## Multiplying Rational Expressions

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.

### How To: Given two rational expressions, multiply them.

1. Factor the numerator and denominator.
2. Multiply the numerators.
3. Multiply the denominators.
4. Simplify.

### Example 2: Multiplying Rational Expressions

Multiply the rational expressions and show the product in simplest form:

$\begin{array}{cc}\frac{\left(x+5\right)\left(x - 1\right)}{3\left(x+6\right)}\cdot \frac{\left(2x - 1\right)}{\left(x+5\right)}\hfill & \text{Factor the numerator and denominator}.\hfill \\ \frac{\left(x+5\right)\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)\left(x+5\right)}\hfill & \text{Multiply numerators and denominators}.\hfill \\ \frac{\cancel{\left(x+5\right)}\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)\cancel{\left(x+5\right)}}\hfill & \text{Cancel common factors to simplify}.\hfill \\ \frac{\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)}\hfill & \hfill \end{array}$

### Solution

$\begin{array}{cc}\frac{\left(x+5\right)\left(x - 1\right)}{3\left(x+6\right)}\cdot \frac{\left(2x - 1\right)}{\left(x+5\right)}\hfill & \text{Factor the numerator and denominator}.\hfill \\ \frac{\left(x+5\right)\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)\left(x+5\right)}\hfill & \text{Multiply numerators and denominators}.\hfill \\ \frac{\cancel{\left(x+5\right)}\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)\cancel{\left(x+5\right)}}\hfill & \text{Cancel common factors to simplify}.\hfill \\ \frac{\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)}\hfill & \hfill \end{array}$

### Try It 2

Multiply the rational expressions and show the product in simplest form:

$\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\cdot \frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}$

Solution

## Dividing Rational Expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite $\frac{1}{x}\div \frac{{x}^{2}}{3}$ as the product $\frac{1}{x}\cdot \frac{3}{{x}^{2}}$. Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

$\frac{1}{x}\cdot \frac{3}{{x}^{2}}=\frac{3}{{x}^{3}}$

### How To: Given two rational expressions, divide them.

1. Rewrite as the first rational expression multiplied by the reciprocal of the second.
2. Factor the numerators and denominators.
3. Multiply the numerators.
4. Multiply the denominators.
5. Simplify.

### Example 3: Dividing Rational Expressions

Divide the rational expressions and express the quotient in simplest form:

$\frac{2{x}^{2}+x - 6}{{x}^{2}-1}\div \frac{{x}^{2}-4}{{x}^{2}+2x+1}$

### Solution

$\begin{array}\text{ }\frac{2x^{2}+x-6}{x^{2}}\cdot\frac{x^{2}+2x+1}{x^{2}-4} \hfill& \text{Rewrite as the first rational expression multiplied by the reciprocal of the second rational expression.} \\ \frac{\left(2\times3\right)\cancel{\left(x+2\right)}}{\cancel{\left(x+1\right)}\left(x-1\right)}\cdot\frac{\cancel{\left(x+1\right)}\left(x+1\right)}{\cancel{\left(x+2\right)}\left(x-2\right)} \hfill& \text{Factor and cancel common factors.} \\ \frac{\left(2x+3\right)\left(x+1\right)}{\left(x-1\right)\left(x-2\right)} \hfill& \text{Multiply numerators and denominators.} \\ \frac{2x^{2}+5x+3}{x^{2}-3x+2} \hfill& \text{Simplify.}\end{array}$

### Try It 3

Divide the rational expressions and express the quotient in simplest form:

$\frac{9{x}^{2}-16}{3{x}^{2}+17x - 28}\div \frac{3{x}^{2}-2x - 8}{{x}^{2}+5x - 14}$

Solution