# Multiply Rational Expressions

Because rational expressions are just fractions with variables, the process of multiplying rational expressions is exactly the same as the process of multiplying fractions. **Multiply the rational expressions just like you multiply fractions (numerator times numerator and denominator times denominator).**

Just like with fractions, you can cross-cancel before you multiply. So, if you have polynomials or polynomial expressions, factor them completely and cross-cancel any terms that you find in both a numerator and a denominator. Remember you can cancel out terms attached with multiplication but if numbers or variable are attached by addition or subtraction they have to stay together.

*Example*:

$$\require{cancel}\eqalign{ \dfrac{x^3+10x^2+25x}{x^2}\times\dfrac{x+1}{x^2-25}=&\\ \dfrac{\overset{x(x+5)(x+5)}{\bcancel{x^3+10x^2+25x}}}{x^2}\times\dfrac{x+1}{\underset{(x+5)(x-5)}{\bcancel{x-25}}}=& \quad&&\text{Factor completely}\\ \dfrac{\overset{\bcancel{x}\bcancel{(x+5)}(x+5)}{\bcancel{x^3+10x^2+25x}}}{\underset{\bcancel{x} \times x}{\bcancel{x^2}}}\times\dfrac{x+1}{\underset{\bcancel{(x+5)}(x-5)}{\bcancel{x^2-25}}}=& \quad&&\text{Cross cancel}\\ \dfrac{\overset{\bcancel{x}\bcancel{(x+5)}(x+5)}{\bcancel{x^3+10x^2+25x}}}{\underset{\bcancel{x}\times x}{\bcancel{x^2}}}\times\dfrac{x+1}{\underset{\bcancel{(x+5)}(x-5)}{\bcancel{x^2-25}}}=&\dfrac{(x+5)(x+1)}{x(x-5)}=\dfrac{x^2+6x+5}{x^2-5x}&&\text{Multiply numerator x numerator & denominator x denominator}}$$

Multiplying rational expressions is easy! Factor completely, cross cancel, and then multiply just like fractions.

#### Practice Problems:

## Multiply Rational Expressions

Simplify the rational expressions:

- $\dfrac{x}{x^2}\times \dfrac{x}{x^2}$
- $\dfrac{x}{3x}\times \dfrac{2}{x^2}$
- $\dfrac{x+1}{x^2}\times \dfrac{3x}{x+1}$
- $\dfrac{9x}{x^2+6x+9}\times \dfrac{x^2}{3x^2+12}$
- $\dfrac{3x+3}{3x^3+6x}\times \dfrac{3x}{x+1}$
- $\dfrac{x^2-x-12}{x^2+4x+4}\times \dfrac{x+2}{x-4}$
- $\dfrac{x^3}{x^2+4x-45}\times \dfrac{x^2+9x}{x^2}$
- $\dfrac{x^3-3x^2-40x}{6x}\times \dfrac{3x+12}{x^2-8x}$

#### Answer Key: