1Cademy - Dividing $$\frac{a^2-b^2}{3ab} \div (a^2+2ab+b^2)$$

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How to Divide Rational Expressions

Example

Dividing $\frac{a^2-b^2}{3ab} \div (a^2+2ab+b^2)$

Divide $\frac{a^2-b^2}{3ab} \div (a^2+2ab+b^2)$ .

Step 1 — Write the second expression as a fraction. Since $(a^2+2ab+b^2)$ is a polynomial (not already written as a fraction), place it over $1$ :

$\frac{a^2-b^2}{3ab} \div \frac{a^2+2ab+b^2}{1}$

Step 2 — Rewrite as multiplication by the reciprocal. Flip the second fraction and change division to multiplication:

$\frac{a^2-b^2}{3ab} \cdot \frac{1}{a^2+2ab+b^2}$

Step 3 — Factor the numerators and denominators, then multiply. Factor $a^2-b^2$ as a difference of squares: $(a-b)(a+b)$ . Factor $a^2+2ab+b^2$ as a perfect square trinomial: $(a+b)(a+b)$ . The expression becomes:

$\frac{(a-b)(a+b) \cdot 1}{3ab \cdot (a+b)(a+b)}$

Step 4 — Simplify by dividing out common factors. Cancel the common factor $(a+b)$ from the numerator and denominator:

$\frac{a-b}{3ab(a+b)}$

This example demonstrates dividing a rational expression by a polynomial that is not written as a fraction. The first step is to express the polynomial as a fraction by placing it over $1$ , after which the standard four-step division procedure applies. The problem combines the difference of squares pattern $a^2-b^2 = (a-b)(a+b)$ with the perfect square trinomial pattern $a^2+2ab+b^2 = (a+b)^2$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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