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E. The Product of Two Polynomials

E. The Product of Two Polynomials

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19



The F-O-I-L Method

By observing the product of two binomials in Example 13(a), we note a pattern that can

make the process more efficient. The product of two binomials can quickly be computed

using the First, Outer, Inner, Last (FOIL) method, an acronym giving the respective

position of each term in a product of binomials in relation to the other terms. We illustrate here using the product 12x Ϫ 12 13x ϩ 22.



WORTHY OF NOTE



Consider the product 1x ϩ 321x ϩ 22

in the context of area. If we view

x ϩ 3 as the length of a rectangle

(an unknown length plus 3 units),

and x ϩ 2 as its width (the same

unknown length plus 2 units), a

diagram of the total area would look

like the following, with the result

x2 ϩ 5x ϩ 6 clearly visible.



Combine like terms

6x2 ϩ x Ϫ 2



Inner

Outer



The first term of the result will always be the product of the first terms from each

binomial, and the last term of the result is the product of their last terms. We also note

that here, the middle term is found by adding the outermost product with the innermost

product. As you practice with the F-O-I-L process, much of the work can be done mentally and you can often compute the entire product without writing anything down

except the answer.





Multiplying Binomials Using F-O-I-L

Compute each product mentally:

a. 15n Ϫ 121n ϩ 22

b. 12b ϩ 32 15b Ϫ 62

a. 15n Ϫ 12 1n ϩ 22:



5n2 ϩ 9n Ϫ 2



product of

first two terms



S







S



Solution



10n ϩ (Ϫ1n) ϭ 9n



S



(x ϩ 3)(x ϩ 2) ϭ x2 ϩ 5x ϩ 6



EXAMPLE 14



S



6



S



2x



S



2



S



3x



S



x2



S



x



First Outer Inner Last



12x Ϫ 12 13x ϩ 22

S



3



6x2 ϩ 4x Ϫ 3x Ϫ 2



Last

First

S



x



The F-O-I-L Method for Multiplying Binomials



sum of

outer and inner

products



product of

last two terms



Ϫ12b ϩ 15b ϭ 3b



product of

first two terms



S



S



S



b. 12b ϩ 3215b Ϫ 62: 10b2 ϩ 3b Ϫ 18

sum of

outer and inner

products



E. You’ve just seen how

we can compute the product

of two polynomials



product of

last two terms



Now try Exercises 101 through 116 ᮣ



F. Special Polynomial Products

Certain polynomial products are considered “special” for two reasons: (1) the product

follows a predictable pattern, and (2) the result can be used to simplify expressions,

graph functions, solve equations, and/or develop other skills.



Binomial Conjugates

Expressions like x ϩ 7 and x Ϫ 7 are called binomial conjugates. For any given

binomial, its conjugate is found by using the same two terms with the opposite sign



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between them. Example 15 shows that when we multiply a binomial and its conjugate,

the “outers” and “inners” sum to zero and the result is a difference of two squares.

EXAMPLE 15







Multiplying Binomial Conjugates

Compute each product mentally:

a. 1x ϩ 721x Ϫ 72

b. 12x Ϫ 5y2 12x ϩ 5y2



2

2

c. ax ϩ b ax Ϫ b

5

5



Ϫ7x ϩ 7x ϭ 0x



Solution







a. 1x ϩ 72 1x Ϫ 72 ϭ x2 Ϫ 49



difference of squares 1x2 2 Ϫ 172 2



10xy ϩ (Ϫ10xy) ϭ 0xy



b. 12x Ϫ 5y212x ϩ 5y2 ϭ 4x2 Ϫ 25y2



difference of squares: 12x2 2 Ϫ 15y2 2



Ϫ 52 x ϩ 25 x ϭ 0



2

2

4

c. ax ϩ b ax Ϫ b ϭ x2 Ϫ

5

5

25



2 2

difference of squares: x2 Ϫ a b

5



Now try Exercises 117 through 124 ᮣ

In summary, we have the following.

The Product of a Binomial and Its Conjugate

Given any expression that can be written in the form A ϩ B, the conjugate of the

expression is A Ϫ B and their product is a difference of two squares:

1A ϩ B2 1A Ϫ B2 ϭ A2 Ϫ B2



Binomial Squares



Expressions like 1x ϩ 72 2 are called binomial squares and are useful for solving many

equations and sketching a number of basic graphs. Note 1x ϩ 72 2 ϭ 1x ϩ 721x ϩ 72 ϭ

x2 ϩ 14x ϩ 49 using the F-O-I-L process. The expression x2 ϩ 14x ϩ 49 is called a

perfect square trinomial because it is the result of expanding a binomial square. If we

write a binomial square in the more general form 1A ϩ B2 2 ϭ 1A ϩ B21A ϩ B2 and

compute the product, we notice a pattern that helps us write the expanded form

more quickly.

LOOKING AHEAD

Although a binomial square can

always be found using repeated

factors and F-O-I-L, learning to

expand them using the pattern is

a valuable skill. Binomial squares

occur often in a study of algebra

and it helps to find the expanded

form quickly.



1A ϩ B2 2 ϭ 1A ϩ B2 1A ϩ B2



repeated multiplication



ϭ A ϩ AB ϩ AB ϩ B



F-O-I-L



ϭ A ϩ 2AB ϩ B



simplify (perfect square trinomial)



2

2



2



2



The first and last terms of the trinomial are squares of the terms A and B. Also, the

middle term of the trinomial is twice the product of these two terms: AB ϩ AB ϭ 2AB.

The F-O-I-L process shows us why. Since the outer and inner products are identical,

we always end up with two. A similar result holds for 1A Ϫ B2 2 and the process can be

summarized for both cases using the Ϯ symbol.



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21



The Square of a Binomial



Given any expression that can be written in the form 1A Ϯ B2 2,

1. 1A ϩ B2 2 ϭ A2 ϩ 2AB ϩ B2

2. 1A Ϫ B2 2 ϭ A2 Ϫ 2AB ϩ B2





CAUTION



EXAMPLE 16







Solution







Note the square of a binomial always results in a trinomial (three terms). In particular,

1A ϩ B2 2 A2 ϩ B2.



Find each binomial square without using F-O-I-L:

a. 1a ϩ 92 2



F. You’ve just seen how

we can compute special

products: binomial conjugates

and binomial squares



b. 13x Ϫ 52 2



a. 1a ϩ 92 2 ϭ a2 ϩ 21a # 92 ϩ 92

ϭ a2 ϩ 18a ϩ 81

b. 13x Ϫ 52 2 ϭ 13x2 2 Ϫ 213x # 52 ϩ 52

ϭ 9x2 Ϫ 30x ϩ 25

c. 13 ϩ 1x2 2 ϭ 9 ϩ 213 # 1x2 ϩ 1 1x2 2

ϭ 9 ϩ 6 1x ϩ x



c. 13 ϩ 1x2 2



1A ϩ B2 2 ϭ A 2 ϩ 2AB ϩ B 2



simplify

1A Ϫ B2 2 ϭ A 2 Ϫ 2AB ϩ B 2

simplify

1A ϩ B2 2 ϭ A 2 ϩ 2AB ϩ B 2

simplify



Now try Exercises 125 through 136 ᮣ

With practice, you will be able to go directly from the binomial square to the

resulting trinomial.



R.2 EXERCISES





CONCEPTS AND VOCABULARY



Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.



1. The equation 1x2 2 3 ϭ x6 is an example of the

property of exponents.

2. The equation 1x3 2 Ϫ2 ϭ

property of



1

is an example of the

x6

exponents.



3. The sum of the “outers” and “inners” for 12x ϩ 52 2

is

, while the sum of the outers and inners

for 12x ϩ 52 12x Ϫ 52 is

.





4. The expression 2x2 Ϫ 3x Ϫ 10 can be classified as

a

of degree

, with a leading

coefficient of

.

5. Discuss/Explain why one of the following

expressions can be simplified further, while the

other cannot: (a) Ϫ7n4 ϩ 3n2; (b) Ϫ7n4 # 3n2.

6. Discuss/Explain why the degree of 2x2y3 is greater

than the degree of 2x 2 ϩ y 3. Include additional

examples for contrast and comparison.



DEVELOPING YOUR SKILLS



Determine each product using the product and/or power properties.



7.



2 2#

n 21n5

3



10. 1Ϫ1.5vy2 2 1Ϫ8v4y2



8. 24g5



# 3 g9

8



11. 1a2 2 4 # 1a3 2 2 # b2 # b5



9. 1Ϫ6p2q2 12p3q3 2

12. d 2 # d 4 # 1c5 2 2 # 1c3 2 2



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Simplify using the product to a power property.



13. 16pq2 2 3



14. 1Ϫ3p2q2 2



15. 13.2hk2 2 3

17. a



16. 1Ϫ2.5h5k2 2



p 2

b

2q



18. a



19. 1Ϫ0.7c 2 110c d 2

4 2



21.



R–22



CHAPTER R A Review of Basic Concepts and Skills



1 34x3y2 2



3 2 2



b 3

b

3a



5m2n3 2

b

2r4



45. a



5p2q3r4



2 2 3



22. 1 45x3 2 2



24. 1 23m2n2 2 # 1 12mn2 2



47.



25. Volume of a cube: The

3x2

formula for the volume of a

cube is V ϭ S3, where S is the

length of one edge. If the

length of each edge is 3x2,

3x2

a. Find a formula for volume

in terms of the variable x.

b. Find the volume of the cube if x ϭ 2.



49.



23. 1Ϫ38x2 2 116xy2 2



26. Area of a circle: The formula

for the area of a circle is

A ϭ ␲r2, where r is the length

of the radius. If the radius is

given as 5x3,

a. Find a formula for area in

terms of the variable x.

b. Find the area of the circle if x ϭ 2.



3x2



5x3



Simplify using the quotient property or the property of

negative exponents. Write answers using positive

exponents only.



27.



Ϫ6w

Ϫ2w2



28.



8z

16z5



29.



Ϫ12a3b5

4a2b4



30.



5m3n5

10mn2



5



7



31. 1 23 2 Ϫ3

33.



2

hϪ3



32. 1 56 2 Ϫ1

34.



3

mϪ2



35. 1Ϫ22 Ϫ3



36. 1Ϫ42 Ϫ2



37.



38.



Ϫ3

1 Ϫ1

2 2



Ϫ2

1 Ϫ2

3 2



Simplify each expression using the quotient to a power

property.



39. a



2p4



41. a



0.2x2 3

b

0.3y3



q3



b



2



40. a



Ϫ5v4 2

b

7w3



42. a



Ϫ0.5a3 2

b

0.4b2



44. a

2



b

Ϫ2pq2r4



46. a



4p3

3x2y



b



3



9p3q2r3

12p5qr



3



b

2



Use properties of exponents to simplify the following.

Write the answer using positive exponents only.



20. 1Ϫ2.5a 2 13a b 2

3 2



43. a



51.



9p6q4

Ϫ12p4q6

20hϪ2

12h5

1a2 2 3

a4 # a5



aϪ3 # b Ϫ4

b

cϪ2

Ϫ612xϪ3 2 2



48.

50.

52.



53. a



54.



55.



56.



57.



10xϪ2



14aϪ3bc0

Ϫ713a2bϪ2c2 3



59. 40 ϩ 50

61. 2Ϫ1 ϩ 5Ϫ1

63. 30 ϩ 3Ϫ1 ϩ 3Ϫ2

65. Ϫ5x0 ϩ 1Ϫ5x2 0



58.



5m5n2

10m5n

5k3

20kϪ2

153 2 4



59

1pϪ4q8 2 2

p5qϪ2



18nϪ3

Ϫ813nϪ2 2 3

Ϫ312x3yϪ4z2 2

18xϪ2yz0



60. 1Ϫ32 0 ϩ 1Ϫ72 0

62. 4Ϫ1 ϩ 8Ϫ1



64. 2Ϫ2 ϩ 2Ϫ1 ϩ 20

66. Ϫ2n0 ϩ 1Ϫ2n2 0



Convert the following numbers to scientific notation.



67. In mid-2009, the U.S. Census Bureau estimated the

world population at nearly 6,770,000,000 people.

68. The mass of a proton is generally given as

0.000 000 000 000 000 000 000 000 001 670 kg.

Convert the following numbers to decimal notation.



69. The smallest microprocessors in common use

measure 6.5 ϫ 10Ϫ9 m across.

70. In 2009, the estimated net worth of Bill Gates, the

founder of Microsoft, was 5.8 ϫ 1010 dollars.

Compute using scientific notation. Show all work.



71. The average distance between the Earth and the

planet Jupiter is 465,000,000 mi. How many hours

would it take a satellite to reach the planet if it

traveled an average speed of 17,500 mi per hour?

How many days? Round to the nearest whole.

72. In fiscal terms, a nation’s debt-per-capita is the

ratio of its total debt to its total population. In the

year 2009, the total U.S. debt was estimated at

$11,300,000,000,000, while the population was

estimated at 305,000,000. What was the U.S. debtper-capita ratio for 2009? Round to the nearest

whole dollar.



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Section R.2 Exponents, Scientific Notation, and a Review of Polynomials



Identify each expression as a polynomial or

nonpolynomial (if a nonpolynomial, state why);

classify each as a monomial, binomial, trinomial, or

none of these; and state the degree of the polynomial.



73. Ϫ35w3 ϩ 2w2 ϩ 1Ϫ12w2 ϩ 14

74. Ϫ2x3 ϩ 23x2 Ϫ 12x ϩ 1.2



4

75. 5nϪ2 ϩ 4n ϩ 117 76. 3 ϩ 2.7r2 ϩ r ϩ 1

r

2

3

77. p Ϫ 5

78. q3 ϩ 2qϪ2 Ϫ 5q

Write each polynomial in standard form and name the

leading coefficient.



79. 7w ϩ 8.2 Ϫ w3 Ϫ 3w2

80. Ϫ2k2 Ϫ 12 Ϫ k

81. c3 ϩ 6 ϩ 2c2 Ϫ 3c



82. Ϫ3v3 ϩ 14 ϩ 2v2 ϩ 1Ϫ12v2

83. 12 Ϫ 23x2



84. 8 ϩ 2n ϩ 7n

2



96. 1s Ϫ 32 15s ϩ 42



97. 1x Ϫ 321x2 ϩ 3x ϩ 92



98. 1z ϩ 52 1z2 Ϫ 5z ϩ 252



99. 1b2 Ϫ 3b Ϫ 282 1b ϩ 22



100. 12h2 Ϫ 3h ϩ 82 1h Ϫ 12

101. 17v Ϫ 4213v Ϫ 52

103. 13 Ϫ m213 ϩ m2



102. 16w Ϫ 1212w ϩ 52

104. 15 ϩ n215 Ϫ n2



105. 1p Ϫ 2.521p ϩ 3.62 106. 1q Ϫ 4.921q ϩ 1.22

107. 1x ϩ 12 2 1x ϩ 14 2



109. 1m ϩ 34 2 1m Ϫ 34 2



108. 1z ϩ 13 21z ϩ 56 2



110. 1n Ϫ 25 21n ϩ 25 2



111. 13x Ϫ 2y212x ϩ 5y2 112. 16a ϩ b21a ϩ 3b2



113. 14c ϩ d213c ϩ 5d2 114. 15x ϩ 3y212x Ϫ 3y2

115. 12x2 ϩ 52 1x2 Ϫ 32 116. 13y2 Ϫ 2212y2 ϩ 12



For each binomial, determine its conjugate and find the

product of the binomial with its conjugate.



117. 4m Ϫ 3



118. 6n ϩ 5



85. 13p3 Ϫ 4p2 ϩ 2p Ϫ 72 ϩ 1p2 Ϫ 2p Ϫ 52



119. 7x Ϫ 10



120. c ϩ 3



121. 6 ϩ 5k



122. 11 Ϫ 3r



87. 15.75b2 ϩ 2.6b Ϫ 1.92 ϩ 12.1b2 Ϫ 3.2b2



123. x ϩ 16



124. p Ϫ 12



Find the indicated sum or difference.



86. 15q2 Ϫ 3q ϩ 42 ϩ 1Ϫ3q2 ϩ 3q Ϫ 42



88. 10.4n2 ϩ 5n Ϫ 0.52 ϩ 10.3n2 Ϫ 2n ϩ 0.752



Find each binomial square.



90. 1 59n2 ϩ 4n Ϫ 12 2 Ϫ 1 23n2 Ϫ 2n ϩ 34 2



127. 14g ϩ 32 2



89. 1 34x2 Ϫ 5x ϩ 22 Ϫ 1 12x2 ϩ 3x Ϫ 42



91. Subtract q5 ϩ 2q4 ϩ q2 ϩ 2q from q6 ϩ 2q5 ϩ

q4 ϩ 2q3 using a vertical format.

92. Find x4 ϩ 2x3 ϩ x2 ϩ 2x decreased by

x4 Ϫ 3x3 ϩ 4x2 Ϫ 3x using a vertical format.

Compute each product.



93. Ϫ3x1x2 Ϫ x Ϫ 62



94. Ϫ2v2 1v2 ϩ 2v Ϫ 152

95. 13r Ϫ 52 1r Ϫ 22





23



125. 1x ϩ 42 2



126. 1a Ϫ 32 2



129. 14p Ϫ 3q2 2



130. 15c ϩ 6d2 2



131. 14 Ϫ 1x2 2



128. 15x Ϫ 32 2



132. 1 1x ϩ 72 2



Compute each product.



133. 1x Ϫ 321y ϩ 22



134. 1a ϩ 321b Ϫ 52



135. 1k Ϫ 521k ϩ 621k ϩ 22



136. 1a ϩ 621a Ϫ 121a ϩ 52



WORKING WITH FORMULAS



137. Medication in the bloodstream: M ‫ ؍‬0.5t 4 ؉ 3t 3 ؊ 97t 2 ؉ 348t

If 400 mg of a pain medication are taken orally, the number of milligrams in the bloodstream is modeled by the

formula shown, where M is the number of milligrams and t is the time in hours, 0 Յ t 6 5. Construct a table of

values for t ϭ 1 through 5, then answer the following.

a. How many milligrams are in the bloodstream after 2 hr? After 3 hr?

b. Based on part a, would you expect the number of milligrams in the bloodstream after 4 hr to be less or more? Why?

c. Approximately how many hours until the medication wears off (the number of milligrams in the bloodstream is 0)?



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Aa

138. Amount of a mortgage payment: M ‫؍‬



r n

r

b a1 ؉ b

12

12



r n

b ؊1

12

The monthly mortgage payment required to pay off (or amortize) a loan is given by the formula shown, where M is

the monthly payment, A is the original amount of the loan, r is the annual interest rate, and n is the term of the loan

in months. Find the monthly payment (to the nearest cent) required to purchase a $198,000 home, if the interest rate

is 6.5% and the home is financed over 30 yr.







a1 ؉



APPLICATIONS



139. Attraction between particles: In electrical theory,

the force of attraction between two particles P and

kPQ

Q with opposite charges is modeled by F ϭ 2 ,

d

where d is the distance between them and k is a

constant that depends on certain conditions. This is

known as Coulomb’s law. Rewrite the formula

using a negative exponent.

140. Intensity of light: The intensity of illumination

from a light source depends on the distance from

k

the source according to I ϭ 2 , where I is the

d

intensity measured in footcandles, d is the distance

from the source in feet, and k is a constant that

depends on the conditions. Rewrite the formula

using a negative exponent.

141. Rewriting an expression: In advanced mathematics,

negative exponents are widely used because they are

easier to work with than rational expressions.

5

3

2

Rewrite the expression 3 ϩ 2 ϩ 1 ϩ 4 using

x

x

x

negative exponents.

142. Swimming pool hours: A swimming pool opens at

8 A.M. and closes at 6 P.M. In summertime, the







number of people in the pool at any time can be

approximated by the formula S1t2 ϭ Ϫt 2 ϩ 10t,

where S is the number of swimmers and t is the

number of hours the pool has been open (8 A.M.:

t ϭ 0, 9 A.M.: t ϭ 1, 10 A.M.: t ϭ 2, etc.).

a. How many swimmers are in the pool at 6 P.M.?

Why?

b. Between what times would you expect the

largest number of swimmers?

c. Approximately how many swimmers are in the

pool at 3 P.M.?

d. Create a table of values for t ϭ 1, 2, 3, 4, . . .

and check your answer to part b.

143. Maximizing revenue: A sporting goods store finds

that if they price their video games at $20, they

make 200 sales per day. For each decrease of $1,

20 additional video games are sold. This means the

store’s revenue can be modeled by the formula

R ϭ 120 Ϫ 1x2 1200 ϩ 20x2, where x is the number

of $1 decreases. Multiply out the binomials and use

a table of values to determine what price will give

the most revenue.

144. Maximizing revenue: Due to past experience, a

jeweler knows that if they price jade rings at $60,

they will sell 120 each day. For each decrease of

$2, five additional sales will be made. This means

the jeweler’s revenue can be modeled by the

formula R ϭ 160 Ϫ 2x2 1120 ϩ 5x2, where x is the

number of $2 decreases. Multiply out the

binomials and use a table of values to determine

what price will give the most revenue.



EXTENDING THE CONCEPT



145. If 13x2 ϩ kx ϩ 12 Ϫ 1kx2 ϩ 5x Ϫ 72 ϩ

12x2 Ϫ 4x Ϫ k2 ϭ Ϫx2 Ϫ 3x ϩ 2, what is the

value of k?



1 2

1

b ϭ 5, then the expression 4x2 ϩ 2

2x

4x

is equal to what number?



146. If a2x ϩ



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R.3



Solving Linear Equations and Inequalities



LEARNING OBJECTIVES



In a study of algebra, you will encounter many families of equations, or groups of

equations that share common characteristics. Of interest to us here is the family of

linear equations in one variable, a study that lays the foundation for understanding

more advanced families. This section will also lay the foundation for solving a formula for

a specified variable, a practice widely used in science, business, industry, and research.



In Section R.3 you will review how to:



A. Solve linear equations



B.



C.

D.

E.



F.



using properties of

equality

Recognize equations that

are identities or

contradictions

Solve linear inequalities

Solve compound

inequalities

Solve basic applications

of linear equations and

inequalities

Solve applications of

basic geometry



CAUTION



A. Solving Linear Equations Using Properties of Equality

An equation is a statement that two expressions are

equal. From the expressions 31x Ϫ 12 ϩ x and

Ϫx ϩ 7, we can form the equation

31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7,



Table R.1

x



31x ؊ 12 ؉ x



؊x ؉ 7



Ϫ2



Ϫ11



9



Ϫ1



Ϫ7



8



which is a linear equation in one variable (the

0

exponent on any variable is a 1). To solve an equa1

tion, we attempt to find a specific input or x-value

2

that will make the equation true, meaning the left3

hand expression will be equal to the right. Using

4

Table R.1, we find that 31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7 is a

true equation when x is replaced by 2, and is a false

equation otherwise. Replacement values that make

the equation true are called solutions or roots of the equation.





Ϫ3



7



1



6



5



5



9



4



13



3



From Section R.1, an algebraic expression is a sum or difference of algebraic terms.

Algebraic expressions can be simplified, evaluated or written in an equivalent form, but

cannot be “solved,” since we’re not seeking a specific value of the unknown.



Solving equations using a table is too time consuming to be practical. Instead we

attempt to write a sequence of equivalent equations, each one simpler than the one before, until we reach a point where the solution is obvious. Equivalent equations are those

that have the same solution set, and can be obtained by using the distributive property to

simplify the expressions on each side of the equation. The additive and multiplicative

properties of equality are then used to obtain an equation of the form x ϭ constant.

The Additive Property of Equality



The Multiplicative Property of Equality



If A, B, and C represent algebraic

expressions and A ϭ B,



If A, B, and C represent algebraic

expressions and A ϭ B,



then A ϩ C ϭ B ϩ C



then AC ϭ BC and



B

A

ϭ , 1C

C

C



02



In words, the additive property says that like quantities, numbers, or terms can be

added to both sides of an equation. A similar statement can be made for the multiplicative property. These properties are combined into a general guide for solving linear

equations, which you’ve likely encountered in your previous studies. Note that not all

steps in the guide are required to solve every equation.



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CHAPTER R A Review of Basic Concepts and Skills



Guide to Solving Linear Equations in One Variable

• Eliminate parentheses using the distributive property, then combine any like terms.

• Use the additive property of equality to write the equation with all variable terms

on one side, and all constants on the other. Simplify each side.

• Use the multiplicative property of equality to obtain an equation of the form

x ϭ constant.

• For applications, answer in a complete sentence and include any units of measure

indicated.

For our first example, we’ll use the equation 31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7 from our

initial discussion.



EXAMPLE 1







Solving a Linear Equation Using Properties of Equality

Solve for x: 31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7.



Solution







31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7

3x Ϫ 3 ϩ x ϭ Ϫx ϩ 7

4x Ϫ 3 ϭ Ϫx ϩ 7

5x Ϫ 3 ϭ 7

5x ϭ 10

xϭ2



original equation

distributive property

combine like terms

add x to both sides (additive property of equality)

add 3 to both sides (additive property of equality)

multiply both sides by 15 or divide both sides by 5

(multiplicative property of equality)



As we noted in Table R.1, the solution is x ϭ 2.

Now try Exercises 7 through 12







To check a solution by substitution means we substitute the solution back into the

original equation (this is sometimes called back-substitution), and verify the lefthand side is equal to the right. For Example 1 we have:

31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7



original equation



312 Ϫ 12 ϩ 2 ϭ Ϫ2 ϩ 7



substitute 2 for x



3112 ϩ 2 ϭ 5

5 ϭ 5✓



simplify

solution checks



If any coefficients in an equation are fractional, multiply both sides by the least

common denominator (LCD) to clear the fractions. Since any decimal number can be

written in fraction form, the same idea can be applied to decimal coefficients.



EXAMPLE 2







Solution







A. You’ve just seen how

we can solve linear equations

using properties of equality



Solving a Linear Equation with Fractional Coefficients

Solve for n: 14 1n ϩ 82 Ϫ 2 ϭ 12 1n Ϫ 62.

1

4 1n ϩ 82

1

4n ϩ 2



Ϫ 2 ϭ 12 1n Ϫ 62

Ϫ 2 ϭ 12n Ϫ 3

1

1

4n ϭ 2n Ϫ 3

1

41 4n2 ϭ 41 12n Ϫ 32

n ϭ 2n Ϫ 12

Ϫn ϭ Ϫ12

n ϭ 12



original equation

distributive property

combine like terms

multiply both sides by LCD ϭ 4

distributive property

subtract 2n

multiply by Ϫ1



Verify the solution is n ϭ 12 using back-substitution.

Now try Exercises 13 through 30







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E. The Product of Two Polynomials

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