Simplifying, Multiplying, and Dividing Rational Expressions

MATH 017

Intermediate Algebra

S. Rook

2

Overview

• Section 6.1 in the textbook– Domain of rational expressions

• Find where a rational expression is undefined

– Simplify rational expressions– Multiply rational expressions– Divide rational expressions

Domain of Rational Expressions

4

Domain of Rational Expressions

• Domain: set of allowable values

• For now, we only care where the rational expression is UNDEFINED

• A rational expression can be viewed as a fraction– When is a fraction undefined?

• An exercise in factoring

5

Domain of Rational Expressions (Example)

Ex 1: Find where the following is undefined:

xxx

x

15196

17423

6

Domain of Rational Expressions (Example)

Ex 2: Find where the following is undefined:

xxx

x

1519

45222

3

Simplify Rational Expressions

8

Simplify Rational Expressions

• Consider simplifying 20 / 30– 2 * 2 * 5 / 2 * 3 * 5– 2 / 3

• Works the same way with rational expressions– Factor the numerator and denominator– Cross out common factors

9

Simplify Rational Expressions (Example)

Ex 3: Simplify

12208

182482

2

xx

xx

10

Simplify Rational Expressions (Example)

Ex 4: Simplify

52312

2592

2

xx

x

Multiply Rational Expressions

12

Multiply Rational Expressions

• Consider multiplying 2 / 8 * 4 / 6– Factor each numerator and denominator

(2) / (2 * 2 * 2) * (2 * 2) / (2 * 3)

– Cancel common factors between numerators and denominators

(2) / (2 * 2 * 2) * (2 * 2) / (2 * 3)

– Multiply to get the final answer1 / 6

13

Multiply Rational Expressions (Continued)

• Same process with rational expressions– Factor the numerator and denominator of

each fraction– Cancel common factors– Multiply the remaining products for the final

answer

14

Multiply Rational Expressions (Example)

Ex 5: Multiply

675

306

1263

822

3

xx

x

xx

x

15

Multiply Rational Expressions (Example)

Ex 6: Multiply

24223

364

722

36122

2

2

2

xx

x

x

xx

Divide Rational Expressions

17

Divide Rational Expressions

• Consider dividing 2 / 8 ÷ 4 / 6– Turn into a multiplication problem by flipping the

second fraction2 / 8 * 6 / 4

– Factor each numerator and denominator(2) / (2 * 2 * 2) * (2 * 3) / (2 * 2)

– Cancel common factors between numerators and denominators

(2) / (2 * 2 * 2) * (2 * 3) / (2 * 2)

– Multiply to get the final answer3 / 8

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Divide Rational Expressions (Continued)

• Same process with rational expressions– Turn into a multiplication problem by flipping

the second rational expression– Factor the numerator and denominator of

each fraction– Cancel common factors– Multiply the remaining products for the final

answer

19

Divide Rational Expressions (Example)

Ex 7: Divide

1862

2

27

323

2

xx

x

x

xx

20

Divide Rational Expressions (Example)

Ex 8: Divide

44

632

63

2482

aa

abab

a

b

21

Summary

• After studying these slides, you should know how to do the following:– Find the values that make a rational

expression undefined– Simplify rational expressions– Multiply rational expressions– Divide rational expressions