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: $$\frac{x - 3}{x - 8}$$

Ready for more?

Watch the Step by Step Video Lesson
Take the Practice Test