Lesson 47: Multiplying and Dividing Rational Expressions
Multiplying and dividing rational expressions follows rules similar to numerical fractions. Simplifying before multiplying or dividing makes calculations easier.
Multiplying Rational Expressions
- Factor all numerators and denominators completely.
- Multiply the numerators together and the denominators together.
- Cancel any common factors between numerator and denominator.
Example 1: Multiply (x² − 9)/(x² − 4) × (x + 2)/(x − 3)
- Factor: (x − 3)(x + 3)/(x − 2)(x + 2) × (x + 2)/(x − 3)
- Cancel common factors: (x + 3)/ (x − 2)
- Simplified result: (x + 3)/(x − 2)
Dividing Rational Expressions
- Factor all numerators and denominators completely.
- Rewrite division as multiplication by the reciprocal.
- Multiply as usual and simplify.
Example 2: Divide (x² − 4)/(x² − x − 6) ÷ (x − 2)/(x + 3)
- Factor: (x − 2)(x + 2)/ (x − 3)(x + 2) ÷ (x − 2)/(x + 3)
- Reciprocal: (x − 2)(x + 2)/ (x − 3)(x + 2) × (x + 3)/(x − 2)
- Cancel common factors: (x + 2)(x + 3)/ (x − 3)(x + 2) → (x + 3)/(x − 3)
Practice Problems
- Multiply: (x² − 16)/(x² − x − 12) × (x − 2)/(x + 4)
- Multiply: (2x² − 8)/(x² − 4x) × (x − 2)/(x + 4)
- Divide: (x² − 9)/(x² − 1) ÷ (x − 3)/(x + 1)
- Divide: (3x² − 12)/(x² − x − 6) ÷ (x − 2)/(x + 3)
- Multiply: (x² + 5x + 6)/(x² + 3x) × (x + 1)/(x + 2)