Multiplying and Dividing Rational Expressions Examples

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Multiplying and Dividing Rational Expressions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Multiplying rational expressions by multiplying numerators together and denominators together (after factoring and canceling). Dividing by multiplying by the reciprocal of the divisor.

It works exactly like multiplying and dividing numeric fractions. To multiply: factor everything, cancel common factors across any numerator and any denominator, then multiply across. To divide: flip the second fraction and multiply. $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$ .

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Treat polynomial fractions like numeric fractions: multiply straight across after canceling, and divide by multiplying by the reciprocal.

Common stuck point: The procedure for multiplying and dividing rational expressions is the easy part; the trap is flipping the wrong fraction. Asking "Are the fractions joined by $\times$ or $\div$ (so I cancel and multiply across) rather than $+$ or $-$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the fractions joined by $\times$ or $\div$ (so I cancel and multiply across) rather than $+$ or $-$ ?

Worked Examples

Example 1

medium

Multiply

\frac{x^2 - 1}{x + 3} \cdot \frac{x + 3}{x + 1}

Answer

x - 1

x \neq -3, -1

First step

Step 1: Factor:

\frac{(x+1)(x-1)}{x+3} \cdot \frac{x+3}{x+1}

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Example 2

hard

Divide

\frac{x^2 - 4}{x^2 + x} \div \frac{x - 2}{x}

Example 3

easy

Multiply

\frac{x-1}{2}\cdot\frac{8}{x-1}

x\neq 1

Example 4

medium

Multiply

\frac{x^2 + 6x + 9}{x^2 - 9}\cdot\frac{x-3}{x+3}

Example 5

medium

Divide

\frac{x^2 - 9}{x^2 + 6x + 9}\div\frac{x-3}{x+3}

Example 6

hard

Simplify

\frac{3x^2 + 6x}{x^2 + 5x + 6}\cdot\frac{x+3}{3x}

, state restrictions.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Multiply

\frac{3}{x} \cdot \frac{x^2}{6}

Example 2

medium

Divide

\frac{x + 5}{x - 1} \div \frac{x + 5}{x^2 - 1}

Example 3

easy

Multiply

\frac{2}{x}\cdot\frac{x}{3}

x\neq0

Example 4

easy

Multiply

\frac{3}{x}\cdot\frac{x^2}{6}

x\neq0

Example 5

easy

Divide

\frac{x}{4}\div\frac{x}{2}

x\neq0

Example 6

easy

Multiply

\frac{x + 1}{2}\cdot\frac{4}{x + 1}

x\neq-1

Example 7

easy

Divide

\frac{6}{x}\div\frac{3}{x^2}

x\neq0

Example 8

easy

Multiply

\frac{x}{x + 2}\cdot\frac{x + 2}{5}

x\neq-2

Example 9

easy

Multiply

\frac{2x}{3}\cdot\frac{9}{4x}

x\neq0

Example 10

easy

Divide

\frac{x + 3}{x}\div\frac{x + 3}{2x}

x\neq0,-3

Example 11

medium

Multiply

\frac{x^2 - 4}{x + 3}\cdot\frac{x + 3}{x - 2}

, state restrictions.

Example 12

medium

Multiply

\frac{x^2 - 9}{x^2 + 4x + 4}\cdot\frac{x + 2}{x - 3}

Example 13

medium

Divide

\frac{x^2 - 1}{x + 2}\div\frac{x - 1}{x + 2}

, state restrictions.

Example 14

medium

Multiply

\frac{x^2 + 5x + 6}{x^2 - 4}\cdot\frac{x - 2}{x + 3}

Example 15

medium

Divide

\frac{2x^2 + x - 3}{x^2 - 1}\div\frac{2x + 3}{x + 1}

Example 16

medium

Multiply

\frac{x}{x - 5}\cdot\frac{x^2 - 25}{x}

x\neq0,5

Example 17

medium

Divide

\frac{x^2 - 6x + 9}{x + 1}\div\frac{x - 3}{x^2 - 1}

Example 18

medium

Divide

\frac{x^2 - 4}{x + 1}\div\frac{x + 2}{x^2 - 1}

Example 19

medium

Multiply

\frac{x^2 + 2x}{x^2 - 9}\cdot\frac{x - 3}{x}

x\neq0,\pm3

Example 20

challenge

Multiply

\frac{x^2 - x - 6}{x^2 + 2x - 8}\cdot\frac{x^2 + 5x + 4}{x^2 - 9}

Example 21

challenge

Divide

\frac{x^3 - 8}{x^2 - 4}\div\frac{x^2 + 2x + 4}{x + 2}

, state restrictions.

Example 22

challenge

Multiply

\frac{2x^2 - 3x - 2}{x^2 - 4}\cdot\frac{x + 2}{2x + 1}

, state restrictions.

Example 23

easy

Multiply

\frac{5}{x}\cdot\frac{x}{10}

x\neq 0

Example 24

easy

Multiply

\frac{x}{x+1}\cdot\frac{x+1}{x^2}

x\neq 0,-1

Example 25

easy

Divide

\frac{x}{6}\div\frac{1}{3}

x

any real.

Example 26

easy

Divide

\frac{5x}{2}\div\frac{x}{4}

x\neq 0

Example 27

medium

Multiply

\frac{x^2 + 3x}{x^2 - 9}\cdot\frac{x-3}{x}

x\neq 0,\pm 3

Example 28

medium

Multiply

\frac{x^2 - 1}{x+3}\cdot\frac{x+3}{x-1}

, state the restrictions.

Example 29

medium

Divide

\frac{x^2 - 4}{x+5}\div\frac{x-2}{x+5}

Example 30

medium

Divide

\frac{x^2 + 7x + 12}{x+2}\div\frac{x+3}{x+2}

Example 31

medium

Multiply

\frac{x^2 - x - 12}{x+2}\cdot\frac{x+2}{x-4}

Example 32

medium

Multiply

\frac{2x^2 + 4x}{x^2 - 1}\cdot\frac{x-1}{2x}

x\neq 0,\pm 1

Example 33

medium

Divide

\frac{4x^2 - 9}{x^2 - 4}\div\frac{2x-3}{x-2}

Example 34

hard

Multiply

\frac{x^3 + 27}{x^2 - 9}\cdot\frac{x-3}{x^2 - 3x + 9}

Example 35

hard

Divide

\frac{x^2 + x - 6}{x^2 + 4x + 4}\div\frac{x-2}{x+2}

Example 36

hard

Multiply

\frac{x^2 - 5x + 6}{x^2 - 2x - 3}\cdot\frac{x^2 - 9}{x^2 - 4}

Example 37

hard

Divide

\frac{x^2 - 16}{x^2 + 8x + 16}\div\frac{x-4}{x+4}

Example 38

hard

Multiply

\frac{3x^2 - 12}{x^2 + x - 6}\cdot\frac{x+3}{6}

Example 39

hard

Divide

\frac{x^2 - 7x + 10}{x^2 - 25}\div\frac{x-2}{x+5}

Example 40

challenge

Simplify

\frac{x^3 - 1}{x^2 - 1}\cdot\frac{x+1}{x^2 + x + 1}

, state restrictions.

Example 41

challenge

Simplify

\frac{6x^2 - x - 1}{2x^2 + 5x + 2}\div\frac{3x+1}{x+2}

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Related Concepts

Simplifying Rational Expressions Factoring Adding and Subtracting Rational Expressions Solving Rational Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

simplifying rational expressionsfactoring