Simplifying Rational Expressions

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Simplify rational expressions by factoring and canceling correctly.
  • Identify and state variable restrictions based on denominators.
  • Rewrite rational expressions in their simplest form.

Key Ideas

A rational expression is a fraction whose numerator and denominator are polynomials.

To simplify a rational expression:

  1. Factor the numerator and denominator completely.
  2. Cancel common factors (never cancel individual terms).
  3. State restrictions—values that make the denominator equal zero.

This process works just like simplifying numeric fractions, but with polynomials.

Common Problem Types

1. Factor–Then–Cancel Expressions

Factor fully before simplifying.

2. Numerical and Variable Cancellation

Cancel common numerical and variable factors only.

3. Restriction Identification

Determine which values make the denominator zero.

4. Expressions Already Partially Simplified

Check for hidden factors you can still factor out.

Strategies

  • Always factor first—canceling before factoring leads to mistakes.
  • Write denominator restrictions before canceling factors.
  • Treat polynomials like numbers: cancel only whole factors, never parts of sums.
  • Look for common binomial factors like \((x - 3)\) or \((x + 4)\).
  • If unsure, rewrite the expression step by step to avoid skipping logic.

Worked Examples

Example 1 — Factor Completely

Simplify: \[ \frac{x^2 - 9}{x^2 - 6x + 9} \]

Solution:
Factor numerator and denominator:

\[ \begin{split} x^2 - 9 &= (x - 3)(x + 3), \\ x^2 - 6x + 9 &= (x - 3)^2 \end{split} \]

Cancel one \((x - 3)\) factor:

\[ \frac{x + 3}{x - 3} \]

Restrictions:
\[ x \ne 3. \]

Example 2 — Cancel Numerical & Variable Factors

Simplify: \[ \frac{4x}{12x^2} \]

Solution:
Cancel numerical factors:

\[ \frac{4}{12} = \frac{1}{3} \]

Cancel \(x\) with one \(x\) in \(x^2\):

\[ \frac{1}{3x} \]

Restriction:
\[ x \ne 0 \]

WarningCommon Mistakes
  • Canceling terms instead of factors (e.g., canceling \(x\) from \(x + 3\)).
  • Forgetting to state restrictions after simplifying.
  • Not factoring completely before canceling.

Practice Problems

  1. \(\dfrac{x^2 - 4}{x^2 - x - 6}\)
  2. \(\dfrac{3x}{9x^2}\)
  3. \(\dfrac{x^2 + 5x}{x}\)
  4. \(\dfrac{5x^2 - 20}{10x}\)
  5. \(\dfrac{y^2 - 16}{y + 4}\)

1.
\[ x^2 - 4 = (x - 2)(x + 2), \quad x^2 - x - 6 = (x - 3)(x + 2) \] Cancel \((x + 2)\): \[ \frac{x - 2}{x - 3}, \quad x \ne 3, -2 \]

2.
\[ \frac{3x}{9x^2} = \frac{1}{3x}, \quad x \ne 0 \]

3.
\[ \frac{x^2 + 5x}{x} = \frac{x(x + 5)}{x} = x + 5, \quad x \ne 0 \]

4.
\[ \frac{5x^2 - 20}{10x} = \frac{5(x^2 - 4)}{10x} = \frac{5(x - 2)(x + 2)}{10x} \] Cancel 5: \[ \frac{(x - 2)(x + 2)}{2x} \] Final simplified: \[ \frac{x - 2}{2}, \quad x \ne 0 \]

5.
\[ y^2 - 16 = (y - 4)(y + 4) \] Cancel \((y + 4)\): \[ y - 4, \quad y \ne -4 \]

Summary

  • Simplify rational expressions by factoring, canceling factors, and stating restrictions.
  • Only cancel complete factors—not individual terms.
  • Restrictions come from values that make the denominator zero.
  • Factor first to avoid incorrect cancellations.
  • Final answers should always reflect excluded values.
  • If you see a polynomial fraction, factor immediately.
  • Always list restrictions before canceling.
  • Look for common binomial factors to simplify quickly.
  • Never cancel part of a sum—factor it first.