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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