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).

Read the full concept explanation →

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: Factor completely, then cancel common factors. Only factors (multiplicative parts) cancel—never cancel terms across addition or subtraction.

Common stuck point: You can only cancel common FACTORS, not individual terms. \frac{x + 3}{x + 5} \neq \frac{3}{5}—the x's are terms, not factors.

Sense of Study hint: Factor the numerator and denominator completely, then cross out only the factors that appear in both.

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: Factor numerator: x^2 - 9 = (x+3)(x-3).
  2. 2
    Step 2: Factor denominator: x^2 + 5x + 6 = (x+2)(x+3).
  3. 3
    Step 3: Cancel common factor (x+3): \frac{(x+3)(x-3)}{(x+2)(x+3)} = \frac{x-3}{x+2}, x \neq -3.
  4. 4
    Check: At x = 1: \frac{1-9}{1+5+6} = \frac{-8}{12} = -\frac{2}{3} and \frac{1-3}{1+2} = -\frac{2}{3} ✓

Answer

\frac{x - 3}{x + 2}, x \neq -3
To simplify a rational expression, factor both numerator and denominator completely, then cancel common factors. Always note the excluded values where the original denominator was zero.

Example 2

easy
Simplify \frac{4x^2}{2x}.

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}.
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Background Knowledge

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

factoringexpressions