Integrated Math 3 / Unit 2: Rational Expressions and Functions
2.2 Multiplying and Dividing Rational Expressions
Notes
Multiply numerators and denominators, then cancel common factors. To divide, multiply by the reciprocal of the second fraction.
- Factor everything first, then cancel.
- $\displaystyle \frac{x}{x + 2} \cdot \frac{x^2 - 4}{x^2} = \frac{x (x - 2)(x + 2)}{(x + 2) x^2} = \frac{x - 2}{x}$.
- Division: $\displaystyle \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$.
Practice
Simplify $\displaystyle \frac{4}{x} \cdot \frac{x^2}{8}$.
$\displaystyle \frac{4x^2}{8x} = \frac{x}{2}$.
Simplify $\displaystyle \frac{x^2 - 1}{x + 3} \div \frac{x + 1}{x^2 - 9}$.
Flip and multiply: $\displaystyle \frac{(x - 1)(x + 1)}{x + 3} \cdot \frac{(x - 3)(x + 3)}{x + 1} = (x - 1)(x - 3)$.
Simplify $\displaystyle \frac{x}{x + 2} \cdot \frac{x^2 - 4}{x^2}$.
$x^2 - 4 = (x - 2)(x + 2)$. Cancel $x + 2$ and one factor of $x$.
Ready for more? Take the Unit 2 test.