Lesson 8.2: Multiply and Divide Rational Expressions

New Concept

This lesson extends your skills with fractions to rational expressions. You'll learn to multiply expressions by multiplying numerators/denominators, and divide by multiplying by the reciprocal. Factoring polynomials is the key to simplifying the result.

What’s next

Get ready to put this into practice! You'll work through interactive examples and a series of practice cards to master multiplying and dividing rational expressions.

Property

If $p, q, r, s$ are polynomials where $q \neq 0$ and $s \neq 0$, then

To multiply rational expressions, multiply the numerators and multiply the denominators. To do this, first factor each numerator and denominator completely. Then, multiply the numerators and denominators. Finally, simplify by dividing out common factors.

Examples

• Multiply: $\frac{5a^2}{4b^3} \cdot \frac{8b}{15a^3}$. Factoring and canceling gives $\frac{5 \cdot a^2 \cdot 8 \cdot b}{4 \cdot b^3 \cdot 15 \cdot a^3} = \frac{5 \cdot a \cdot a \cdot 2 \cdot 4 \cdot b}{4 \cdot b \cdot b \cdot b \cdot 3 \cdot 5 \cdot a \cdot a \cdot a} = \frac{2}{3ab^2}$.
• Multiply: $\frac{x^2-4}{3x^2} \cdot \frac{9x}{x^2+x-6}$. Factor to get $\frac{(x-2)(x+2)}{3x^2} \cdot \frac{9x}{(x+3)(x-2)}$. Multiply and cancel common factors: $\frac{3(x+2)}{x(x+3)}$.
• Multiply: $\frac{25-x^2}{x^2-2x-3} \cdot \frac{x+1}{x-5}$. Factor $25-x^2$ as $-1(x-5)(x+5)$ to get $\frac{-1(x-5)(x+5)}{(x-3)(x+1)} \cdot \frac{x+1}{x-5}$. Simplify to $-\frac{x+5}{x-3}$.

Explanation

Multiplying rational expressions is just like multiplying fractions. The trick is to factor everything first! This lets you cancel out common factors from the top and bottom, which makes the problem much simpler before you even multiply.

Property

If $p, q, r, s$ are polynomials where $q \neq 0, r \neq 0, s \neq 0$, then

To divide rational expressions, multiply the first fraction by the reciprocal of the second. This is done by first rewriting the division as multiplication. Then, factor the numerators and denominators, multiply, and simplify by dividing out common factors.

Examples

• Divide: $\frac{x-5}{x^2-49} \div \frac{x^2-25}{x+7}$. First, flip the second fraction: $\frac{x-5}{(x-7)(x+7)} \cdot \frac{x+7}{(x-5)(x+5)}$. After canceling, the result is $\frac{1}{(x-7)(x+5)}$.
• Divide: $\frac{y^2-36}{4y} \div (y^2-12y+36)$. Rewrite the second part as a fraction and flip: $\frac{(y-6)(y+6)}{4y} \cdot \frac{1}{(y-6)(y-6)}$. Simplify to $\frac{y+6}{4y(y-6)}$.
• Simplify the complex fraction $\frac{\frac{x^2-1}{x^2+3x+2}}{\frac{x^2-2x+1}{x+2}}$. Rewrite as division and flip: $\frac{(x-1)(x+1)}{(x+1)(x+2)} \cdot \frac{x+2}{(x-1)(x-1)}$. This simplifies to $\frac{1}{x-1}$.

Explanation

Don't let division scare you! It's just multiplication in disguise. Remember the rule: 'Keep, Change, Flip.' Keep the first fraction, change division to multiplication, and flip the second fraction. After that, it's just a multiplication problem!