Dividing Rational Expressions
April 29, 2025
Introduction
Back in elementary, we learned that dividing fractions involves flipping (taking the reciprocal of) the second fraction and then multiplying. The example below shows how this works step by step.
Simplify: \dfrac{3}{5} \div \dfrac{5}{9}
Solution: \displaystyle \frac{3}{5} \div \frac{5}{9} = \frac{3}{5} \times \frac{9}{5} = \frac{3 \cdot 9}{5 \cdot 5} = \frac{27}{25}
Dividing rational expressions works just like dividing fractions. First, rewrite the division as multiplication by taking the reciprocal of the second expression. Then, factor all terms in the numerators and denominators completely. Cancel any common factors. Finally, multiply the remaining factors.
Suppose P, \space Q, \space R, \text{ and } S are polynomials. To divide two rational expressions, multiply the first rational expression by the reciprocal of the second:
where Q \neq 0, \space R \neq 0, \text{ and } S \neq 0.
For simplicity, dividing rational expressions are done using these simple steps:
Step 1: Rewrite the division as multiplication by flipping (taking the reciprocal of) the second rationalexpression.
Step 2: Factor all numerators and denominators completely.
Step 3: Cancel any common factors between numerators and denominators.
Step 4: Multiply the remaining factors across the numerator and denominator