Multiplying & Dividing Rational Expressions

TipLearning Objectives

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

  • Multiply rational expressions using factoring and canceling.
  • Divide rational expressions by multiplying by the reciprocal.
  • Identify and state all variable restrictions throughout the process.

Key Ideas

Multiplying and dividing rational expressions follow the same logic as numerical fractions—just with polynomial factors.

To multiply rational expressions:

  1. Factor all numerators and denominators.
  2. Cancel common factors (never cancel terms).
  3. Multiply the remaining factors.
  4. State restrictions based on all original denominators.

To divide rational expressions:

  • Multiply by the reciprocal of the divisor, then follow the same steps as multiplication.

Common Problem Types

1. Multiplying Fractions With Polynomial Factors

Look for matching factors that cancel.

2. Dividing by a Rational Expression

Rewrite division as multiplication by the reciprocal.

3. Factoring First

Expressions may hide common factors—always factor completely.

4. Restrictions From All Denominators

Identify all \(x\)-values that would make any denominator zero.

Strategies

  • Factor everything before canceling—this prevents incorrect cancellations.
  • After flipping the divisor in division problems, treat it just like multiplication.
  • Keep a running list of restrictions each time a denominator appears.
  • Combine constants and variables only when they are complete factors.
  • Leave answers in factored form unless simplification is required.

Worked Examples

Example 1 — Multiplying

Simplify: \[ \frac{x}{x - 4} \cdot \frac{x - 4}{x + 3} \]

Solution:
Cancel the common factor \((x - 4)\):

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

Restrictions: \[ x \ne 4, -3. \]

Example 2 — Dividing

Simplify: \[ \frac{x^2 - 9}{x + 5} \div \frac{x - 3}{x + 1} \]

Solution:
Factor and rewrite as multiplication:

\[ \frac{(x - 3)(x + 3)}{x + 5} \cdot \frac{x + 1}{x - 3} \]

Cancel \((x - 3)\):

\[ \frac{(x + 3)(x + 1)}{x + 5} \]

Restrictions: \[ x \ne -5, 3, -1. \]

WarningCommon Mistakes
  • Forgetting to flip the second fraction when dividing.
  • Canceling parts of sums instead of full factors.
  • Ignoring restrictions from all denominators in the problem.

Practice Problems

  1. \(\dfrac{2x}{x + 1} \cdot \dfrac{x + 1}{3x}\)
  2. \(\dfrac{x^2 - 16}{x - 4} \cdot \dfrac{1}{x + 4}\)
  3. \(\dfrac{3x}{x^2 - 9} \div \dfrac{1}{x - 3}\)
  4. \(\dfrac{x}{2x - 4} \cdot \dfrac{x - 2}{3}\)
  5. \(\dfrac{y^2}{y + 2} \div \dfrac{y}{y - 2}\)

1.
\[ \frac{2x}{x + 1} \cdot \frac{x + 1}{3x} = \frac{2}{3} \]

2.
\[ x^2 - 16 = (x - 4)(x + 4) \] \[ \frac{(x - 4)(x + 4)}{x - 4} \cdot \frac{1}{x + 4} = 1, \quad x \ne 4, -4 \]

3.
\[ \frac{3x}{(x - 3)(x + 3)} \div \frac{1}{x - 3} = \frac{3x}{(x - 3)(x + 3)} \cdot (x - 3) = \frac{3x}{x + 3} \]

4.
\[ \frac{x}{2(x - 2)} \cdot \frac{x - 2}{3} = \frac{x}{6} \]

5.
\[ \frac{y^2}{y + 2} \div \frac{y}{y - 2} = \frac{y^2}{y + 2} \cdot \frac{y - 2}{y} = \frac{y(y - 2)}{y + 2} \]

Summary

  • Multiply rational expressions by factoring, canceling, then multiplying remaining factors.
  • Divide by multiplying by the reciprocal of the divisor.
  • Only cancel full factors—never individual terms.
  • Restrictions come from all denominators before simplification.
  • Answers should be written in simplified factored form with restrictions stated.
  • Factor first—cancel only after expressions are fully factored.
  • When dividing, flip the second fraction every time.
  • Track restrictions from all denominators, even if factors cancel later.
  • Leave final answers in simplified factored form unless combining is necessary.