Simplifying Rational Expressions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Simplifying 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

Simplifying a rational expression $\frac{p(x)}{q(x)}$ by factoring both the numerator and denominator, then canceling common factors. The domain must exclude values that make any original denominator zero.

Just like simplifying the fraction $\frac{6}{8} = \frac{3}{4}$ by canceling the common factor of 2, you can simplify $\frac{x^2 - 4}{x - 2}$ by factoring the top as $(x+2)(x-2)$ and canceling the common $(x-2)$ factor. But remember: you can only cancel FACTORS (things being multiplied), not TERMS (things being added).

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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: Reduce a polynomial fraction by factoring numerator and denominator and canceling common factors, never common terms.

Common stuck point: The procedure for simplifying rational expressions is the easy part; the trap is canceling terms instead of factors. Asking "Are the things I want to cancel FACTORS (multiplied) and not TERMS (added)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the things I want to cancel FACTORS (multiplied) and not TERMS (added)?

Worked Examples

Example 1

medium

Simplify

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

Answer

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

x \neq -3

First step

Step 1: Factor numerator:

x^2 - 9 = (x+3)(x-3)

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Example 2

easy

Simplify

\frac{4x^2}{2x}

Example 3

medium

Simplify

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

and state restrictions.

Example 4

hard

Simplify

\frac{x^2 - 2x - 8}{x^2 - 16}

x \neq \pm 4

Example 5

hard

Simplify

\frac{x^2 - 3x - 10}{x^2 - 7x + 10}

x \neq 2, 5

Practice Problems

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

Example 1

easy

Simplify

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

Example 2

hard

Simplify

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

Example 3

easy

Simplify

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

x\neq2

Example 4

easy

Simplify

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

x\neq0

Example 5

easy

Simplify

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

x\neq-3

Example 6

easy

Simplify

\frac{2x + 6}{x + 3}

x\neq-3

Example 7

easy

Simplify

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

x\neq0

Example 8

easy

Simplify

\frac{4x^2}{2x}

x\neq0

Example 9

easy

Simplify

\frac{x - 5}{5 - x}

x\neq5

Example 10

easy

Simplify

\frac{6x^3}{9x}

x\neq0

Example 11

medium

Simplify

\frac{x^2 - x - 6}{x^2 - 9}

and state restrictions.

Example 12

medium

Simplify

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

, state restrictions.

Example 13

medium

Simplify

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

Example 14

medium

Simplify

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

Example 15

medium

Simplify

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

, state restrictions.

Example 16

medium

Simplify

\frac{2x^2 + 7x + 3}{x^2 + 6x + 9}

Example 17

medium

Simplify

\frac{x^2 - 2x - 8}{x^2 - 16}

, state restrictions.

Example 18

medium

Simplify

\frac{x^2 - 25}{x^2 - 10x + 25}

, state restrictions.

Example 19

medium

Simplify

\frac{3x^2 - 12}{x^2 - x - 2}

, state restrictions.

Example 20

challenge

Simplify

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

, state restrictions.

Example 21

challenge

Simplify

\frac{2x^2 - 5x - 3}{2x^2 + 5x + 2}

, state restrictions.

Example 22

challenge

Simplify

\frac{x^2 - y^2}{x^2 - 2xy + y^2}

x\neq y

Example 23

easy

Simplify

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

x \neq -4

Example 24

easy

Simplify

\frac{5x^2}{10x}

x \neq 0

Example 25

easy

Simplify

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

x \neq 1

Example 26

medium

Simplify

\frac{x^2 + 7x + 12}{x^2 + 4x}

x \neq 0, -4

Example 27

medium

Simplify

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

x \neq -2

Example 28

medium

Simplify

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

x \neq \pm 2

Example 29

medium

Simplify

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

x \neq 2, 3

Example 30

medium

Simplify

\frac{3 - x}{x - 3}

x \neq 3

Example 31

medium

Simplify

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

x \neq \pm 1

Example 32

medium

Simplify

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

x \neq \pm 3

Example 33

hard

Simplify

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

x \neq \pm 1

Example 34

hard

Simplify

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

x \neq \pm 2

Example 35

hard

Simplify

\frac{2x^2 + 7x + 3}{x^2 + 2x - 3}

x \neq -3, 1

Example 36

hard

Simplify

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

x \neq 2, -1

Example 37

hard

Simplify

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

x

such that denominator

\neq 0

Example 38

medium

Simplify

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

x \neq 0

Example 39

hard

Simplify

\frac{x^2 + 2x - 15}{x^2 + 6x + 5}

x \neq -1, -5

Example 40

hard

Simplify

\frac{2x^3 - 18x}{x^2 - 9}

x \neq \pm 3

Example 41

challenge

Simplify

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

Example 42

challenge

Simplify

\frac{a^3 - b^3}{a^2 - b^2}

a \neq \pm b

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Related Concepts

Factoring Expressions Multiplying and Dividing Rational Expressions Adding and Subtracting Rational Expressions

Background Knowledge

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

factoringexpressions