1Cademy - Simplifying $$\frac{14-2x}{x^2-49}$$

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Opposites in a Rational Expression

Example

Simplifying $\frac{14-2x}{x^2-49}$

Simplify the rational expression $\frac{14-2x}{x^2-49}$ by factoring the numerator and denominator, recognizing opposite factors, and simplifying.

Step 1 — Factor the numerator and denominator. The numerator $14 - 2x$ has a GCF of $2$ : $14 - 2x = 2(7 - x)$ . The denominator $x^2 - 49$ is a difference of squares, since $x^2 = x^2$ and $49 = 7^2$ : $x^2 - 49 = (x - 7)(x + 7)$ . The expression becomes:

$\frac{2(7 - x)}{(x - 7)(x + 7)}$

Step 2 — Recognize the opposite factors. The factor $(7 - x)$ in the numerator and the factor $(x - 7)$ in the denominator are opposites of each other, since $7 - x = -(x - 7)$ . By the opposite factors property, $\frac{7 - x}{x - 7} = -1$ .

Step 3 — Simplify. Replace the ratio of opposite factors with $-1$ :

$\frac{-2}{x + 7}$

The simplified result is $\frac{-2}{x + 7}$ . This example extends the basic opposite-factors simplification $\frac{x - 8}{8 - x} = -1$ to a more complex expression where the opposite factors do not make up the entire numerator and denominator — they are embedded among other factors. After factoring the GCF from the numerator and applying the difference of squares pattern to the denominator, the opposite pair $(7 - x)$ and $(x - 7)$ becomes visible and contributes a factor of $-1$ to the simplified result.

Updated 2026-04-21

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OpenStax Elementary Algebra

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