Chapter Chapter 77Section Section 22

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

11

22

33

Adding and Subtracting Rational Expressions

Add and subtract rational expressions with the same denominator.Find a least common denominator.Add and subtract rational expressions with different denominators.

7.27.27.27.2

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Objective 1

Add and subtract rational expressions with the same denominator.

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Adding or Subtracting Rational ExpressionsStep 1 If the denominators are the same, add or

subtract the numerators. Place the result over the common denominator.

If the denominators are different, first find the least common denominator. Write all rational expressions with this least common denominator, and then add or subtract the numerators. Place the result over the common denominator.

Step 2 Simplify. Write all answers in lowest terms.

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2 2

6

3 18 3 18

x

x x x x

2 2 2 2

r t

r t r t

3 3

8 14

3 3x x

7

9 9

x y

Add or subtract as indicated.

EXAMPLE 1

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Objective 2

Find a least common denominator.

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Finding the Least Common DenominatorStep 1 Factor each denominator.

Step 2 Find the least common denominator. The LCD is the product of all of the different factors from each denominator, with each factor raised to the greatest power that occurs in any denominator.

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EXAMPLE 2

Find the LCD for each group of denominators.

a. 3 5 2 610 ,15a b a b

b. z, z + 6

Factor. 10a3b5 = 2 • 5 • a3 • b5

15a2b6 = 3 • 5 • a2 • b6

LCD = 2 • 3 • 5 • a3 • b5 = 30a3b6

Each denominator is already factored.

LCD = z(z + 6)

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continued

c. m2 – 16 , m2 + 8m + 16

Factor. = (m +4)(m – 4)

LCD

m2 – 16

m2 + 8m + 16 = (m +4)2

= (m +4)2(m – 4)

d. x2 – 2x + 1, x2 – 4x + 3, 4x – 4

x2 – 2x + 1 = (x – 1)(x – 1)

x2 – 4x + 3 = (x – 1)(x – 3)

4x – 4 = 4(x – 1)

LCD = 4(x – 1)2(x – 3)

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Objective 3

Add and subtract rational expressions with different denominators.

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Add or subtract as indicated.

a. 6 1

4m m

b. 2 1

4y y

6 1

4

4

4m m

24 1

4 4m m

24 1

4m

25

4m

4

4

2 1

4y y y

yy

y

2 4

4 4

y y

y y y y

2 8

4

y y

y y

8

4

y

y y

Example 3

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EXAMPLE 4

Subtract.

a. 5 7 14

2 7 2 7

x x

x x

The denominators are already the same for both rational expressions. The subtraction sign must be applied to both terms in the numerator of the second rational expression.

15 7

2 7

( 4)xx

x

5 7 1

7

4

2

x x

x

6 21

2 7

x

x

2 73

2 7

x

x

3

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2 3

2 1

r

r r

continued

b.

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Add:2 1

.3 3x x

To get a common denominator of x – 3, multiply both the numerator and denominator of the second expression by –1.

1 12

3 3 1x x

2 1

3 3x x

2 1

3x

1

3x

The equivalent answer is

1 1

1 3 x

1.

3x

Example 5

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Simplify.

Example 6

x

x

xx

365

32

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Simplify

Example 7

2

4 2 10

5 5x x x x

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Simplify.

Example 8

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Simplify.

Example 9

45

3

56

422

xxxx

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Subtract

Example 10

2 2

4

3 4 7 12

a a

a a a a

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Add

Example 11

2 2

4 1

6 9 2 15p p p p