -32$ and $-8 + 4 = -4$ . So $x^2 - 4x - 32 = (x - 8)(x + 4)$ . The denominator $64 - x^2$ is a difference of squares written with the constant first: $64 = 8^2$ and $x^2 = x^2$ , so $64 - x^2 = (8 - x)(8 + x)$ . The expression becomes:

$\frac{(x - 8)(x + 4)}{(8 - x)(8 + x)}$

Step 2 — Recognize the opposite factors. The factor $(x - 8)$ in the numerator and the factor $(8 - x)$ in the denominator are opposites, since $8 - x = -(x - 8)$ . By the opposite factors property, their ratio equals $-1$ .

Step 3 — Simplify. Replace the ratio of opposite factors with $-1$ and note that $(8 + x) = (x + 8)$ :

$\frac{-(x + 4)}{x + 8}$

The simplified result is $\frac{-(x + 4)}{x + 8}$ . Unlike Example 8.15 where the numerator required only GCF factoring, this example requires trinomial factoring for the numerator and the difference of squares pattern for the denominator. The denominator $64 - x^2$ is written in "backwards" order (constant first), which produces the factor $(8 - x)$ — the opposite of $(x - 8)$ from the numerator. Recognizing opposite factors within a fully factored expression containing multiple binomial factors is the key skill demonstrated here.

Updated 2026-04-21

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OpenStax Elementary Algebra

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