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OBJECTIVES:

1. Add and subtract rational expressions with the same denominator.

2. Find a least common denominator.

3. Add and subtract rational expressions with different denominators.

1. Add and subtract rational expressions with the same denominator.

2. Find a least common denominator.

3. Add and subtract rational expressions with different denominators.

1.
Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 8.2 - 1

Sec 8.2 - 1

2.
Chapter 8

Rational Expressions and

Functions

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Sec 8.2 - 2

Rational Expressions and

Functions

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Sec 8.2 - 2

3.
8.2

Adding and Subtracting

Rational Expressions

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Sec 8.2 - 3

Adding and Subtracting

Rational Expressions

Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 8.2 - 3

4.
8.2 Adding and Subtracting Rational Expressions

Objectives

1. Add and subtract rational expressions with the

same denominator.

2. Find a least common denominator.

3. Add and subtract rational expressions with

different denominators.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 4

Objectives

1. Add and subtract rational expressions with the

same denominator.

2. Find a least common denominator.

3. Add and subtract rational expressions with

different denominators.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 4

5.
8.2 Adding and Subtracting Rational Expressions

Rational Expressions

Adding or Subtracting Rational Expressions

Step 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 LCD, and then

add or subtract the numerators. Place the result over the commo

denominator.

Step 2 Simplify. Write all answers in lowest terms.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 5

Rational Expressions

Adding or Subtracting Rational Expressions

Step 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 LCD, and then

add or subtract the numerators. Place the result over the commo

denominator.

Step 2 Simplify. Write all answers in lowest terms.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 5

6.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 1 Adding and Subtracting Rational Expressions

with the Same Denominator

Add or subtract as indicated.

4m 5n 4m + 5n Add the numerators.

(a) + =

7 7 7 Keep the common denominator.

1 – 5 1 – 5 Subtract the numerators; keep the

(b) =

g3 g3 g3 common denominator.

= – 4 Simplify.

g3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 6

EXAMPLE 1 Adding and Subtracting Rational Expressions

with the Same Denominator

Add or subtract as indicated.

4m 5n 4m + 5n Add the numerators.

(a) + =

7 7 7 Keep the common denominator.

1 – 5 1 – 5 Subtract the numerators; keep the

(b) =

g3 g3 g3 common denominator.

= – 4 Simplify.

g3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 6

7.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 1 Adding and Subtracting Rational Expressions

with the Same Denominator

Add or subtract as indicated.

a – b a–b Subtract the numerators; keep

(c) =

a2 – b2 a2 – b 2 a2 – b 2 the common denominator.

a–b

= Factor.

(a – b)(a + b)

1

= a+b Lowest terms

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 7

EXAMPLE 1 Adding and Subtracting Rational Expressions

with the Same Denominator

Add or subtract as indicated.

a – b a–b Subtract the numerators; keep

(c) =

a2 – b2 a2 – b 2 a2 – b 2 the common denominator.

a–b

= Factor.

(a – b)(a + b)

1

= a+b Lowest terms

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 7

8.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 1 Adding and Subtracting Rational Expressions

with the Same Denominator

Add or subtract as indicated.

5 + k

k2 + 2k – 15 k2 + 2k – 15

5+k Add.

=

k2 + 2k – 15

5+k

= Factor.

(k – 3)(k + 5)

1

= Lowest terms

k–3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 8

EXAMPLE 1 Adding and Subtracting Rational Expressions

with the Same Denominator

Add or subtract as indicated.

5 + k

k2 + 2k – 15 k2 + 2k – 15

5+k Add.

=

k2 + 2k – 15

5+k

= Factor.

(k – 3)(k + 5)

1

= Lowest terms

k–3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 8

9.
8.2 Adding and Subtracting Rational Expressions

The Least Common Denominator

Finding the Least Common Denominator

Step 1 Factor each denominator.

Step 2 Find the least common denominator. The LCD is the product of

all different factors from each denominator, with each factor raise

to the greatest power that occurs in the denominator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 9

The Least Common Denominator

Finding the Least Common Denominator

Step 1 Factor each denominator.

Step 2 Find the least common denominator. The LCD is the product of

all different factors from each denominator, with each factor raise

to the greatest power that occurs in the denominator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 9

10.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(a) 4m3n2, 6m2n5

Factor each denominator.

22 · m3 · n 2

2 · 3 · m2 · n 5

Choose the factors with

LCD = 22 · 3 · m3 · n5 the greatest exponents.

= 12m3n5

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 10

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(a) 4m3n2, 6m2n5

Factor each denominator.

22 · m3 · n 2

2 · 3 · m2 · n 5

Choose the factors with

LCD = 22 · 3 · m3 · n5 the greatest exponents.

= 12m3n5

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 10

11.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(b) y – 5, y

Each denominator is already factored. The LCD, an expression

divisible by both y – 5 and y is

y(y – 5).

It is usually best to leave a least common denominator in factored form.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 11

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(b) y – 5, y

Each denominator is already factored. The LCD, an expression

divisible by both y – 5 and y is

y(y – 5).

It is usually best to leave a least common denominator in factored form.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 11

12.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(c) n2 – 3n – 10, n2 – 8n + 15

Factor the denominators.

n2 – 3n – 10 = (n – 5)(n + 2)

Factor.

n2 – 8n + 15 = (n – 5)(n – 3)

The LCD, divisible by both polynomials, is (n – 5)(n + 2)(n – 3).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 12

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(c) n2 – 3n – 10, n2 – 8n + 15

Factor the denominators.

n2 – 3n – 10 = (n – 5)(n + 2)

Factor.

n2 – 8n + 15 = (n – 5)(n – 3)

The LCD, divisible by both polynomials, is (n – 5)(n + 2)(n – 3).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 12

13.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(d) 4h2 – 12h, 3h – 9

4h2 – 12h = 4h(h – 3)

Factor.

3h – 9 = 3(h – 3)

The LCD is 4h·3·(h – 3) = 12h(h – 3).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 13

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(d) 4h2 – 12h, 3h – 9

4h2 – 12h = 4h(h – 3)

Factor.

3h – 9 = 3(h – 3)

The LCD is 4h·3·(h – 3) = 12h(h – 3).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 13

14.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(e) g2 – 2g + 1, g2 + 3g – 4, 5g + 20

g2 – 2g + 1 = (g – 1)2

Factor.

g2 + 3g – 4 = (g – 1)(g + 4)

5g + 20 = 5(g + 4)

The LCD is 5(g – 1)2(g + 4).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 14

EXAMPLE 2 Finding Least Common Denominators

Assume that the given expressions are denominators of fractions. Find the

LCD for each group.

(e) g2 – 2g + 1, g2 + 3g – 4, 5g + 20

g2 – 2g + 1 = (g – 1)2

Factor.

g2 + 3g – 4 = (g – 1)(g + 4)

5g + 20 = 5(g + 4)

The LCD is 5(g – 1)2(g + 4).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 14

15.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 3 Adding and Subtracting Rational

Expressions with Different Denominators

Add or subtract as indicated. The LCD of 3z and 9z is 9z.

(a) 7 1

+

3z 9z

7 1 = 7·3 1 Fundamental property

+ +

3z 9z 3z · 3 9z

= 21 1

+

9z 9z

= 21 + 1 Add the numerators.

9z

= 22

9z

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 15

EXAMPLE 3 Adding and Subtracting Rational

Expressions with Different Denominators

Add or subtract as indicated. The LCD of 3z and 9z is 9z.

(a) 7 1

+

3z 9z

7 1 = 7·3 1 Fundamental property

+ +

3z 9z 3z · 3 9z

= 21 1

+

9z 9z

= 21 + 1 Add the numerators.

9z

= 22

9z

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 15

16.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 3 Adding and Subtracting Rational

Expressions with Different Denominators

Add or subtract as indicated. The LCD is y(y – 4).

(b) 2 – 3 2(y – 4) – y·3 Fundamental property

=

y y–4 y(y – 4) y(y – 4)

2y – 8 – 3y Distributive and

=

y(y – 4) y(y – 4) commutative properties

2y – 8 – 3y Subtract the

=

y(y – 4) numerators.

–y – 8 Combine like terms in

=

y(y – 4) the numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 16

EXAMPLE 3 Adding and Subtracting Rational

Expressions with Different Denominators

Add or subtract as indicated. The LCD is y(y – 4).

(b) 2 – 3 2(y – 4) – y·3 Fundamental property

=

y y–4 y(y – 4) y(y – 4)

2y – 8 – 3y Distributive and

=

y(y – 4) y(y – 4) commutative properties

2y – 8 – 3y Subtract the

=

y(y – 4) numerators.

–y – 8 Combine like terms in

=

y(y – 4) the numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 16

17.
8.2 Adding and Subtracting Rational Expressions

Caution with Subtracting Rational Expressions

One of the most common sign errors in algebra occurs when a rational

expression with two or more terms in the numerator is being subtracted.

In this situation, the subtraction sign must be distributed to every term

in the numerator of the fraction that follows it. Carefully study the

example below to see how this is done.

Subtract the numerators;

keep the common denominator.

8d – d–6 = 8d – (d – 6) = 8d – d + 6

d+5 d+5 d+5 d+5

= 7d + 6

Combine terms in the numerator.

d+5

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 17

Caution with Subtracting Rational Expressions

One of the most common sign errors in algebra occurs when a rational

expression with two or more terms in the numerator is being subtracted.

In this situation, the subtraction sign must be distributed to every term

in the numerator of the fraction that follows it. Carefully study the

example below to see how this is done.

Subtract the numerators;

keep the common denominator.

8d – d–6 = 8d – (d – 6) = 8d – d + 6

d+5 d+5 d+5 d+5

= 7d + 6

Combine terms in the numerator.

d+5

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 17

18.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 4 Using the Distributive Property When

Subtracting Rational Expressions

Add or subtract as indicated. The LCD of (e – 2) and (e + 2) is (e – 2)(e + 2).

2 – 9 = 2(e + 2) – 9(e – 2) Fundamental

e–2 e+2 (e – 2)(e + 2) (e + 2)(e – 2) property

2(e + 2) – 9(e – 2)

= Subtract.

(e – 2)(e + 2)

2e + 4 – 9e + 18 Distributive property

=

(e – 2)(e + 2)

= –7e + 22 Combine terms in

(e – 2)(e + 2) the numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 18

EXAMPLE 4 Using the Distributive Property When

Subtracting Rational Expressions

Add or subtract as indicated. The LCD of (e – 2) and (e + 2) is (e – 2)(e + 2).

2 – 9 = 2(e + 2) – 9(e – 2) Fundamental

e–2 e+2 (e – 2)(e + 2) (e + 2)(e – 2) property

2(e + 2) – 9(e – 2)

= Subtract.

(e – 2)(e + 2)

2e + 4 – 9e + 18 Distributive property

=

(e – 2)(e + 2)

= –7e + 22 Combine terms in

(e – 2)(e + 2) the numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 18

19.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 5 Adding Rational Expressions with

Denominators That Are Opposites

Add. 4 + x = 4 + x(–1)

x–5 5–x x–5 (5 – x)(–1)

= 4 + –x

Opposites

x–5 x–5

To get a common denominator of x – 5, multiply the second

expression by –1 in both the numerator and the denominator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 19

EXAMPLE 5 Adding Rational Expressions with

Denominators That Are Opposites

Add. 4 + x = 4 + x(–1)

x–5 5–x x–5 (5 – x)(–1)

= 4 + –x

Opposites

x–5 x–5

To get a common denominator of x – 5, multiply the second

expression by –1 in both the numerator and the denominator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 19

20.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 5 Adding Rational Expressions with

Denominators That Are Opposites

Add. 4 + x = 4 + x(–1)

x–5 5–x x–5 (5 – x)(–1)

= 4 + –x

Opposites

x–5 x–5

= 4–x Add the numerators.

x–5

If we had used 5 – x as the common denominator and rewritten

the first expression, we would have obtained

x–4 ,

5–x

an equivalent answer. Verify this.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 20

EXAMPLE 5 Adding Rational Expressions with

Denominators That Are Opposites

Add. 4 + x = 4 + x(–1)

x–5 5–x x–5 (5 – x)(–1)

= 4 + –x

Opposites

x–5 x–5

= 4–x Add the numerators.

x–5

If we had used 5 – x as the common denominator and rewritten

the first expression, we would have obtained

x–4 ,

5–x

an equivalent answer. Verify this.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 20

21.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 6 Adding and Subtracting Three Rational

Expressions

Add. 3 + 2 + 4 3n 2 4(n + 3)

= + +

n+3 n2 + 3n n(n + 3)

n n(n + 3) n(n + 3)

Fundamental property

The denominator of the second rational expression factors as

n(n + 3), which is the LCD for the three rational expressions.

3n + 2 + 4(n + 3) Add the numerators.

=

n(n + 3)

3n + 2 + 4n + 12 Distributive property

=

n(n + 3)

7n + 14 Combine terms.

=

n(n + 3)

7(n + 2)

= Factor the numerator.

n(n + 3)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 21

EXAMPLE 6 Adding and Subtracting Three Rational

Expressions

Add. 3 + 2 + 4 3n 2 4(n + 3)

= + +

n+3 n2 + 3n n(n + 3)

n n(n + 3) n(n + 3)

Fundamental property

The denominator of the second rational expression factors as

n(n + 3), which is the LCD for the three rational expressions.

3n + 2 + 4(n + 3) Add the numerators.

=

n(n + 3)

3n + 2 + 4n + 12 Distributive property

=

n(n + 3)

7n + 14 Combine terms.

=

n(n + 3)

7(n + 2)

= Factor the numerator.

n(n + 3)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 21

22.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 7 Subtracting Rational Expressions

Add. a–2 – a+2 a–2 – a+2

=

a2 + 2a – 3 a2 – 5a + 4 (a – 1)(a + 3) (a – 1)(a – 4)

Factor each denominator.

The LCD is (a – 1)(a + 3)(a – 4).

(a – 2)(a – 4) (a + 2)(a + 3) Fundamental

= –

(a – 1)(a + 3)(a – 4) (a – 1)(a – 4)(a + 3) property

(a – 2)(a – 4) – (a + 2)(a + 3)

= Subtract.

(a – 1)(a + 3)(a – 4)

a2 – 6a + 8 – (a2 + 5a + 6) Multiply in the

=

(a – 1)(a + 3)(a – 4) numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 22

EXAMPLE 7 Subtracting Rational Expressions

Add. a–2 – a+2 a–2 – a+2

=

a2 + 2a – 3 a2 – 5a + 4 (a – 1)(a + 3) (a – 1)(a – 4)

Factor each denominator.

The LCD is (a – 1)(a + 3)(a – 4).

(a – 2)(a – 4) (a + 2)(a + 3) Fundamental

= –

(a – 1)(a + 3)(a – 4) (a – 1)(a – 4)(a + 3) property

(a – 2)(a – 4) – (a + 2)(a + 3)

= Subtract.

(a – 1)(a + 3)(a – 4)

a2 – 6a + 8 – (a2 + 5a + 6) Multiply in the

=

(a – 1)(a + 3)(a – 4) numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 22

23.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 7 Subtracting Rational Expressions

Add. a–2 – a+2 a–2 – a+2

=

a2 + 2a – 3 a2 – 5a + 4 (a – 1)(a + 3) (a – 1)(a – 4)

(a – 2)(a – 4) – (a + 2)(a + 3)

=

(a – 1)(a + 3)(a – 4) (a – 1)(a – 4)(a + 3)

(a – 2)(a – 4) – (a + 2)(a + 3)

= Subtract.

(a – 1)(a + 3)(a – 4)

a2 – 6a + 8 – (a2 + 5a + 6) Multiply in the

=

(a – 1)(a + 3)(a – 4) numerator.

a2 – 6a + 8 – a2 – 5a – 6) Distributive

=

(a – 1)(a + 3)(a – 4) property

–11a + 2 Combine terms in

=

(a – 1)(a + 3)(a – 4) the numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 23

EXAMPLE 7 Subtracting Rational Expressions

Add. a–2 – a+2 a–2 – a+2

=

a2 + 2a – 3 a2 – 5a + 4 (a – 1)(a + 3) (a – 1)(a – 4)

(a – 2)(a – 4) – (a + 2)(a + 3)

=

(a – 1)(a + 3)(a – 4) (a – 1)(a – 4)(a + 3)

(a – 2)(a – 4) – (a + 2)(a + 3)

= Subtract.

(a – 1)(a + 3)(a – 4)

a2 – 6a + 8 – (a2 + 5a + 6) Multiply in the

=

(a – 1)(a + 3)(a – 4) numerator.

a2 – 6a + 8 – a2 – 5a – 6) Distributive

=

(a – 1)(a + 3)(a – 4) property

–11a + 2 Combine terms in

=

(a – 1)(a + 3)(a – 4) the numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 23

24.
8.2 Adding and Subtracting Rational Expressions

EXAMPLE 8 Adding Rational Expressions

Add. 2 + 8 2 + 8

=

b2 + 4b + 4 b2 + 3a + 2 (b + 2)2 (b + 1)(b + 2)

Factor each denominator.

The LCD is (b + 2)2(b + 1)

2(b + 1) 8(b + 2) Fundamental

= +

(b + 2)2(b + 1) (b + 2)2(b + 1) property

2(b + 1) + 8(b + 2)

= Add.

(b + 2)2(b + 1)

2b + 2 + 8b + 16 Distributive

=

(b + 2)2(b + 1) property

10b + 18 Combine like terms

=

(b + 2)2(b + 1) in the numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 24

EXAMPLE 8 Adding Rational Expressions

Add. 2 + 8 2 + 8

=

b2 + 4b + 4 b2 + 3a + 2 (b + 2)2 (b + 1)(b + 2)

Factor each denominator.

The LCD is (b + 2)2(b + 1)

2(b + 1) 8(b + 2) Fundamental

= +

(b + 2)2(b + 1) (b + 2)2(b + 1) property

2(b + 1) + 8(b + 2)

= Add.

(b + 2)2(b + 1)

2b + 2 + 8b + 16 Distributive

=

(b + 2)2(b + 1) property

10b + 18 Combine like terms

=

(b + 2)2(b + 1) in the numerator.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 24