Lesson Objectives
• Demonstrate an understanding of how to simplify a rational expression
• Learn how to multiply rational expressions
• Learn how to divide rational expressions

## How to Multiply & Divide Rational Expressions

In the last lesson, we introduced rational expressions. When we multiply or divide rational expressions, we follow the same rules we used with fractions.

### Multiplying Rational Expressions

• Factor all numerators and all denominators
• Cancel any common factors other than 1 between the numerators and denominators
• Multiply the remaining factors in the numerators and the remaining factors in the denominators
• We may choose to leave the rational expression in factored form
Let's look at a few examples.
Example 1: Find each product. $$\frac{7x - 56}{5x^2 + 45x + 40}\cdot \frac{40x^2 + 40x}{8x}$$ Step 1) Factor all numerators and all denominators: $$\frac{7(x - 8)}{5(x + 1)(x + 8)}\cdot \frac{40x(x + 1)}{8x}$$ Step 2) Cancel any common factors other than 1 between the numerators and denominators: $$\require{cancel}\frac{7(x - 8)}{\cancel{5}\cancel{(x + 1)}(x + 8)}\cdot \frac{\cancel{40x}\cancel{(x + 1)}}{\cancel{8x}}$$ Step 3) Multiply the remaining factors in the numerators and the remaining factors in the denominators: $$\frac{7x - 56}{x + 8}$$ It's also valid to report your answer in factored form. $$\frac{7(x - 8)}{x + 8}$$ Example 2: Find each product. $$\frac{-15x^2 - 13x + 72}{15x^2 - 12x - 27}\cdot \frac{3x^2 - 6x - 9}{3x + 8}$$ Step 1) Factor all numerators and all denominators: $$\frac{-1(5x - 9)(3x + 8)}{3(5x - 9)(x + 1)}\cdot \frac{3(x - 3)(x + 1)}{(3x + 8)}$$ Step 2) Cancel any common factors other than 1 between the numerators and denominators: $$\frac{-1\cancel{(5x - 9)}\cancel{(3x + 8)}}{\cancel{3}\cancel{(5x - 9)}\cancel{(x + 1)}}\cdot \frac{\cancel{3}(x - 3)\cancel{(x + 1)}}{\cancel{(3x + 8)}}$$ Step 3) Multiply the remaining factors in the numerators and the remaining factors in the denominators: $$-1(x -3)$$ $$-x + 3$$

### Dividing Rational Expressions

When we divide rational expressions, we multiply the first rational expression (leftmost) by the reciprocal of the second (rightmost). Let's look at an example.
Example 3: Find each quotient. $$\frac{15x^2 + 31x + 10}{9x^2 + 30x + 25}÷ \frac{5x^2 - 38x - 16}{3x^2 - 4x - 15}$$ Step 1) Set up the division problem as the multiplication of the first rational expression by the reciprocal of the second: $$\frac{15x^2 + 31x + 10}{9x^2 + 30x + 25}\cdot \frac{3x^2 - 4x - 15}{5x^2 - 38x - 16}$$ Now we can follow our procedure for multiplying rational expressions.
Step 2) Factor all numerators and all denominators: $$\frac{(5x + 2)(3x + 5)}{(3x + 5)(3x + 5)}\cdot \frac{(x - 3)(3x + 5)}{(x - 8)(5x + 2)}$$ Step 3) Cancel any common factors other than 1 between the numerators and denominators: $$\frac{\cancel{(5x + 2)}\cancel{(3x + 5)}}{\cancel{(3x + 5)}\cancel{(3x + 5)}}\cdot \frac{(x - 3)\cancel{(3x + 5)}}{(x - 8)\cancel{(5x + 2)}}$$ Step 4) Multiply the remaining factors in the numerators and the remaining factors in the denominators: $$\frac{x - 3}{x - 8}$$

#### Skills Check:

Example #1

Simplify each. $$\frac{16x - 104}{-22x^{2}+ 151x - 52}\cdot \frac{33x - 12}{8x - 80}$$

A
$$\frac{11}{x - 12}$$
B
$$\frac{6(x - 13)}{x + 13}$$
C
$$\frac{2(x + 8)}{x - 1}$$
D
$$\frac{3}{x}$$
E
$$-\frac{3}{x - 10}$$

Example #2

Simplify each. $$\frac{3x + 1}{70x^{3}- 130x^{2}}\cdot \frac{7x^{2}- 6x - 13}{21x + 7}$$

A
$$\frac{x + 11}{x + 6}$$
B
$$\frac{x + 1}{70x^{2}}$$
C
$$\frac{x + 11}{4(x - 11)}$$
D
$$\frac{x + 5}{x - 14}$$
E
$$\frac{x - 1}{35x^{2}}$$

Example #3

Simplify each. $$\frac{20x^{3}+ 44x^{2}}{35x^{2}+ 27x - 110}\div \frac{5x}{63x - 90}$$

A
$$\frac{x - 5}{8x}$$
B
$$-1$$
C
$$\frac{36x}{5}$$
D
$$\frac{8x}{(x - 4)(x + 14)}$$
E
$$\frac{x}{x - 2}$$