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.

Example 1: Find each product. Step 1) Factor all numerators and all denominators: Step 2) Cancel any common factors other than 1 between the numerators and denominators: 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. Step 1) Factor all numerators and all denominators: Step 2) Cancel any common factors other than 1 between the numerators and denominators: Step 3) Multiply the remaining factors in the numerators and the remaining factors in the denominators: $$-1(x -3)$$ $$-x + 3$$

Example 3: Find each quotient. Step 1) Set up the division problem as the multiplication of the first rational expression by the reciprocal of the second: Now we can follow our procedure for multiplying rational expressions.

Step 2) Factor all numerators and all denominators: Step 3) Cancel any common factors other than 1 between the numerators and denominators: Step 4) Multiply the remaining factors in the numerators and the remaining factors in the denominators: $$\frac{x - 3}{x - 8}$$

### 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

Example 1: Find each product. Step 1) Factor all numerators and all denominators: Step 2) Cancel any common factors other than 1 between the numerators and denominators: 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. Step 1) Factor all numerators and all denominators: Step 2) Cancel any common factors other than 1 between the numerators and denominators: 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. Step 1) Set up the division problem as the multiplication of the first rational expression by the reciprocal of the second: Now we can follow our procedure for multiplying rational expressions.

Step 2) Factor all numerators and all denominators: Step 3) Cancel any common factors other than 1 between the numerators and denominators: 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. $$\frac{16x - 104}{-22x^{2}+ 151x - 52}\cdot \frac{33x - 12}{8x - 80}$$

Please choose the best answer.

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

Please choose the best answer.

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

Please choose the best answer.

A

$$\frac{x - 5}{8x}$$

B

$$-1$$

C

$$\frac{36x}{5}$$

D

$$\frac{8x}{(x - 4)(x + 14)}$$

E

$$\frac{x}{x - 2}$$

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