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2: Multiplying and Dividing Algebraic Fractions

2: Multiplying and Dividing Algebraic Fractions

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7.2 • Multiplying and Dividing Algebraic Fractions



Classroom Example

Multiply and simplify

d

d 2 ϩ 8d ϩ 7

.

и

2

3

d ϩ 7d



EXAMPLE 2



Classroom Example

Multiply and simplify

x2 Ϫ 4

12x2 ϩ 13x Ϫ 4

и 3x2 ϩ 4x .

xϪ2



x

x ϩ 2x

2



x2 ϩ 10x ϩ 16

.

5



и



Solution

x

2

x ϩ 2x



Classroom Example

Multiply and simplify

x2 Ϫ x x2 Ϫ 4x Ϫ 21

и x2 Ϫ 8x ϩ 7 .

xϩ3



Multiply and simplify



275



и



x1x ϩ 221x ϩ 82

x2 ϩ 10x ϩ 16

xϩ8

ϭ

ϭ

5

x1x ϩ 22152

5



EXAMPLE 3



Multiply and simplify



Solution

a2 Ϫ 3a

aϩ5



и



a2 Ϫ 3a

aϩ5



и



a2 ϩ 3a Ϫ 10

.

a2 Ϫ 5a ϩ 6



a1a Ϫ 321a ϩ 521a Ϫ 22

a2 ϩ 3a Ϫ 10

ϭ

ϭa

2

1a ϩ 521a Ϫ 221a Ϫ 32

a Ϫ 5a ϩ 6



EXAMPLE 4



Multiply and simplify



Solution

6n2 ϩ 7n Ϫ 3

nϩ1



и



6n2 ϩ 7n Ϫ 3

nϩ1



и



n2 Ϫ 1

.

2n2 ϩ 3n



12n ϩ 3213n Ϫ 121n ϩ 121n Ϫ 12

n2 Ϫ 1

ϭ

2

1n ϩ 121n212n ϩ 32

2n ϩ 3n

ϭ



13n Ϫ 12 1n Ϫ 12

n



Dividing Algebraic Fractions

a

form, we invert the divisor and multiply.

b

d

a

c a d

c

Symbolically we express this as Ϭ ϭ и . Furthermore, we call the numbers and

c

c

b

d b

d

reciprocals of each other because their product is 1. Thus we can also describe division as to

divide by a fraction, multiply by its reciprocal. We define division of algebraic fractions in the

same way using the same vocabulary.

Recall that to divide two rational numbers in



Definition 7.2

If



C

A

and are rational expressions with B ϶ 0, D ϶ 0, and C ϶ 0, then

B

D

A

C

A

Ϭ ϭ

B

D

B



и



D

AD

ϭ

C

BC



Consider some examples.

4x

6x2

4x

1.

Ϭ

ϭ

2

7y

7y

14y



и



22



14y2



ϭ



6x2



18a3

Ϫ8ab

Ϫ8ab

2.

Ϭ

ϭ

9b

9b

15a2b

x2y3

5xy2

x2y3

3.

Ϭ

ϭ

2

4ab

4ab

Ϫ9a b



y



22



и 14 и x и y2 ϭ 4y

3x

7 и 6 и x2 и y

33

x



4



44



и



3



99



x



и



b



55



20b

15a2b

8 и 15 и a3 и b2

ϭϪ

ϭϪ

3

27

9 и 18 и a3 и b

18a

y



a



9 и x2 и y3 и a2 и b

9axy

Ϫ9a2b

ϭϪ

ϭϪ

2

20

4 и 5 и a и b и x и y2

5xy



276



Chapter 7 • Algebraic Fractions



The key idea when dividing fractions is to first convert to an equivalent multiplication

problem and then proceed to factor numerator and denominator completely and look for common factors.



Classroom Example

Divide and simplify:

c3 ϩ c2



Ϭ



cd 2



d2 Ϫ 9

d 2 Ϫ 3d



EXAMPLE 5



Divide and simplify



Solution

x2 Ϫ 4x

x2 Ϫ 16

x2 Ϫ 4x

Ϭ 3

ϭ

xy

xy

y ϩ y2

ϭ

ϭ



Classroom Example

Divide and simplify:

4x2 ϩ 24

x2 ϩ 6



Ϭ



1

x2 Ϫ 2x Ϫ 8



x2 Ϫ 16

x2 Ϫ 4x

Ϭ 3

.

xy

y ϩ y2



EXAMPLE 6



y3 ϩ y2



и

y



x2 Ϫ 16



x1x Ϫ 421y2 21y ϩ 12



xy1x ϩ 421x Ϫ 42

y1y ϩ 12

xϩ4



Divide and simplify



a2 ϩ 3a Ϫ 18

1

Ϭ 2

.

2

a ϩ4

3a ϩ 12



Solution

a2 ϩ 3a Ϫ 18

1

a2 ϩ 3a Ϫ 18

Ϭ 2

ϭ

2

a ϩ4

3a ϩ 12

a2 ϩ 4

ϭ



и



3a2 ϩ 12

1



1a ϩ 621a Ϫ 32132 1a2 ϩ 42

a2 ϩ 4



ϭ 31a ϩ 621a Ϫ 32



Classroom Example

Divide and simplify:

5x2 Ϫ 16x ϩ 3

10x2 ϩ 23x Ϫ 5



Ϭ (x Ϫ 3)



EXAMPLE 7



Divide and simplify



2n2 Ϫ 7n Ϫ 4

Ϭ 1n Ϫ 42 .

6n2 ϩ 7n ϩ 2



Solution

2n2 Ϫ 7n Ϫ 4

2n2 Ϫ 7n Ϫ 4

Ϭ 1n Ϫ 42 ϭ 2

2

6n ϩ 7n ϩ 2

6n ϩ 7n ϩ 2



и



1

nϪ4



12n ϩ 121n Ϫ 42



ϭ



12n ϩ 1213n ϩ 221n Ϫ 42



ϭ



1

3n ϩ 2



In a problem such as Example 7, it may be helpful to write the divisor with a denominator

nϪ4

1

of 1. Thus we can write n Ϫ 4 as

; its reciprocal then is obviously

.

1

nϪ4



Concept Quiz 7.2

For Problems 1–10, answer true or false.

1. To multiply two rational numbers in fraction form, we need to change to equivalent

fractions with a common denominator.

2. When multiplying rational expressions that contain polynomials, the polynomials are

factored so that common factors can be divided out.



7.2 • Multiplying and Dividing Algebraic Fractions



3. In the division problem



2x2y

4x3

4x3

Ϭ 2 , the fraction 2 is the divisor.

3z

5y

5y



2

3

4. The numbers Ϫ and are multiplicative inverses.

3

2

5. To divide two numbers in fraction form, we invert the divisor and multiply.

4xy

3y

6y2

6. If x ϶ 0, then a

ba b ϭ

.

x

x

2x

3

4

Ϭ ϭ 1.

4

3



7.



5x2y

10x2

3

Ϭ

ϭ .

2y

3y

4

xϪy

1

9. If x ϶ y, then

Ϭ (x Ϫ y) ϭ .

2

2

8. If x ϶ 0 and y ϶ 0, then



10. If x ϶ Ϫ2 and x ϶ 2, then



xϪ2

xϩ2

x2 Ϫ 4

Ϭ

ϭ 2

.

xϩ2

xϪ2

x ϩ4



Problem Set 7.2

For Problems 1–40, perform the indicated multiplications

and divisions and express your answers in simplest form.



23.



2x2 ϩ xy

xy



24.



x2 ϩ y2

xϪy



(Objectives 1 and 2)



5

1.

9



и



3

10



7

2.

8



и



12

14



и



и



y

10x ϩ 5y



x2 Ϫ xy

3



3

6

3. aϪ b a b

4

7



5

4

4. a baϪ b

6

15



25.



6ab

7a Ϫ 7b

Ϭ 2

4ab ϩ 4b2

a Ϫ b2



5. a



6. aϪ



15

13

bϬa b

7

14



26.



4ab

ab ϩ b

Ϭ

3a Ϫ 3b

2a Ϫ 2ab



20xy

18x



27.



x2 ϩ 11x ϩ 30

x2 ϩ 4



и



5x2 ϩ 20

x2 ϩ 14x ϩ 45



28.



x2 ϩ 15x ϩ 54

x2 ϩ 2



и



3x2 ϩ 6

x ϩ 10x ϩ 9



7.



17

19

b Ϭ aϪ b

9

9



8xy

12y



9. aϪ



6x

14y



и



5n2 27n

ba

b

18n 25



8.



9x

15y



10. a



и



4ab

30a

baϪ

b

10

22b



11.



3a2

6a

Ϭ

7

28



12.



12x

4x

Ϭ

11y

33



29.



13.



18a2b2

Ϫ9a

Ϭ

Ϫ27a

5b



14.



24ab2

Ϫ12ab

Ϭ

25b

15a2



30.



15. 24x3 Ϭ



16x

y



16. 14xy2 Ϭ



7y

9



2x2 ϩ xy Ϫ y2

x2y



Ϭ



6x2y2



5x2 ϩ 4xy Ϫ y2

y

1Ϫa

1 Ϫ a2



2a2 Ϫ 11a Ϫ 21

3a2 ϩ a



и



3a2 Ϫ 11a Ϫ 4

2a2 Ϫ 5a Ϫ 12



Ϫ2

1

Ϭ

2 3

7a b

9ab4



32.



19.



18rs

Ϭ (9r)

34



20.



8rs

Ϭ (6s)

3



33.



21.



y

xϩy



22.



x2 Ϫ 9

6



34.



8

xϪ3



4x2y



x2 Ϫ y2



и



18.



и



Ϭ



a ϩ a2

15a2 ϩ 11a ϩ 2



1

Ϫ1

Ϭ

3

12a

15ab



x2 Ϫ y2

xy



2x2 Ϫ 3xy ϩ y2



2



31.



17.



и



2



2x2 Ϫ 2xy

x ϩ 4x Ϫ 32

2



x3 ϩ 3x2

x ϩ 4x ϩ 4

2



и



и



x2 Ϫ 16

5xy Ϫ 5y2



x2 Ϫ 5x Ϫ 14

x2 ϩ 3x



277



278



Chapter 7 • Algebraic Fractions



4x2 Ϫ 12xy ϩ 9y2

10x Ϫ 15y



multiplications and divisions are done in the order that they

appear from left to right. (Objectives 1 and 2)



x2 Ϫ 4y2



41.



6

30x

Ϭ

9y

12y2



12t2 ϩ 5t Ϫ 3

37.

Ϭ

45t Ϫ 15

20t ϩ 5



42.



5xy2

12y



5t Ϫ 3t Ϫ 2

5t ϩ 32t ϩ 12

38.

Ϭ

1t Ϫ 12 2

4t2 Ϫ 3t Ϫ 1



43.



8x2

xy Ϫ xy2



и



xy

xϪ1

Ϭ

2

2

xϩy

8x Ϫ 8y



44.



5x Ϫ 20

x2 Ϫ 9



и



xϩ3

15

Ϭ

xϪ4

xϪ3



45.



x2 ϩ 9x ϩ 18

x2 ϩ 3x



46.



4x

3x ϩ 6y



35.



36.



2x2 Ϫ xy Ϫ 3y2

1x ϩ y2



Ϭ



2



x2 ϩ 4xy ϩ 4y2



Ϭ



x2



x2 Ϫ 2xy



13t Ϫ 12 2

2



2



n3 Ϫ n

39. 2

n ϩ 7n ϩ 6

40.



4n ϩ 24

n2 Ϫ n



и



2x2 Ϫ 6x Ϫ 36

x2 ϩ 2x Ϫ 48



и



x2 ϩ 5x Ϫ 24

2x2 Ϫ 18



For Problems 41–46, perform the indicated operations and

express the answers in simplest form. Remember that



и



5xy

4



и



18x2

3

Ϭ

15y

2xy



и



и



x2 ϩ 5x

x2 ϩ 8x

Ϭ

x2 Ϫ 25

x2 ϩ 3x Ϫ 40



5xy

x Ϫ4

2



Ϭ



10

x ϩ 4x ϩ 4

2



Thoughts Into Words

47. Give a step-by-step description of how to do the following multiplication problem:

x2 Ϫ x

x2 Ϫ 1



и



x2 ϩ x Ϫ 6

x ϩ 4x Ϫ 12

2



x

xϪ2

Ϭ

is undefined

xϩ1

xϪ1

for x ϭ Ϫ1, x ϭ 1, and x ϭ 0 but is defined for

x ϭ 2.



49. Explain why the quotient



x

xϪ1

1

x

1

xϪ1

Ϭ

bϬ ϭ

Ϭ b?

Ϭ a

x

x xϩ1

x

x

xϩ1

Justify your answer.



48. Is a



Answers to the Concept Quiz

1. False

2. True

3. True

4. False

9. True

10. False



7.3



5. True



6. True



7. False



8. False



Adding and Subtracting Algebraic Fractions



OBJECTIVES



1



Combine rational expressions with common denominators



2



Find the lowest common denominator



3



Add and subtract rational expressions with different denominators



In Chapter 2 we defined addition and subtraction of rational numbers as



a

c aϩc

ϩ ϭ

and

b

b

b



a

c

aϪc

Ϫ ϭ

, respectively. These definitions extend to algebraic fractions in general.

b

b

b



7.3 • Adding and Subtracting Algebraic Fractions



279



Definition 7.3

If



A

C

and are rational expressions with B ϶ 0, then

B

B



A

C

AϩC

ϩ ϭ

B

B

B



and



C

AϪC

A

Ϫ ϭ

B

B

B



Thus if the denominators of two algebraic fractions are the same, we can add or subtract the

fractions by adding or subtracting the numerators and placing the result over the common

denominator. Here are some examples:

5

7

5ϩ7

12

ϩ ϭ

ϭ

x

x

x

x

8

3

8Ϫ3

5

Ϫ

ϭ

ϭ

xy

xy

xy

xy

14

15

14 ϩ 15

29

ϩ

ϭ

ϭ

2x ϩ 1

2x ϩ 1

2x ϩ 1

2x ϩ 1

3

4

3Ϫ4

Ϫ1

or

Ϫ

ϭ

ϭ

aϪ1

aϪ1

aϪ1

aϪ1



Ϫ



1

aϪ1



In the next examples, notice how we put to use our previous work with simplifying

polynomials.

1x ϩ 32 ϩ 12x Ϫ 32

xϩ3

2x Ϫ 3

3x

ϩ

ϭ

ϭ

4

4

4

4

1x

ϩ

52

Ϫ

1x

ϩ

22

xϩ5

xϩ2

xϩ5ϪxϪ2

3

Ϫ

ϭ

ϭ

ϭ

7

7

7

7

7

13x ϩ 12 ϩ 12x ϩ 32

3x ϩ 1

2x ϩ 3

5x ϩ 4

ϩ

ϭ

ϭ

xy

xy

xy

xy

213n ϩ 12

31n Ϫ 12

213n ϩ 12 Ϫ 31n Ϫ 12

6n ϩ 2 Ϫ 3n ϩ 3

3n ϩ 5

Ϫ

ϭ

ϭ

ϭ

n

n

n

n

n



It may be necessary to simplify the fraction that results from adding or subtracting two

fractions.

14x Ϫ 32 ϩ 12x ϩ 32

4x Ϫ 3

2x ϩ 3

6x

3x

ϩ

ϭ

ϭ

ϭ

8

8

8

8

4

13n

Ϫ

12

Ϫ

1n

Ϫ

52

3n Ϫ 1

nϪ5

3n Ϫ 1 Ϫ n ϩ 5

Ϫ

ϭ

ϭ

12

12

12

12

ϭ



21n ϩ 22

2n ϩ 4

nϩ2

ϭ

ϭ

12

12

6



1Ϫ2x ϩ 32 ϩ 13x Ϫ 12

Ϫ2x ϩ 3

3x Ϫ 1

xϩ2

ϩ 2

ϭ

ϭ 2

2

x Ϫ4

x Ϫ4

x2 Ϫ 4

x Ϫ4

ϭ



xϩ2

1x ϩ 221x Ϫ 22



ϭ



1

xϪ2



Recall that to add or subtract rational numbers with different denominators, we first

change them to equivalent fractions that have a common denominator. In fact, we found that

by using the least common denominator (LCD), our work was easier. Let’s carefully review

the process because it will also work with algebraic fractions in general.



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