Algebra 2 / Unit 6: Rational Functions
6.2 Simplifying, Multiplying and Dividing Rational Expressions
Notes
A rational expression is a fraction of polynomials. It is undefined where the denominator is 0.
- Simplify: factor the top and bottom completely, then cancel common factors (never cancel terms).
- Example: $\displaystyle \frac{x^2 - 4}{x^2 + x - 6} = \frac{(x - 2)(x + 2)}{(x - 2)(x + 3)} = \frac{x + 2}{x + 3}$, with $x \ne 2, -3$.
- Multiply: factor, cancel, then multiply across.
- Divide: multiply by the reciprocal of the second fraction.
Practice
Simplify $\displaystyle \frac{x^2 - 9}{x^2 + 5x + 6}$.
$\displaystyle \frac{(x - 3)(x + 3)}{(x + 2)(x + 3)}$; cancel $x + 3$.
Multiply $\displaystyle \frac{x}{x + 1} \cdot \frac{x^2 - 1}{x^2}$.
$\displaystyle \frac{x(x - 1)(x + 1)}{(x + 1)x^2}$; cancel $x + 1$ and one $x$.
Divide $\displaystyle \frac{2x}{x - 4} \div \frac{6x^2}{x^2 - 16}$.
$\displaystyle \frac{2x}{x - 4} \cdot \frac{(x - 4)(x + 4)}{6x^2} = \frac{x + 4}{3x}$.
Ready for more? Take the Unit 6 test.