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

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. $$\frac{5x^3 + 20x^2}{5x^2 + 25x}÷ \frac{5x^2}{x^2 + 6x + 5}$$

Please choose the best answer.

A

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

B

$$\frac{x - 3}{x - 5}$$

C

$$\frac{x + 6}{4}$$

D

$$\frac{(x + 4)(x + 1)}{5x}$$

E

$$\frac{(x - 1)(x + 2)}{x}$$

Example #2

Simplify each. $$\frac{x^2 - 15x + 56}{x^2 - 16x + 64}\cdot \frac{7x - 56}{7x - 42}$$

Please choose the best answer.

A

$$x - 2$$

B

$$\frac{14x}{x - 4}$$

C

$$\frac{5x}{2}$$

D

$$\frac{x - 7}{x - 6}$$

E

$$\frac{x + 1}{x - 7}$$

Example #3

Simplify each. $$\frac{x^2 - x - 30}{x^2 - 13x + 42}\cdot \frac{x - 2}{x^2 + x - 6}$$

Please choose the best answer.

A

$$\frac{5}{x + 7}$$

B

$$\frac{x + 5}{(x - 7)(x + 3)}$$

C

$$1$$

D

$$x - 4$$

E

$$\frac{x + 1}{(x - 2)(x + 3)}$$

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