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Section 7.4 Partial Fractions 533

IntroductionIn this section, you will learn to write a rational expression as the sum of two ormore simpler rational expressions. For example, the rational expression

can be written as the sum of two fractions with first-degree denominators. That is,

Partial fraction decomposition

of

Partial Partialfraction fraction

Each fraction on the right side of the equation is a partial fraction, and togetherthey make up the partial fraction decomposition of the left side.

x � 7

x2 � x � 6�

2

x � 3�

�1

x � 2.

x � 7x2 � x � 6

x � 7x2 � x � 6

What you should learn• Recognize partial fraction

decompositions of rationalexpressions.

• Find partial fraction decompo-sitions of rational expressions.

Why you should learn itPartial fractions can help youanalyze the behavior of a rational function. For instance,in Exercise 57 on page 540, youcan analyze the exhaust temper-atures of a diesel engine usingpartial fractions.

Partial Fractions

© Michael Rosenfeld/Getty Images

7.4

Section A.4, shows you how tocombine expressions such as

The method of partial fractionsshows you how to reverse thisprocess.

�?

x � 2�

?x � 3

5�x � 2��x � 3�

�5

�x � 2��x � 3�.1

x � 2�

�1x � 3

Decomposition of into Partial Fractions

1. Divide if improper: If is an improper fraction degree ofdivide the denominator into the numerator to

obtain

and apply Steps 2, 3, and 4 below to the proper rational expressionNote that is the remainder from the division of

by

2. Factor the denominator: Completely factor the denominator into factorsof the form

and

where is irreducible.

3. Linear factors: For each factor of the form the partial frac-tion decomposition must include the following sum of fractions.

4. Quadratic factors: For each factor of the form the par-tial fraction decomposition must include the following sum of fractions.

B1x � C1

ax2 � bx � c�

B2x � C2

�ax2 � bx � c�2� . . . �

Bnx � Cn

�ax2 � bx � c�n

n�ax2 � bx � c�n,

A1

� px � q��

A2

� px � q�2� . . . �

Am

� px � q�m

m� px � q�m,

�ax2 � bx � c�

�ax2 � bx � c�n� px � q�m

D�x�.N�x�N1�x�N1�x��D�x�.

N�x�D�x�

� �polynomial� �N1�x�D�x�

N�x� ≥ degree of D�x��,�N�x��D�x�

N�x�/D�x�

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Partial Fraction DecompositionAlgebraic techniques for determining the constants in the numerators of partialfractions are demonstrated in the examples that follow. Note that the techniquesvary slightly, depending on the type of factors of the denominator: linear orquadratic, distinct or repeated.

Distinct Linear Factors

Write the partial fraction decomposition of

SolutionThe expression is proper, so be sure to factor the denominator. Because

you should include one partial fraction with aconstant numerator for each linear factor of the denominator. Write the form ofthe decomposition as follows.

Write form of decomposition.

Multiplying each side of this equation by the least common denominator,leads to the basic equation

Basic equation

Because this equation is true for all you can substitute any convenient values ofthat will help determine the constants and Values of that are especially

convenient are ones that make the factors and equal to zero. Forinstance, let Then

Substitute for

To solve for let and obtain

Substitute 3 for

So, the partial fraction decomposition is

Check this result by combining the two partial fractions on the right side of theequation, or by using your graphing utility.

Now try Exercise 15.

x � 7

x2 � x � 6�

2

x � 3�

�1

x � 2

2 � A.

10 � 5A

10 � A�5� � B�0�

x. 3 � 7 � A�3 � 2� � B�3 � 3�

x � 3A,

�1 � B.

5 � �5B

5 � A�0� � B��5�

x.�2 �2 � 7 � A��2 � 2� � B��2 � 3�

x � �2.�x � 3��x � 2�

xB.Axx,

x � 7 � A�x � 2� � B�x � 3�.

�x � 3��x � 2�,

x � 7

x2 � x � 6�

A

x � 3�

B

x � 2

x2 � x � 6 � �x � 3��x � 2�,

x � 7

x2 � x � 6.

534 Chapter 7 Systems of Equations and Inequalities

You can use a graphing utility tocheck graphically the decomposi-tion found in Example 1. To dothis, graph

and

in the same viewing window. Thegraphs should be identical, asshown below.

9−9

−6

6

y2 �2

x � 3�

�1x � 2

y1 �x � 7

x2 � x � 6

Techno logy

Example 1

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The next example shows how to find the partial fraction decomposition of arational expression whose denominator has a repeated linear factor.

Repeated Linear Factors


SolutionThis rational expression is improper, so you should begin by dividing the numer-ator by the denominator to obtain

Because the denominator of the remainder factors as

you should include one partial fraction with a constant numerator for each powerof and and write the form of the decomposition as follows.


Multiplying by the LCD, leads to the basic equation

Basic equation

Letting eliminates the - and -terms and yields

Letting eliminates the - and -terms and yields

At this point, you have exhausted the most convenient choices for so to findthe value of use any other value for along with the known values of and

So, using and



x �6x

��1

x � 1�

9�x � 1�2 .

x4 � 2x3 � 6x2 � 20x � 6x3 � 2x2 � x

�

�1 � B.

�2 � 2B

31 � 6�4� � 2B � 9

5�1�2 � 20�1� � 6 � 6�1 � 1�2 � B�1��1 � 1� � 9�1�

C � 9,A � 6,x � 1,C.AxB,

x,

6 � A.

6 � A�1� � 0 � 0

5�0�2 � 20�0� � 6 � A�0 � 1�2 � B�0��0 � 1� � C�0�CBx � 0

C � 9.

5 � 20 � 6 � 0 � 0 � C

5��1�2 � 20��1� � 6 � A��1 � 1�2 � B��1��1 � 1� � C��1�

BAx � �1

5x2 � 20x � 6 � A�x � 1�2 � Bx�x � 1� � Cx.

x�x � 1�2,

5x2 � 20x � 6

x�x � 1�2�

A

x�

B

x � 1�

C

�x � 1�2

�x � 1�x

x3 � 2x2 � x � x�x2 � 2x � 1� � x�x � 1�2

x �5x2 � 20x � 6x3 � 2x2 � x

.

x4 � 2x3 � 6x2 � 20x � 6x3 � 2x2 � x

.


Example 2

Have students check some of theiranswers to Exercises 15–52 or theanswers to some of the examples bycombining the partial fractions.

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The procedure used to solve for the constants in Examples 1 and 2 works wellwhen the factors of the denominator are linear. However, when the denominatorcontains irreducible quadratic factors, you should use a different procedure, whichinvolves writing the right side of the basic equation in polynomial form andequating the coefficients of like terms. Then you can use a system of equations to solve for the coefficients.

Distinct Linear and Quadratic Factors


SolutionThis expression is proper, so factor the denominator. Because the denominatorfactors as

you should include one partial fraction with a constant numerator and one partialfraction with a linear numerator and write the form of the decomposition as follows.


Multiplying by the LCD, yields the basic equation

Basic equation

Expanding this basic equation and collecting like terms produces

Polynomial form

Finally, because two polynomials are equal if and only if the coefficients of liketerms are equal, you can equate the coefficients of like terms on opposite sides ofthe equation.

Equate coefficients of like terms.

You can now write the following system of linear equations.

From this system you can see that and Moreover, substitutinginto Equation 1 yields



3x2 � 4x � 4

x3 � 4x�

1

x�

2x � 4

x2 � 4.

1 � B � 3 ⇒ B � 2.

A � 1C � 4.A � 1

Equation 1

Equation 2

Equation 3� A

4A

� B

C

�

�

�

344

3x2 � 4x � 4 � �A � B�x2 � Cx � 4A

� �A � B�x2 � Cx � 4A.

3x2 � 4x � 4 � Ax2 � 4A � Bx2 � Cx

3x2 � 4x � 4 � A�x2 � 4� � �Bx � C �x.

x�x2 � 4�,

3x2 � 4x � 4

x3 � 4x�

A

x�

Bx � C

x2 � 4

x3 � 4x � x�x2 � 4�

3x2 � 4x � 4

x3 � 4x.

Example 3

Historical NoteJohn Bernoulli (1667–1748),a Swiss mathematician, intro-duced the method of partialfractions and was instrumentalin the early development ofcalculus. Bernoulli was a pro-fessor at the University of Baseland taught many outstandingstudents, the most famous ofwhom was Leonhard Euler.

The

Gra

ng

er C

olle

ctio

n

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The next example shows how to find the partial fraction decomposition of arational expression whose denominator has a repeated quadratic factor.

Repeated Quadratic Factors


SolutionYou need to include one partial fraction with a linear numerator for each powerof


Multiplying by the LCD, yields the basic equation

Basic equation

Polynomial form

Equating coefficients of like terms on opposite sides of the equation

produces the following system of linear equations.

Finally, use the values and to obtain the following.

Substitute 8 for A in Equation 3.

Substitute 0 for B in Equation 4.

So, using and the partial fraction decompositionis

Check this result by combining the two partial fractions on the right side of theequation, or by using your graphing utility.


8x3 � 13x

�x2 � 2�2�

8x

x2 � 2�

�3x

�x2 � 2�2.

D � 0,C � �3,B � 0,A � 8,

D � 0

2�0� � D � 0

C � �3

2�8� � C � 13

B � 0A � 8

Equation 1

Equation 2

Equation 3

Equation 4�

A

2A �

B

2B �

C D

�

�

�

�

80

130

8x3 � 0x2 � 13x � 0 � Ax3 � Bx2 � �2A � C �x � �2B � D�

� Ax3 � Bx2 � �2A � C�x � �2B � D�.

� Ax3 � 2Ax � Bx2 � 2B � Cx � D

8x3 � 13x � �Ax � B��x2 � 2� � Cx � D

�x2 � 2�2,

8x3 � 13x

�x2 � 2�2�

Ax � B

x2 � 2�

Cx � D

�x2 � 2�2

�x2 � 2�.

8x3 � 13x

�x2 � 2�2.

Additional Examples

Partial fraction decompositions can beconfusing to students. You may want togo over several additional examples.

a.

b.

c.

2x � 2�x2 � 1�2

3x3 � x2 � 5x � 1�x2 � 1�2 �

3x � 1x2 � 1

�

3x2 �

1x2 � 2

� x3 � 4x2 � 2x � 6x2�x2 � 2� � �

1x

�

3�x � 1�2 �

1�x � 1�3

2x2 � 7x � 4�x � 1�3 �

2x � 1

�

Check students’ understanding. Why do you use two different procedures?When is each procedure more effective?Is it mathematically correct (althoughnot necessarily as efficient) to use eitherprocedure regardless of whether thefactors of the denominator are linear orquadratic?

Example 4

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Keep in mind that for improper rational expressions such as

you must first divide before applying partial fraction decomposition.

N�x�D�x� �

2x3 � x2 � 7x � 7x2 � x � 2

Activities

1. Write the partial fractiondecomposition of

Answer:

2. Write the partial fractiondecomposition of

Answer: 4x � 1 �3x

�7

x � 2

4x3 � 9x2 � 2x � 6x�x � 2� .

4x � 2

�3

2x � 1

5x � 10�x � 2��2x � 1�.

Guidelines for Solving the Basic EquationLinear Factors

1. Substitute the zeros of the distinct linear factors into the basic equation.

2. For repeated linear factors, use the coefficients determined in Step 1 torewrite the basic equation. Then substitute other convenient values of and solve for the remaining coefficients.

Quadratic Factors

1. Expand the basic equation.

2. Collect terms according to powers of

3. Equate the coefficients of like terms to obtain equations involving and so on.

4. Use a system of linear equations to solve for A, B, C, . . . .

C,A, B,

x.

x

W RITING ABOUT MATHEMATICS

Error Analysis You are tutoring a student in algebra. In trying to find a partial fraction decomposition, the student writes the following.

Basic equation

By substituting and into the basic equation, the student concludesthat and However, in checking this solution, the student obtainsthe following.

What has gone wrong?

�x2 � 1

x�x � 1�

�x � 1

x�x � 1�

�1

x�

2

x � 1�

��1��x � 1� � 2�x�x�x � 1�

B � 2.A � �1x � 1x � 0

x 2 � 1 � A�x � 1� � Bx

x 2 � 1

x�x � 1��

A�x � 1�x�x � 1�

�Bx

x�x � 1�

x 2 � 1

x�x � 1��

A

x�

B

x � 1

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In Exercises 1– 4, match the rational expression with theform of its decomposition. [The decompositions arelabeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

1. 2.

3. 4.

In Exercises 5–14, write the form of the partial fractiondecomposition of the rational expression. Do not solve forthe constants.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

In Exercises 15–38, write the partial fraction decompositionof the rational expression. Check your result algebraically.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

In Exercises 39–44, write the partial fraction decompositionof the improper rational expression.

39. 40.

41. 42.

43. 44.16x4

�2x � 1�3

x4

�x � 1�3

x3 � 2x2 � x � 1x2 � 3x � 4

2x3 � x2 � x � 5x2 � 3x � 2

x2 � 4xx2 � x � 6

x2 � xx2 � x � 1

x2 � 4x � 7

�x � 1��x2 � 2x � 3�x2 � 5

�x � 1��x2 � 2x � 3�

x � 1

x3 � x

x

16x4 � 1

2x2 � x � 8

�x2 � 4�2

x2

x4 � 2x2 � 8

x � 6x3 � 3x2 � 4x � 12

xx3 � x2 � 2x � 2

x

�x � 1��x2 � x � 1�x2 � 1

x�x2 � 1�

6x2 � 1

x2�x � 1�2

3x

�x � 3�2

2x � 3

�x � 1�2

4x2 � 2x � 1

x2�x � 1�

x � 2

x�x � 4�x2 � 12x � 12

x3 � 4x

x � 1

x2 � 4x � 3

3

x2 � x � 2

5

x 2 � x � 6

1

2x2 � x

3

x2 � 3x

1

x2 � x

1

4x2 � 9

1

x2 � 1

x � 4x2�3x � 1�2

x � 1x�x2 � 1�2

x � 62x3 � 8x

2x � 3

x3 � 10x

6x � 5�x � 2�4

4x2 � 3�x � 5�3

x2 � 3x � 24x3 � 11x2

12

x3 � 10x2

x � 2

x2 � 4x � 3

7

x2 � 14x

3x � 1x�x2 � 4�

3x � 1x�x2 � 4�

3x � 1x2�x � 4�

3x � 1x�x � 4�

Ax

�Bx � Cx2 � 4

Ax

�Bx2 �

Cx � 4

Ax

�B

x � 4Ax

�B

x � 2�

Cx � 2

Exercises 7.4

VOCABULARY CHECK: Fill in the blanks.

1. The process of writing a rational expression as the sum or difference of two or more simpler rational expressions is called ________ ________ ________.

2. If the degree of the numerator of a rational expression is greater than or equal to the degree of the denominator, then the fraction is called ________.

3. In order to find the partial fraction decomposition of a rational expression, the denominator must be completely factored into ________ factors of the form and ________ factors of the form

which are ________ over the rationals.

4. The ________ ________ is derived after multiplying each side of the partial fraction decomposition form by the least common denominator.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

�ax2 � bx � c�n,�px � q�m

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In Exercises 45–52, write the partial fraction decompositionof the rational expression. Use a graphing utility to checkyour result graphically.

45. 46.

47. 48.

49. 50.

51. 52.

Graphical Analysis In Exercises 53–56, (a) write the partialfraction decomposition of the rational function, (b) identifythe graph of the rational function and the graph of eachterm of its decomposition, and (c) state any relationshipbetween the vertical asymptotes of the graph of therational function and the vertical asymptotes of the graphsof the terms of the decomposition. To print an enlargedcopy of the graph, go to the website www.mathgraphs.com.

53. 54.

55. 56.

Synthesis58. Writing Describe two ways of solving for the constants

in a partial fraction decomposition.

True or False? In Exercises 59 and 60, determine whetherthe statement is true or false. Justify your answer.

59. For the rational expression the partial

fraction decomposition is of the form

60. When writing the partial fraction decomposition of the

expression the first step is to factor the

denominator.

In Exercises 61–64, write the partial fraction decompositionof the rational expression. Check your result algebraically.Then assign a value to the constant a to check the resultgraphically.

61. 62.

63. 64.

Skills Review

In Exercises 65–70, sketch the graph of the function.

65. 66.

67. 68.

69. 70. f �x� �3x � 1

x2 � 4x � 12f �x� �

x2 � x � 6x � 5

f �x� �12x3 � 1f �x� � �x2�x � 3�

f �x� � 2x2 � 9x � 5f �x� � x2 � 9x � 18

1

�x � 1��a � x�1

y�a � y�

1

x�x � a�1

a2 � x2

x 3 � x � 2x2 � 5x � 14

Ax � 10

�B

�x � 10�2 .

x�x � 10��x � 10�2

–4 4 8–4

4

8

12

x

y

4 8x

y

−4

−8

y �2�4x2 � 15x � 39�x2�x2 � 10x � 26�

y �2�4x � 3�

x2 � 9

2 4

4

x

y

−4

4 8

−8

−4

4

8

x

y

y �2(x � 1)2

x�x2 � 1�y �

x � 12

x�x � 4�

x3 � x � 3

x2 � x � 2

2x3 � 4x2 � 15x � 5

x2 � 2x � 8

x3

�x � 2�2�x � 2�2

x2 � x � 2

�x2 � 2�2

4x2 � 1

2x�x � 1�2

x � 1

x3 � x2

3x2 � 7x � 2

x3 � x

5 � x

2x2 � x � 1


57. Thermodynamics The magnitude of the range ofexhaust temperatures (in degrees Fahrenheit) in anexperimental diesel engine is approximated by themodel

0 < x ≤ 1R �2000�4 � 3x�

�11 � 7x��7 � 4x�,

R

Model It

Model It (cont inued)

where is the relative load (in foot-pounds).

(a) Write the partial fraction decomposition of theequation.

(b) The decomposition in part (a) is the difference oftwo fractions. The absolute values of the terms givethe expected maximum and minimum temperaturesof the exhaust gases for different loads.

Write the equations for Ymax and Ymin.

(c) Use a graphing utility to graph each equation frompart (b) in the same viewing window.

(d) Determine the expected maximum and minimumtemperatures for a relative load of 0.5.

Ymin � 2nd termYmax � 1st term

x

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