1Cademy - Dividing $$\frac{2x^2+5x-12}{x^2-16} \div \frac{2x^2-13x+15}{x^2-8x+16}$$

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How to Divide Rational Expressions

Example

Dividing $\frac{2x^2+5x-12}{x^2-16} \div \frac{2x^2-13x+15}{x^2-8x+16}$

Divide $\frac{2x^2+5x-12}{x^2-16} \div \frac{2x^2-13x+15}{x^2-8x+16}$ .

Step 1 — Rewrite as multiplication by the reciprocal. Flip the second fraction and change division to multiplication:

$\frac{2x^2+5x-12}{x^2-16} \cdot \frac{x^2-8x+16}{2x^2-13x+15}$

Step 2 — Factor the numerators and denominators completely. Each polynomial requires a different factoring technique:

$2x^2+5x-12 = (2x-3)(x+4)$ — trinomial with leading coefficient $2$ , factored by trial and error or the ac method.
$x^2-16 = (x-4)(x+4)$ — difference of squares, since $x^2 = x^2$ and $16 = 4^2$ .
$x^2-8x+16 = (x-4)(x-4)$ — perfect square trinomial, since $x^2 - 2(x)(4) + 4^2 = (x-4)^2$ .
$2x^2-13x+15 = (2x-3)(x-5)$ — trinomial with leading coefficient $2$ .

The expression becomes:

$\frac{(2x-3)(x+4)(x-4)(x-4)}{(x-4)(x+4)(2x-3)(x-5)}$

Step 3 — Multiply the numerators and denominators (already shown combined above).

Step 4 — Simplify by dividing out common factors. Cancel the shared factors $(2x-3)$ , $(x+4)$ , and $(x-4)$ from the numerator and denominator:

$\frac{x-4}{x-5}$

This example illustrates dividing rational expressions in which all four polynomials require factoring — combining the difference of squares pattern, the perfect square trinomial pattern, and trinomial factoring with a leading coefficient other than $1$ . After converting the division to multiplication by the reciprocal, the factored forms reveal three common factors that cancel, leaving a simple rational expression.

Updated 2026-04-21

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OpenStax Elementary Algebra

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