Factoring Polynomials - · PDF fileSection P.5 Factoring Polynomials 53 Factoring by ... Factoring the Sum or Difference of Two Cubes The last type of special factoring

Section P.5 Factoring Polynomials 51

Factoring Polynomials with Common FactorsNow you will switch from the process of multiplying polynomials to the reverseprocess—factoring polynomials. This section and the next section deal onlywith polynomials that have integer coefficients. Remember that in Section P.3you used the Distributive Property to multiply and remove parentheses, asfollows.

Distributive Property

In this and the next section, you will use the Distributive Property in the reversedirection to factor and create parentheses.


Factoring an expression (by the Distributive Property) changes a sum of termsinto a product of factors. Later you will see that this is an important strategy forsolving equations and for simplifying algebraic expressions.

To be efficient in factoring, you need to understand the concept of thegreatest common factor (or GCF). Recall from arithmetic that every integer canbe factored into a product of prime numbers. The greatest common factor of twoor more integers is the greatest integer that divides evenly into each integer. Tofind the greatest common factor of two integers or two expressions, begin by writ-ing each as a product of prime factors. The greatest common factor is the productof the common prime factors. For instance, from the factorizations

you can see that the common prime factors of 18 and 42 are 2 and 3. So, it fol-lows that the greatest common factor is or 6.

Example 1 � Finding the Greatest Common Factor

Find the greatest common factors of (a) 36, 8x, 64y and (b)

Solution(a) From the factorizations

you can conclude that the greatest common factor is 4.

(b) From the factorizations

you can conclude that the greatest common factor is

Now try Exercise 5.

6x3.

12x3 � 2 � 2 � 3 � x � x � x � �6x3��2� 30x4 � 2 � 3 � 5 � x � x � x � x � �6x3��5x� 6x5 � 2 � 3 � x � x � x � x � x � �6x3��x2�

64y � 2 � 2 � 2 � 2 � 2 � 2 � y � 4�16y� 8x � 2 � 2 � 2 � x � 4�2x� 36 � 2 � 2 � 3 � 3 � 4�9�

12x3.30x4,6x5,

2 � 3

42 � 2 � 3 � 718 � 2 � 3 � 3

12x � 15x2 � 3x�4 � 5x�

3x�4 � 5x� � 12x � 15x2

Factoring PolynomialsP.5

• Factor polynomials with commonfactors

• Factor polynomials by groupingterms

• Factor the difference of twosquares

• Factor the sum or difference oftwo cubes

• Factor polynomials completely

In some cases, factoring a polyno-mial enables you to determineunknown quantities. For example, in Exercise 113 on page 59, you willfactor the expression for the area ofa rectangle to determine the lengthof the rectangle.

What you should learn:

Why you should learn it:

52 Chapter P Prerequisites

Consider the three terms given in Example 1(b) as terms of the polynomial

The greatest common factor, of these terms is called the greatest commonmonomial factor of the polynomial. When you use the Distributive Property toremove this factor from each term of the polynomial, you are factoring out thegreatest common monomial factor.

Factor each term.

If a polynomial in x (with integer coefficients) has a greatest commonmonomial factor of the form the statements below must be true.

1. The coefficient a of the greatest common monomial factor must be the greatestinteger that divides each of the coefficients in the polynomial.

2. The variable factor of the greatest common monomial factor has the lowestpower of x of all terms of the polynomial.

Example 2 � Factoring out a Greatest Common Monomial Factor

Factor out the greatest common monomial factor from

SolutionFor the terms and 8 is the greatest integer factor of 24 and 32 and isthe highest-powered variable factor common to and So, the greatestcommon monomial factor of and is You can factor the givenpolynomial as follows.

Now try Exercise 19.

The greatest common monomial factor of a polynomial is usually consideredto have a positive coefficient. However, sometimes it is convenient to factor anegative number out of a polynomial, as shown in the next example.

Example 3 � A Negative Common Monomial Factor

Factor the polynomial in two ways.

(a) Factor out a 3. (b) Factor out a

Solution

(a) By factoring out the common monomial factor of 3, you obtain

Factor each term.

Factored form

(b) By factoring out the common monomial factor of you obtain

Factor each term.

Factored form


� �3�x2 � 4x � 6�. �3x2 � 12x � 18 � �3�x2� � ��3��4x� � ��3��6�

�3,

� 3��x2 � 4x � 6�. �3x2 � 12x � 18 � 3��x2� � 3�4x� � 3��6�

�3.

�3x2 � 12x � 18

24x3 � 32x2 � �8x2��3x� � �8x2��4� � 8x2�3x � 4�

8x2.32x224x3x2.x3

x232x2,24x3

24x3 � 32x2.

xn

axn,

� 6x3�x2 � 5x � 2� 6x5 � 30x4 � 12x3 � 6x3�x2� � 6x3�5x� � 6x3�2�

6x3,

6x5 � 30x4 � 12x3.

Factor out common monomial factor.

Study Tip

Whenever you are factoring apolynomial, remember that youcan check your results by multi-plying. That is, if you multiplythe factors, you should obtainthe original polynomial.


Factoring by GroupingSome polynomials have common factors that are not simple monomials. Forinstance, the polynomial has the common binomialfactor Factoring out this common factor produces

This type of factoring is part of a more general procedure called factoring bygrouping.

Example 4 � A Common Binomial Factor

Factor the polynomial

SolutionEach of the terms of this polynomial has a binomial factor of Factoringthis binomial out of each term produces


In Example 4, the given polynomial was already grouped, and so it was easyto determine the common binomial factor. In practice, you will have to do thegrouping as well as the factoring. To see how this works, consider the expression

and try to factor it. Note first that there is no commonmonomial factor to take out of all four terms. But suppose you group the first twoterms together and the last two terms together. Then you have

Group terms.

Example 5 � Factoring by Grouping


Solution

Group terms.



Note that in Example 5 the polynomial is factored by grouping the first andsecond terms and the third and fourth terms. You could just as easily have groupedthe first and third terms and the second and fourth terms, as follows.

� �x2 � 1��x � 5� � x�x2 � 1� � 5�x2 � 1�

x3 � 5x2 � x � 5 � �x3 � x� � �5x2 � 5�

� �x � 5��x2 � 1� � x2�x � 5� � 1�x � 5�

x3 � 5x2 � x � 5 � �x3 � 5x2� � �x � 5�

x3 � 5x2 � x � 5.

� �x � 3��x2 � 5�. � x2�x � 3� � 5�x � 3�

x3 � 3x2 � 5x � 15 � �x3 � 3x2� � �5x � 15�

x3 � 3x2 � 5x � 15

5x2�6x � 5� � 2�6x � 5� � �6x � 5��5x2 � 2�.

�6x � 5�.

5x2�6x � 5� � 2�6x � 5�.

x2�2x � 3� � 4�2x � 3� � �2x � 3��x2 � 4�.

�2x � 3�.x2�2x � 3� � 4�2x � 3�

Factor out common mono-mial factor in each group.

Factor out common mono-mial factor in each group.

Factor out common binomial factor.


Factoring the Difference of Two SquaresSome polynomials have special forms that you should learn to recognize so thatthey can be factored easily. Here are some examples of forms that you should beable to recognize by the time you have completed this section.

Difference of two squares

Sum of two cubes

Difference of two cubes

One of the easiest special polynomial forms to recognize and to factor is theform called a difference of two squares. This form arises from thespecial product in Section P.4.

To recognize perfect squares, look for coefficients that are squares of integersand for variables raised to even powers. Here are some examples.

Original Think: DifferencePolynomial of Squares Factored Form

Example 6 � Factoring the Difference of Two Squares

Factor (a) and (b)

Solution(a) Because and 64 are both perfect squares, you can recognize this polyno-

mial as the difference of two squares. So, the polynomial factors as follows.

Write as difference of two squares.

Factored form

(b) Because both and 49 are perfect squares, you can recognize the differenceof two squares. So, the polynomial factors as follows.


Factored form


� �2y � 7��2y � 7� 4y2 � 49 � �2y�2 � 72

4y2

� �x � 8��x � 8� x2 � 64 � x2 � 82

x2

4y2 � 49.x2 � 64

�x5 � 4y��x5 � 4y��x5�2 � �4y�2x10 � 16y2

�5 � 7x��5 � 7x�52 � �7x�225 � 49x2

�2x � 5��2x � 5��2x�2 � 524x2 � 25

�x � 2��x � 2�x2 � 22x2 � 4

�u � v��u � v�u2 � v2,

x3 � 1 � �x � 1��x2 � x � 1� x3 � 8 � �x � 2��x2 � 2x � 4� x2 � 9 � �x � 3��x � 3�

Difference of Two Squares

Let u and v be real numbers, variables, or algebraic expressions. Then theexpression can be factored using the following pattern.

Difference Opposite signs

u2 � v2 � �u � v��u � v�

u2 � v2



SolutionBecause and are both perfect squares, you can recognize thispolynomial as the difference of two squares. So, the polynomial factors as fol-lows.


Factored form


Remember that the rule applies to polynomials orexpressions in which u and v are themselves expressions. The next example illus-trates this possibility.


Factor the difference of two squares.

(a) (b)

Solution

(a) Write as difference of two squares.

Factored form

Simplify.

(b) Write as difference of two squares.

Factored form

Simplify.

To check these results, write the original polynomial in standard form. Thenmultiply the factored form to see that you obtain the same standard form.


Factoring the Sum or Difference of Two CubesThe last type of special factoring discussed in this section is factoring of the sum or difference of two cubes. The patterns for these two special forms are sum-marized below. In these patterns, pay particular attention to the signs of the terms.

� �x � 1��x � 9� � ��x � 5� � 4��x � 5� � 4�

�x � 5�2 � 16 � �x � 5�2 � 42

� �x � 5��x � 1� � ��x � 2� � 3��x � 2� � 3�

�x � 2�2 � 9 � �x � 2�2 � 32

�x � 5�2 � 16�x � 2�2 � 9

u2 � v2 � �u � v��u � v�

� �7x � 9y��7x � 9y� 49x2 � 81y2 � �7x�2 � �9y�2

81y249x2

49x2 � 81y2.

Sum or Difference of Two Cubes

Let u and v be real numbers, variables, or algebraic expressions. Then theexpressions and can be factored as follows.

Like signs Like signs

Unlike signs Unlike signs

u3 � v3 � �u � v��u2 � uv � v2�u3 � v3 � �u � v��u2 � uv � v2�

u3 � v3u3 � v3


Study Tip

Remember that you can check afactoring result by multiplying.For instance, you can check thefirst two results in Example 9 asfollows.

(a)

(b)


Example 9 � Factoring Sums and Differences of Cubes

Factor each polynomial.

(a) (b) (c)

Solution

(a) This polynomial is the difference of two cubes, because is the cube of xand 125 is the cube of 5. So, you can factor the polynomial as follows.

Factored form

Simplify.

(b) This polynomial is the sum of two cubes, because is the cube of 2y and 1is its own cube. So, you can factor the polynomial as follows.

Factored form

Simplify.

(c) Because is the cube of 3x and is the cube of 4y, this polynomial isthe difference of two cubes. So, you can factor the polynomial as follows.

Factored form

Simplify.


Factoring CompletelySometimes the difference of two squares can be hidden by the presence of acommon monomial factor. Remember that with all factoring techniques, youshould first factor out any common monomial factors.

Example 10 � Factoring out a Common Monomial Factor First

Factor the polynomial completely.

SolutionThe polynomial has a common monomial factor of 5. After factoringout this factor, the remaining polynomial is the difference of two squares.



Factored form


The polynomial in Example 10 is said to be completely factored becausenone of its factors can be factored further using integer coefficients.

� 5�5x � 4��5x � 4�

� 5��5x�2 � 42� 125x2 � 80 � 5�25x2 � 16�

125x2 � 80

125x2 � 80

� �3x � 4y��9x2 � 12xy � 16y2� � �3x � 4y��3x�2 � �3x��4y� � �4y�2�

27x3 � 64y3 � �3x�3 � �4y�3

64y327x3

� �2y � 1��4y2 � 2y � 1� � �2y � 1��2y�2 � �2y��1� � 12�

8y3 � 1 � �2y�3 � 13

8y3

� �x � 5��x2 � 5x � 25� � �x � 5��x2 � 5x � 52�

x3 � 125 � x3 � 53

x3

27x3 � 64y38y3 � 1x3 � 125

Write as differenceof two cubes.

Write as sum oftwo cubes.

Write as differenceof two cubes.

EXPLORATIONFind a formula for completelyfactoring using theformulas from this section. Useyour formula to factor and x6 64 completely.�

x6 � 1

u6 � v 6

x3

x3

�

x2

5x2

�5x2

�

�

�

5x

x

25x

25x

�

�

�

�

25

5

125

125

8y3

8y3

�

4y2

4y2

4y2

�

�

�

2y

2y

2y

2y

�

�

�

�

1

1

1

1


Example 11 � Factoring out a Common Monomial Factor First

Factor the polynomial completely.

Solution


Write as sum of two cubes.

Factored form


To factor a polynomial completely, always check to see whether the factorsobtained might themselves be factorable. For instance, after factoring the poly-nomial once as the difference of two squares

you can see that the second factor is itself the difference of two squares. So, tofactor the polynomial completely, you must continue factoring, as follows.

Example 12 � Factoring Completely

Factor (a) and (b) completely.

Solution

(a) Recognizing as a difference of two squares, you can write

Note that the second factor is itself a difference of two squares andyou therefore obtain

(b) Recognizing as a difference of two squares, you can write

Note that the second factor is itself a difference of two squares andyou therefore obtain


Example 13 � Using Factoring to Find the Length of a Rectangle

The area of a rectangle of width 5x is given by as shown in FigureP.15. Factor this expression to determine the length of the rectangle.

SolutionFactoring out the common monomial factor of 5x from the area polynomial

results in the polynomial So, the length of the rectangle isTo check this result, multiply to see that you obtain the area

polynomial


15x2 � 40x.5x�3x � 8�3x � 8.3x � 8.15x2 � 40x

15x2 � 40x,

81m4 � 1 � �9m2 � 1��9m2 � 1� � �9m2 � 1��3m � 1��3m � 1�.

�9m2 � 1�

81m4 � 1 � �9m2�2 � �1�2 � �9m2 � 1��9m2 � 1�.

81m4 � 1

x4 � y4 � �x2 � y2��x2 � y2� � �x2 � y2��x � y��x � y�.

�x2 � y2�

x4 � y4 � �x2�2 � �y2�2 � �x2 � y2��x2 � y2�.

x4 � y4

81m4 � 1x4 � y4

x4 � 16 � �x2 � 4��x2 � 4� � �x2 � 4��x � 2��x � 2�

x4 � 16 � �x2�2 � 42 � �x2 � 4��x2 � 4�

x4 � 16

� 3x�x � 3��x2 � 3x � 9� � 3x�x3 � 33�

3x4 � 81x � 3x�x3 � 27�

3x4 � 81x

Area = 15x2 + 40x 5x

Length

Figure P.15

Study Tip

Note in Example 12 that thesum of two squares does notfactor further. A second-degreepolynomial that is the sum oftwo squares, such as or

is not factorable,which means that it is notfactorable using integercoefficients.

9m2 � 1,x2 � y2


In Exercises 1–12, find the greatest common factor of theexpressions.

1. 48, 90 2. 36, 150, 100

3. 4.

5. 6.

7. 8.

9.

10.

11.

12.

In Exercises 13–34, factor out the greatest common mono-mial factor. Some of the polynomials have no commonmonomial factor other than 1 or

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33.

34.

In Exercises 35–42, factor a negative real number from thepolynomial and then write the polynomial factor in standardform.

35. 36.

37. 38.

39. 40.

41. 42.

In Exercises 43–46, fill in the missing factor.

43.

44.

45.

46. 712u �

218 v �

724��

56x �

59y �

518��

13x �

56 �

16��

32x �

54 �

14��

16 � 32s2 � 5s42t � 15 � 4t212x � 6x2 � 188 � 4x � 2x2

15 � 5x7 � 14x

32 � 4x10 � 5x

18a2b4 � 24a2b2 � 12a2b

3x3y3 � 2x2y2 � 5xy

24y4z2 � 12y2z29x3y � 6xy2

8y3 � 6y3z � 2y2z217x5y3 � xy2 � 34y2

25z6 � 15z4 � 35z314x4 � 21x3 � 9x2

9 � 27y � 15y228x2 � 16x � 8

4uv � 6u2v23x2y2 � 15y

16x2 � 3y311u2 � 9

36y4 � 24y221u2 � 14u

�a3 � 4a2x2 � x

14z3 � 2124x2 � 18

5x � 58z � 8

�1.��

8�x � 5�2�x � 5�,x2�1 � z�34x�1 � z�2,

44�3 � y�266�3 � y�,63�x � 8�342�x � 8�2,

36x2y284xy2,16x2y,42a2b514a2b3,28ab2,

150y345y,12z330z2,

18x327x4,12x3x2,

P.5 � Exercises

Writing About Mathematics

Factoring Higher-Degree Polynomials

Have each person in your group create a cubic polynomial by multiplying threepolynomials of the form

Then write the result in standard form on a piece of paper and describe the stepsused to create the polynomials and the steps used to write the polynomial in stan-dard form. Give it to another person in the group to factor. When each person hasfactored his or her polynomial, compare the results with the original factors usedto create the polynomials.

�x � a��x � a��x � b�.

VOCABULARY CHECK: Fill in the blanks.

1. The ________ of a group of expressions is the product of the common prime factors.

2. In the expression is the ________.

3. The process of writing a polynomial as a product is called ________.

4. A polynomial of the form is called a ________ of ________.

5. A polynomial of the form is called a ________ of ________.

6. A polynomial is considered ________ if each of its factors cannot be factored further using integer coefficients.

u3 � v3

u2 � v2

5x5x2 � 10x,


In Exercises 47–52, factor the polynomial by factoring outthe common binomial factor.

47.

48.

49.

50.

51.

52.

In Exercises 53–64, factor the polynomial by grouping.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

In Exercises 65–82, factor the difference of two squares.

65. 66.

67. 68.

69. 70.

71. 72.

73. 74.

75. 76.

77. 78.

79. 80.

81. 82.

In Exercises 83–94, factor the sum or difference of twocubes.

83. 84.

85. 86.

87. 88.

89. 90.

91. 92.

93. 94.

In Exercises 95–106, factor the polynomial completely.

95. 96.

97. 98.

99. 100.

101. 102.

103. 104.

105. 106.

In Exercises 107 and 108, factor the expression. (Assume)

107. 108.

In Exercises 109 and 110, show all the different groupingsthat can be used to factor the polynomial completely. Carryout the various factorizations to show that they yield thesame result.

109.

110.

111. Simple Interest The total amount of money froma principal of P invested at r% simple interest for tyears is given by Factor this expression.

112. Revenue The revenue from selling x units of aproduct at a price of p dollars per unit is given byxp. For a particular commodity, the revenue is

Factor the expression for the revenue and determinean expression for the price in terms of x.

113. Geometry A rectangle has area and width as indi-cated in the figure. Factor the expression for area tofind the length of the rectangle.

Figure for 113 Figure for 114

114. Geometry A rectangle has area and length as indi-cated in the figure. Find the width of the rectangle.

115. Geometry The area of a rectangle of width w isgiven by Factor this expression to deter-mine the length of the rectangle. Draw a diagram toillustrate your answer.

116. Geometry The area of a rectangle of length l isgiven by Factor this expression to deter-mine the width of the rectangle. Draw a diagram toillustrate your answer.

45l � l2.

32w � w2.

A = 27t2 − 18t ?

9t

?

A = w2 + 8w w

R � 800x � 0.25x2.

R

P � Prt.

6x3 � 8x2 � 9x � 12

3x3 � 4x2 � 3x � 4

81 � 16y4n4x2n � 25

n > 0.

7b3 � 563a3 � 192

5y3 � 6252x3 � 54

u4 � 256v4y4 � 81x4

a4 � 625b4 � 16

a3 � 16ax3 � 144x

3z2 � 19250x2 � 8

z3 � 125w3y3 � 64z3

a3 � 216b38x3 � y3

125x3 � 34327s3 � 64

64a3 � 18t3 � 27

z3 � 216y3 � 125

t3 � 27x3 � 8

100 � �y � 3�281 � �z � 5�2

�x � 3�2 � 4�a � 4�2 � 49

u2v2 � 25a2b2 � 16

y10 � 64a8 � 36

81a2 � b6x2 � 4y2

9z2 � 25w216y2 � 9z2

81 � z2121 � y2

625 � 49x2100 � 9y2

y2 � 144x2 � 25

u2 � uv � 4u � 4v

cd � 3c � 3d � 9

4u4 � 2u3 � 6u � 3

z4 � 3z3 � 2z � 6

3s3 � 6s2 � 5s � 10

a3 � 4a2 � 2a � 8

t3 � 11t2 � t � 11

x3 � 2x2 � x � 2

7t3 � 5t2 � 35t � 25

14z3 � 21z2 � 6z � 9

4x3 � 2x2 � 6x � 3

y3 � 6y2 � 2y � 12

�5x � y��x � y� � 5x�x � y�a�a � 6� � �2a � 5��a � 6�3a�a2 � 3� � 10�a2 � 3�5t�t2 � 1� � 4�t2 � 1�7t�s � 9� � 6�s � 9�2y�y � 3� � 5�y � 3�


117. Geometry The area of a rectangle of width isgiven by Factor this expression todetermine the length of the rectangle. Draw a dia-gram to illustrate your answer.

118. Geometry The area of a rectangle of length is given by Factor this expression todetermine the width of the rectangle. Draw a dia-gram to illustrate your answer.

119. Geometry The surface area of a right circularcylinder is (see figure). Factorthe expression for the surface area.

120. Manufacturing A washer on the drive train of acar has an inside radius of r centimeters and an out-side radius of R centimeters (see figure). Find thearea of one of the flat surfaces of the washer andwrite the area in factored form.

121. Chemistry The rate of change of an autocatalyticchemical reaction is where Q is theamount of the original substance, x is the amount ofsubstance formed, and k is a constant of propor-tionality. Factor the expression for this rate ofchange.

Synthesis

122. Writing Explain the relationship between usingthe Distributive Property to multiply polynomialsand using the Distributive Property to factor alge-braic expressions.

123. Writing Explain what is meant when it is said thata polynomial is in factored form.

124. Writing Explain how the word factor can be usedas a noun and as a verb.

Writing In Exercises 125 and 126, explain how theproduct could be obtained mentally using the sample as amodel.

125.

126.

127. Exploration The cube shown in the figure isformed by four solids: I, II, III, and IV.

(a) Explain how you could determine each expres-sion for volume.

Volume

Entire cube

Solid I

Solid II

Solid III

Solid IV

(b) Add the volumes of solids I, II, and III. Factorthe result to show that the total volume can bewritten as

(c) Explain why the total volume of solids I, II, andIII can also be written as Then explainhow the figure can be used as a geometricmodel for the difference of two cubes factoringpattern.

I

II

IIIIV

a b

a b−

a b−b

a

b

a b−

a

a3 � b3.

�a � b��a2 � ab � b2�.

b3

�a � b�b2

�a � b�ab

�a � b�a2

a3

18 � 22 � 396

79 � 81 � 6399

48 � 52 � �50 � 2��50 � 2� � 502 � 22 � 2496

kQx � kx2,

r

R

h

r

S � 2�r2 � 2�rhS

3y2 � 15y.y � 5

10x3 � 4x2.2x2

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Factoring Polynomials - · PDF fileSection P.5 Factoring Polynomials 53 Factoring by ... Factoring the Sum or Difference of Two Cubes The last type of special factoring