algebra & trigonometry graphs & models 3rd ed - marvin l. bittinger

R
ina wants to establish a college fund for
her newborn daughter that will have
accumulated $120,000 at the end of
18 yr. If she can count on an interest rate of 6%,
compounded monthly, how much should she deposit
each month to accomplish this?
This problem appears as Exercise 95 in Section R.2.
G
Basic Concepts
of Algebra
R.1 The Real-Number System
R.2 Integer Exponents, Scientific Notation,
and Order of Operations
R.3 Addition, Subtraction, and
Multiplication of Polynomials
R.4 Factoring
R.5 Rational Expressions
R.6 Radical Notation and Rational Exponents
R.7 The Basics of Equation Solving
SUMMARY AND REVIEW
TEST
APPLICATION
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 1
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
2.1
Polynomial
Functions and
Modeling
2 Chapter R • Basic Concepts of Algebra
R.1

The Real-Number
System
Identify various kinds of real numbers.
Use interval notation to write a set of numbers.
Identify the properties of real numbers.
Find the absolute value of a real number.
Real Numbers
In applications of algebraic concepts, we use real numbers to represent
quantities such as distance, time, speed, area, profit, loss, and tempera-
ture. Some frequently used sets of real numbers and the relationships
among them are shown below.
Real
numbers
Rational
numbers
Negative integers:
−1, −2, −3, …
Natural numbers
(positive integers):
1, 2, 3, …
Zero: 0
−, − −, −−, −−, 8.3,
0.56, …
2
3
4
5
19
−5
−7

8
−
Whole numbers:
0, 1, 2, 3, …
Rational numbers
that are not integers:
Integers:
…, −3, −2, −1, 0,
1, 2, 3, …
Irrational numbers:
−4.030030003…, …
√2, p, −√3, √27,
54
Numbers that can be expressed in the form , where p and q are in-
tegers and , are rational numbers.Decimal notation for rational
numbers either terminates (ends) or repeats.Each of the following is a
rational number.
a) 0
for any nonzero integer a
b) Ϫ7 ,or
c) Te r minating decimal
d) Repeating decimalϪ
5
11
෇ Ϫ0.45
1
4
෇ 0.25
7
؊1

؊7 ؍
؊7
1
0 ؍
0
a
q  0
p͞q
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
The real numbers that are not rational are irrational numbers.Decimal
notation for irrational numbers neither terminates nor repeats. Each of the
following is an irrational number.
a)
There is no repeating block of digits.
and 3.14 are rational approximations of the irrational number
b)
There is no repeating block of digits.
c) Although there is a pattern, there is no
repeating block of digits.
The set of all rational numbers combined with the set of all irrational
numbers gives us the set of real numbers.The real numbers are modeled
using a number line, as shown below.
Each point on the line represents a real number, and every real number
is represented by a point on the line.
The order of the real numbers can be determined from the number
line. If a number a is to the left of a number b, then a is less than b
.Similarly, a is greater than b if a is to the right of b on
the number line. For example, we see from the number line above that
,because Ϫ2.9 is to the left of . Also, , because

is to the right of .
The statement , read “a is less than or equal to b,” is true if either
is true or is true.
The symbol ʦ is used to indicate that a member, or element,belongs to
a set. Thus if we let represent the set of rational numbers, we can see from
the diagram on page 2 that . We can also write to indi-
cate that is not an element of the set of rational numbers.
When all the elements of one set are elements of a second set, we say that
the first set is a subset of the second set. The symbol
ʕ
is used to denote this.
For instance, if we let represent the set of real numbers, we can see from
the diagram that (read “ is a subset of ”).
Interval Notation
Sets of real numbers can be expressed using interval notation.For example,
for real numbers a and b such that , the open interval is the set
of real numbers between, but not including, a and b.That is,
.
The points a and b are endpoints of the interval. The parentheses indicate
that the endpoints are not included in the interval.
Some intervals extend without bound in one or both directions. The
interval , for example, begins at a and extends to the right without
bound. That is,
.
The bracket indicates that a is included in the interval.
͓a,
ϱ͒ ෇ ͕x ͉x Ն a͖
͓a,
ϱ͒
͑a, b͒ ෇ ͕x ͉a Ͻ x Ͻ b͖

͑a, b͒a Ͻ b
ޒޑޑ
ʕ
ޒ
ޒ
͙
2
͙
2  ޑ0.56 ʦ ޑ
ޑ
a ෇ ba Ͻ b
a Յ b
͙
3
17
4
17
4
Ͼ
͙
3Ϫ
3
5
Ϫ2.9 ϽϪ
3
5
͑a Ͼ b͒͑a Ͻ b͒
Ϫ2.9 ϪE ͙3 p *
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 123450
Ϫ6.12122122212222

͙
2
෇ 1.414213562
␲
.
͒͑
22
7
␲
෇ 3.1415926535
Section R.1 • The Real-Number System 3
(
)
ab
(a, b)
[
a
[a, ∞)
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 3
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
The various types of intervals are listed below.
4 Chapter R • Basic Concepts of Algebra
The interval , graphed below, names the set of all real num-
bers, .
EXAMPLE 1 Write interval notation for each set and graph the set.
a) b)
c) d)
Solution
a) ;
b) ;

c) ;
01Ϫ4Ϫ5 Ϫ2 Ϫ1Ϫ3 2345
͕x͉Ϫ5 Ͻ x ՅϪ2͖ ෇ ͑Ϫ5,Ϫ2͔
01Ϫ4Ϫ5 Ϫ2 Ϫ1Ϫ3 2345
͕x͉x Ն 1.7͖ ෇ ͓1.7,ϱ͒
01Ϫ4Ϫ5 Ϫ2 Ϫ1Ϫ3 2345
͕x͉Ϫ4 Ͻ x Ͻ 5͖ ෇ ͑Ϫ4, 5͒
͕
x ͉ x Ͻ
͙
5
͖
͕x͉Ϫ5 Ͻ x ՅϪ2͖
͕x͉x Ն 1.7͖͕x͉Ϫ4 Ͻ x Ͻ 5͖
ޒ
͑Ϫ
ϱ, ϱ͒
Intervals: Types, Notation, and Graphs
INTERVAL SET
TYPE NOTATION NOTATION GRAPH
Open
Closed
Half-open
Half-open
Open
Half-open
Open
Half-open
]
b

͕x ͉ x Յ b͖͑Ϫϱ, b͔
)
b
͕x ͉ x Ͻ b͖͑Ϫϱ, b͒
[
a
͕x ͉ x Ն a͖͓a, ϱ͒
(
a
͕x ͉ x Ͼ a͖͑a, ϱ͒
(
]
ab
͕x ͉ a Ͻ x Յ b͖͑a, b͔
[
)
ab
͕x ͉ a Յ x Ͻ b͖͓a, b͒
[
]
ab
͕x ͉ a Յ x Յ b͖͓a, b͔
(
)
ab
͕x ͉ a Ͻ x Ͻ b͖͑a, b͒
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 4
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.1 • The Real-Number System 5
d) ;

Properties of the Real Numbers
The following properties can be used to manipulate algebraic expressions as
well as real numbers.
Properties of the Real Numbers
For any real numbers a, b, and c:
and Commutative properties of
addition and multiplication
and Associative properties of
addition and multiplication
Additive identity property
Additive inverse property
Multiplicative identity property
Multiplicative inverse property
Distributive property
Note that the distributive property is also true for subtraction since
.
EXAMPLE 2 State the property being illustrated in each sentence.
a) b)
c) d)
e)
Solution
SENTENCE PROPERTY
a) Commutative property of multiplication:
b) Associative property of addition:
c) Additive inverse property:
d) Multiplicative identity property:
e) Distributive property:
a͑b ϩ c͒ ෇ ab ϩ ac
2͑a Ϫ b͒ ෇ 2a Ϫ 2b
a и 1 ෇ 1 и a ෇ a

6 и 1 ෇ 1 и 6 ෇ 6
a ϩ ͑Ϫa͒ ෇ 014 ϩ ͑Ϫ14͒ ෇ 0
a ϩ ͑b ϩ c͒ ෇ ͑a ϩ b͒ ϩ c
5 ϩ ͑m ϩ n͒ ෇ ͑5 ϩ m͒ ϩ n
ab ෇ ba
8 и 5 ෇ 5 и 8
2͑a Ϫ b͒ ෇ 2a Ϫ 2b
6 и 1 ෇ 1 и 6 ෇ 614 ϩ ͑Ϫ14͒ ෇ 0
5 ϩ ͑m ϩ n͒ ෇ ͑5 ϩ m͒ ϩ n8 и 5 ෇ 5 и 8
a͑b Ϫ c͒ ෇ a͓b ϩ ͑Ϫc͔͒ ෇ ab ϩ a͑Ϫc͒ ෇ ab Ϫ ac
a͑b ϩ c͒ ෇ ab ϩ ac
͑a  0͒a и
1
a
෇
1
a
и a ෇ 1
a и 1 ෇ 1 и a ෇ a
Ϫa ϩ a ෇ a ϩ ͑Ϫa͒ ෇ 0
a ϩ 0 ෇ 0 ϩ a ෇ a
a͑bc͒ ෇ ͑ab͒c
a ϩ ͑b ϩ c͒ ෇ ͑a ϩ b͒ ϩ c
ab ෇ ba
a ϩ b ෇ b ϩ a
01Ϫ4Ϫ5 Ϫ2 Ϫ1Ϫ32345
͕
x ͉ x Ͻ
͙
5

͖
෇
͑
Ϫϱ,
͙
5
͒
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 5
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
6 Chapter R • Basic Concepts of Algebra
ab
͉a Ϫ b͉ ϭ ͉b Ϫ a͉
Absolute Value
The number line can be used to provide a geometric interpretation of
absolute value.The absolute value of a number a,denoted , is its dis-
tance from 0 on the number line. For example, , because the
distance of Ϫ5 from 0 is 5. Similarly, , because the distance of
from 0 is .
Absolute Value
For any real number a,
When a is nonnegative, the absolute value of a is a.When a is negative,
the absolute value of a is the opposite, or additive inverse, of a.Thus,
is never negative; that is, for any real number a,.
Absolute value can be used to find the distance between two points on
the number line.
Distance Between Two Points on the Number Line
For any real numbers a and b, the distance between a and b is
,or equivalently, .
EXAMPLE 3 Find the distance between Ϫ2 and 3.
Solution The distance is

,or equivalently,
.
We can also use the absolute-value operation on a graphing calculator to
find the distance between two points. On many graphing calculators, ab-
solute value is denoted “abs” and is found in the
MATH NUM menu and also
in the
CATALOG.
5
abs (3Ϫ(
Ϫ
2))
5
abs (
Ϫ
2Ϫ3)
͉3 Ϫ ͑Ϫ2͉͒ ෇ ͉3 ϩ 2͉ ෇ ͉5͉ ෇ 5
͉Ϫ2 Ϫ 3͉ ෇ ͉Ϫ5͉ ෇ 5
͉b Ϫ a͉͉a Ϫ b͉
͉a͉ Ն 0͉a͉
͉a͉ ෇
ͭ
a,
Ϫa,
if a Ն 0,
if a Ͻ 0.
3
4
3
4

Խ
3
4
Խ
෇
3
4
͉Ϫ5͉ ෇ 5
͉a͉
GCM
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 6
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.1 • The Real-Number System 7
In Exercises 1– 10, consider the numbers Ϫ12,,,
,,0,,,,Ϫ1.96, 9,
,,,.
1. Which are whole numbers? ,0,9,
2. Which are integers?
Ϫ12, , 0,9,
3. Which are irrational numbers?
4. Which are natural numbers?
,9,
5. Which are rational numbers?
6. Which are real numbers? All of them
7. Which are rational numbers but not integers?
8. Which are integers but not whole numbers? Ϫ12
9. Which are integers but not natural numbers? Ϫ12, 0
10. Which are real numbers but not integers? Ճ
Write interval notation. Then graph the interval.
11. Ճ 12. Ճ

13. Ճ 14. Ճ
15. Ճ 16. Ճ
17. Ճ 18. Ճ
19. Ճ 20. Ճ
Write interval notation for the graph.
21.
22.
23.
24.
25.
26.
27.
28.
In Exercises 29 –46, the following notation is used:
the set of natural numbers, the set of whole
numbers, the set of integers, the set of
rational numbers, the set of irrational numbers, and
the set of real numbers. Classify the statement as
true or false.
29. Tr ue 30. Tr ue
31. False 32. Tr ue
33. Tr ue 34. False
35. False 36. False
37. False 38. Tr ue
39. True 40. Tr ue
41. Tr ue 42. False
43. True 44. Tr u e
45. False 46. False
Name the property illustrated by the sentence.
47. Commutative property of

multiplication
48. Associative property
of addition
49. 50. Ճ
Multiplicative identity property
51. Ճ 52.
Distributive property
4͑y Ϫ z͒ ෇ 4y Ϫ 4z5͑ab͒ ෇ ͑5a͒b
x ϩ 4 ෇ 4 ϩ xϪ3 и 1 ෇ Ϫ3
3 ϩ ͑x ϩ y͒ ෇ ͑3 ϩ x͒ ϩ y
6 и x ෇ x и 6
ޑ
ʕ
މޒ
ʕ
ޚ
ޚ
ʕ
ޑޑ
ʕ
ޒ
ޚ
ʕ
ގޗ
ʕ
ޚ
ގ
ʕ
ޗ1.089  މ
1 ʦ ޚ24  ޗ

Ϫ1 ʦ ޗ
͙
11  ޒ
Ϫ
͙
6 ʦ ޑϪ
11
5
ʦ ޑ
Ϫ10.1 ʦ ޒ3.2 ʦ ޚ
0  ގ6 ʦ ގ
ޒ ෇
މ ෇
ޑ ෇ޚ ෇
ޗ ෇ގ ෇
q
]
͑Ϫϱ, q͔
p
(
͑ p, ϱ͒
(
]
xx ϩ h
͑x, x ϩ h͔
[
]
xx ϩ h
͓x, x ϩ h͔
Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ3 Ϫ2 Ϫ1 012Ϫ4

(
]
͑Ϫ9, Ϫ5͔
Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ3 Ϫ2 Ϫ1 012Ϫ4
[
)
͓Ϫ9, Ϫ4͒
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 1234560
[
]
͓Ϫ1, 2͔
Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 1234560
(
)
͑0, 5͒
͕x ͉Ϫ3 Ͼ x͖͕x ͉7 Ͻ x͖
͕
x ͉x Ն
͙
3
͖
͕x ͉x Ͼ 3.8͖
͕x ͉x ϾϪ5͖͕x͉ x ՅϪ2͖
͕x ͉1 Ͻ x Յ 6͖͕x ͉Ϫ4 Յ x ϽϪ1͖
͕x ͉Ϫ4 Ͻ x Ͻ 4͖͕x ͉Ϫ3 Յ x Յ 3͖
͙
25
͙
3
8

͙
25
͙
3
8
͙
25
͙
3
8
5
7
͙
3
4
͙
254
2
3
͙
5
5Ϫ
͙
145.242242224 . . .
͙
3
8Ϫ
7
3
5.3

͙
7
Exercise Set
R.1
,,
,,͙
3
4
͙
5
5Ϫ
͙
14
5.242242224 . . .
͙
7
,,Ϫ1.96,
,
5
7
4
2
3
Ϫ
7
3
5.3
Ϫ12, , , , 0,
Ϫ1.96, 9, , ,
5

7
͙
254
2
3
͙
3
8Ϫ
7
3
5.3
Ճ Answers to Exercises 10–20, 50, and 51 can be found on p. IA-1.
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 7
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
53. 54.
Commutative property of multiplication
55. Commutative property
of addition
56. Additive identity property
57. Multiplicative inverse property
58. Distributive property
Simplify.
59. 7.1 60. 86.2
61. 347 62. 54
63. 64.
65.
0 66. 15
67. 68.
Find the distance between the given pair of points on
the number line.

69. Ϫ5, 6 11 70. Ϫ2.5, 0 2.5
71. Ϫ8, Ϫ2 6 72. ,
73. 6.7, 12.1 5.4 74. Ϫ14, Ϫ3 11
75. , 76. Ϫ3.4, 10.2 13.6
77. Ϫ7, 0 7 78. 3, 19 16
Collaborative Discussion and Writing
To the student and the instructor: The Collaborative
Discussion and Writing exercises are meant to be
answered with one or more sentences. These exercises
can also be discussed and answered collaboratively by
the entire class or by small groups. Because of their
open-ended nature, the answers to these exercises do
21
8
15
8
Ϫ
3
4
1
24
23
12
15
8
͙
3
Խ
Ϫ
͙

3
Խ
5
4
͉
5
4
͉
͉15͉͉0͉
12
19
͉
12
19
͉
͙
97
Խ
Ϫ
͙
97
Խ
͉Ϫ54͉͉347͉
͉Ϫ86.2͉͉Ϫ7.1͉
9x ϩ 9y ෇ 9͑x ϩ y͒
8 и
1
8
෇ 1
t ϩ 0 ෇ t

Ϫ6͑m ϩ n͒ ෇ Ϫ6͑n ϩ m͒
Ϫ7 ϩ 7 ෇ 02͑a ϩ b͒ ෇ ͑a ϩ b͒2
8 Chapter R • Basic Concepts of Algebra
Ճ Answer to Exercise 85 can be found on p. IA-1.
not appear at the back of the book. They are denoted
by the words “Discussion and Writing.”
79. How would you convince a classmate that division is
not associative?
80. Under what circumstances is a rational number?
Synthesis
To the student and the instructor: The Synthesis
exercises found at the end of every exercise set challenge
students to combine concepts or skills studied in that
section or in preceding parts of the text.
Between any two (different) real numbers there are
many other real numbers. Find each of the following.
Answers may vary.
81. An irrational number between 0.124 and 0.125
Answers may vary;
82. A rational number between and
Answers may vary; Ϫ1.415
83. A rational number between and
Answers may vary; Ϫ0.00999
84. An irrational number between and
Answers may vary;
85. The hypotenuse of an isosceles right triangle with
legs of length 1 unit can be used to “measure” a
value for by using the Pythagorean theorem,
as shown.
c

1
1
͙
2
͙
5.995
͙
6
͙
5.99
Ϫ
1
100
Ϫ
1
101
Ϫ
͙
2Ϫ
͙
2.01
0.124124412444 . . .
͙
a
c ෇
͙
2
c
2
෇ 2

c
2
෇ 1
2
ϩ 1
2
Draw a right triangle that could be used to
“measure” units. Ճ
͙
10
Additive inverse property
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 8
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2 • Integer Exponents, Scientific Notation, and Order of Operations 9
R.2
Integer
Exponents,
Scientific
Notation, and
Order of
Operations
Simplify expressions with integer exponents.
Solve problems using scientific notation.
Use the rules for order of operations.
Integers as Exponents
When a positive integer is used as an exponent, it indicates the number of
times a factor appears in a product. For example, means and
means 5.
For any positive integer n,
,

n factors
where a is the base and n is the exponent.
Zero and negative-integer exponents are defined as follows.
For any nonzero real number a and any integer m,
and .
EXAMPLE 1 Simplify each of the following.
a) b)
Solution
a) b)
EXAMPLE 2 Write each of the following with positive exponents.
a) b) c)
Solution
a)
b)
c)
x
Ϫ3
y
Ϫ8
෇ x
Ϫ3
и
1
y
Ϫ8
෇
1
x
3
и y

8
෇
y
8
x
3
1
͑0.82͒
Ϫ7
෇ ͑0.82͒
Ϫ͑Ϫ7͒
෇ ͑0.82͒
7
4
Ϫ5
෇
1
4
5
x
Ϫ3
y
Ϫ8
1
͑0.82͒
Ϫ7
4
Ϫ5
͑Ϫ3.4͒
0

෇ 16
0
෇ 1
͑Ϫ3.4͒
0
6
0
a
Ϫm
෇
1
a
m
a
0
෇ 1
a
n
෇ a и a и a иииa
5
1
7 и 7 и 77
3








BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 9
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
10 Chapter R • Basic Concepts of Algebra
The results in Example 2 can be generalized as follows.
For any nonzero numbers a and b and any integers m and n,
.
(A factor can be moved to the other side of the fraction bar if the
sign of the exponent is changed.)
EXAMPLE 3 Write an equivalent expression without negative exponents:
.
Solution Since each exponent is negative, we move each factor to the other
side of the fraction bar and change the sign of each exponent:
.
The following properties of exponents can be used to simplify
expressions.
Properties of Exponents
For any real numbers a and b and any integers m and n, assuming 0 is
not raised to a nonpositive power:
Product rule
Quotient rule
Power rule
Raising a product to a power
Raising a quotient to a power
EXAMPLE 4 Simplify each of the following.
a) b)
c) d)
e)
ͩ
45x
Ϫ4

y
2
9z
Ϫ8
ͪ
Ϫ3
͑2s
Ϫ2
͒
5
͑t
Ϫ3
͒
5
48x
12
16x
4
y
Ϫ5
и y
3
͑b  0͒
ͩ
a
b
ͪ
m
෇
a

m
b
m
͑ab͒
m
෇ a
m
b
m
͑a
m
͒
n
෇ a
mn
͑a  0͒
a
m
a
n
෇ a
mϪn
a
m
и a
n
෇ a
mϩn
x
Ϫ3

y
Ϫ8
z
Ϫ10
෇
z
10
x
3
y
8
x
Ϫ3
y
Ϫ8
z
Ϫ10
a
Ϫm
b
Ϫn
෇
b
n
a
m
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 10
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2 • Integer Exponents, Scientific Notation, and Order of Operations 11
Solution

a) ,or
b)
c) ,or
d) ,or
e)
,or
Scientific Notation
We can use scientific notation to name very large and very small positive
numbers and to perform computations.
Scientific Notation
Scientific notation for a number is an expression of the type
,
where , N is in decimal notation, and m is an integer.
Keep in mind that in scientific notation positive exponents are used for
numbers greater than or equal to 10 and negative exponents for numbers
between 0 and 1.
EXAMPLE 5 Undergraduate Enrollment. In a recent year, there were
16,539,000 undergraduate students enrolled in post-secondary institutions
in the United States (Source:U.S. National Center for Education Statistics).
Convert this number to scientific notation.
Solution We want the decimal point to be positioned between the 1 and
the 6, so we move it 7 places to the left. Since the number to be converted is
greater than 10, the exponent must be positive.
16,539,000 ෇ 1.6539 ϫ 10
7
1 Յ N Ͻ 10
N ϫ 10
m
x
12

125y
6
z
24
෇
5
Ϫ3
x
12
y
Ϫ6
z
24
෇
x
12
5
3
y
6
z
24

ͩ
45x
Ϫ4
y
2
9z
Ϫ8

ͪ
Ϫ3
෇
ͩ
5x
Ϫ4
y
2
z
Ϫ8
ͪ
Ϫ3
32
s
10
͑2s
Ϫ2
͒
5
෇ 2
5
͑s
Ϫ2
͒
5
෇ 32s
Ϫ10
1
t
15

͑t
Ϫ3
͒
5
෇ t
Ϫ3и5
෇ t
Ϫ15
48x
12
16x
4
෇
48
16
x
12Ϫ4
෇ 3x
8
1
y
2
y
Ϫ5
и y
3
෇ y
Ϫ5ϩ3
෇ y
Ϫ2

BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 11
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
12 Chapter R • Basic Concepts of Algebra
EXAMPLE 6 Mass of a Neutron. The mass of a neutron is about
0.00000000000000000000000000167 kg. Convert this number to scien-
tific notation.
Solution We want the decimal point to be positioned between the 1 and
the 6, so we move it 27 places to the right. Since the number to be converted
is between 0 and 1, the exponent must be negative.
EXAMPLE 7 Convert each of the following to decimal notation.
a) b)
Solution
a) The exponent is negative, so the number is between 0 and 1. We move the
decimal point 4 places to the left.
b) The exponent is positive, so the number is greater than 10. We move the
decimal point 5 places to the right.
Most calculators make use of scientific notation. For example, the num-
ber 48,000,000,000,000 might be expressed in one of the ways shown below.
EXAMPLE 8 Distance to a Star. The nearest star, Alpha Centauri C, is
about 4.22 light-years from Earth. One light-year is the distance that light
travels in one year and is about miles. How many miles is it
from Earth to Alpha Centauri C? Express your answer in scientific notation.
Solution
This is not scientific
notation because
.
miles Writing scientific
notation
෇ 2.48136 ϫ 10
13

෇ 2.48136 ϫ ͑10
1
ϫ 10
12
͒
෇ ͑2.48136 ϫ 10
1
͒ ϫ 10
12
24.8136 w 10
෇ 24.8136 ϫ 10
12
4.22 ϫ ͑5.88 ϫ 10
12
͒ ෇ ͑4.22 ϫ 5.88͒ ϫ 10
12
5.88 ϫ 10
12
4.8 134.8E13
9.4 ϫ 10
5
෇ 940,000
7.632 ϫ 10
Ϫ4
෇ 0.0007632
9.4 ϫ 10
5
7.632 ϫ 10
Ϫ4
0.00000000000000000000000000167 ෇ 1.67 ϫ 10

Ϫ27
2.48136E13
4.22ء5.88E12
GCM
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 12
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2 • Integer Exponents, Scientific Notation, and Order of Operations 13
Order of Operations
Recall that to simplify the expression , first we multiply 4 and 5 to
get 20 and then add 3 to get 23. Mathematicians have agreed on the follow-
ing procedure, or rules for order of operations.
Rules for Order of Operations
1. Do all calculations within grouping symbols before operations
outside. When nested grouping symbols are present, work from
the inside out.
2. Evaluate all exponential expressions.
3. Do all multiplications and divisions in order from left to right.
4. Do all additions and subtractions in order from left to right.
EXAMPLE 9 Calculate each of the following.
a) b)
Solution
a)
Doing the calculation within
parentheses
Evaluating the exponential expression
Multiplying
Subtracting
b)
Note that fraction bars act as grouping symbols. That is, the given ex-
pression is equivalent to .

We can also enter these computations on a graphing calculator as shown
below.
To confirm that it is essential to include parentheses around the numer-
ator and around the denominator when the computation in Example 9(b) is
entered in a calculator, enter the computation without using these parenthe-
ses. What is the result?
1
(10/(8Ϫ6)ϩ9ء4)/(2
ˆ
5ϩ3
2
)
44
8(5Ϫ3)
ˆ
3Ϫ20
͓10 Ϭ ͑8 Ϫ 6͒ ϩ 9 и 4͔ Ϭ ͑2
5
ϩ 3
2
͒
෇
5 ϩ 36
41
෇
41
41
෇ 1

10 Ϭ ͑8 Ϫ 6͒ ϩ 9 и 4

2
5
ϩ 3
2
෇
10 Ϭ 2 ϩ 9 и 4
32 ϩ 9
෇ 44
෇ 64 Ϫ 20
෇ 8 и 8 Ϫ 20
8 ͑5 Ϫ 3͒
3
Ϫ 20 ෇ 8 и 2
3
Ϫ 20
10 Ϭ ͑8 Ϫ 6͒ ϩ 9 и 4
2
5
ϩ 3
2
8͑5 Ϫ 3͒
3
Ϫ 20
3 ϩ 4 и 5
GCM
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 13
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
EXAMPLE 10 Compound Interest. If a principal P is invested at an
interest rate r,compounded n times per year, in t years it will grow to
an amount A given by

.
Suppose that $1250 is invested at 4.6% interest, compounded quarterly. How
much is in the account at the end of 8 years?
Solution We have , , or 0.046, , and . Sub-
stituting, we find that the amount in the account at the end of 8 years is
given by
.
Next, we evaluate this expression:
Dividing
Adding
Multiplying in the exponent
Evaluating the exponential expression
Multiplying
. Rounding to the nearest cent
The amount in the account at the end of 8 years is $1802.26.
Ϸ 1802.26
Ϸ 1802.263969
Ϸ 1250͑1.441811175͒
෇ 1250͑1.0115͒
32
෇ 1250͑1.0115͒
4и8
A ෇ 1250͑1 ϩ 0.0115͒
4и8
A ෇ 1250
ͩ
1 ϩ
0.046
4
ͪ

4и8
t ෇ 8n ෇ 4r ෇ 4.6%P ෇ 1250
A ෇ P
ͩ
1 ϩ
r
n
ͪ
nt
14 Chapter R • Basic Concepts of Algebra
Ճ Answers to Exercises 15–20 can be found on p. IA-1.
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 14
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
14 Chapter R • Basic Concepts of Algebra
R.2
Exercise Set
Simplify.
1. 1 2. 1
3. 4.
5. ,or 25 6. ,or
7. 1 8. 1
9. ,or 10.
11. ,or 12.
13. 14.
15.
Ճ 16. Ճ
17. Ճ 18. Ճ
19. Ճ 20. Ճ
21. 22.
23. 24.

25. 26.
27. ,or 28. ,or
29. ,or 30. ,or
31. ,or 32. ,or
4b
a
2
4a
Ϫ2
b
20a
5
b
Ϫ2
5a
7
b
Ϫ3
8x
y
5
8xy
Ϫ5
32x
Ϫ4
y
3
4x
Ϫ5
y

8
x
4
y
5
x
4
y
Ϫ5
x
3
y
Ϫ3
x
Ϫ1
y
2
x
3
y
3
x
3
y
Ϫ3
x
2
y
Ϫ2
x

Ϫ1
y
1
y
3
y
Ϫ3
y
Ϫ24
y
Ϫ21
1
x
21
x
Ϫ21
x
Ϫ5
x
16
a
7
a
39
a
32
b
3
b
40

b
37
288x
7
͑2x͒
5
͑3x͒
2
Ϫ200n
5
͑Ϫ2n͒
3
͑5n͒
2
432y
5
͑4y͒
2
͑3y͒
3
72x
5
͑2x͒
3
͑3x͒
2
͑8ab
7
͒͑Ϫ7a
Ϫ5

b
2
͒͑6x
Ϫ3
y
5
͒͑Ϫ7x
2
y
Ϫ9
͒
͑4xy
2
͒͑3x
Ϫ4
y
5
͒͑5a
2
b͒͑3a
Ϫ3
b
4
͒
͑Ϫ6b
Ϫ4
͒͑2b
Ϫ7
͒͑Ϫ3a
Ϫ5

͒͑5a
Ϫ7
͒
12y
7
3y
4
и 4y
3
6x
5
2x
3
и 3x
2
3
5
3
6
и 3
Ϫ5
и 3
4
1
7
7
Ϫ1
7
3
и 7

Ϫ5
и 7
b
8
b
Ϫ4
и b
12
1
y
4
y
Ϫ4
y
3
и y
Ϫ7
n
9
и n
Ϫ9
m
Ϫ5
и m
5
1
6
5
6
Ϫ5

6
2
и 6
Ϫ7
5
2
5
8
и 5
Ϫ6
a
4
a
0
и a
4
x
9
x
9
и x
0
͑
Ϫ
4
3
͒
0
18
0

Ճ Answers to Exercises 15–20 can be found on p. IA-1.
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 14
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
33. 34.
35. 36.
37. 38.
39. Ճ 40.
41. Ճ 42. Ճ
43. Ճ 44. Ճ
Convert to scientific notation.
45. 405,000 46. 1,670,000
47. 0.00000039 48. 0.00092
49. 234,600,000,000 50. 8,904,000,000
51. 0.00104 52. 0.00000000514
53. One cubic inch is approximately equal to
.
54. The United States government collected
$1,137,000,000,000 in individual income taxes in a
recent year (Source:U.S. Internal Revenue Service).
Convert to decimal notation.
55. 0.000083 56. 4,100,000
57. 20,700,000 58.
0.00000315
59. 60.
34,960,000,000 840,900,000,000
61. 62.
0.0000000541 0.000000000627
63. The amount of solid waste generated in the United
States in a recent year was tons (Source:
Franklin Associates, Ltd.). 231,900,000

64. The mass of a proton is about g.
0.00000000000000000000000167
Compute. Write the answer using scientific notation.
65.
66.
67.
68.
69. 70.
71. 72.
Solve. Write the answer using scientific notation.
73. Distance to Pluto. The distance from Earth to the
sun is defined as 1 astronomical unit,or AU.It is
about 93 million miles. The average distance from
Earth to the planet Pluto is 39 AUs. Find this
distance in miles.
74. Parsecs. One parsec is about 3.26 light-years and
1 light-year is about Find the
number of miles in 1 parsec.
75. Nanowires. A nanometer is 0.000000001 m.
Scientists have developed optical nanowires to
transmit light waves short distances. A nanowire
with a diameter of 360 nanometers has been used in
experiments on :the transmission of light (Source:
New York Times, January 29, 2004). Find the
diameter of such a wire in meters.
76. iTunes. In the first quarter of 2004, Apple
Computer was selling 2.7 million songs per week on
iTunes, its online music service (Source:Apple
Computer). At $0.99 per song, what is the revenue
during a 13-week period?

77. Chesapeake Bay Bridge-Tunnel. The 17.6-mile-long
Chesapeake Bay Bridge-Tunnel was completed in
1964. Construction costs were $210 million. Find
the average cost per mile.
78. Personal Space in Hong Kong. The area of Hong
Kong is 412 square miles. It is estimated that the
population of Hong Kong will be 9,600,000 in 2050.
Find the number of square miles of land per person
in 2050. sq mi
79. Nuclear Disintegration. One gram of radium
produces 37 billion disintegrations per second. How
many disintegrations are produced in 1 hr?
disintegrations
80. Length of Earth’s Orbit. The average distance from
the earth to the sun is 93 million mi. About how far
does the earth travel in a yearly orbit? (Assume a
circular orbit.) mi5.8 ϫ 10
8
1.332 ϫ 10
14
4.3 ϫ 10
Ϫ5
$1.19 ϫ 10
7
$3.4749 ϫ 10
7
3.6 ϫ 10
Ϫ7
m
1.91688 ϫ 10

13
mi
5.88 ϫ 10
12
mi.
3.627 ϫ 10
9
mi
2.5 ϫ 10
Ϫ7
1.3 ϫ 10
4
5.2 ϫ 10
10
2.5 ϫ 10
5
1.8 ϫ 10
Ϫ3
7.2 ϫ 10
Ϫ9
5.5 ϫ 10
30
1.1 ϫ 10
Ϫ40
2.0 ϫ 10
Ϫ71
8 ϫ 10
Ϫ14
6.4 ϫ 10
Ϫ7

8.0 ϫ 10
6
2.368 ϫ 10
8
͑6.4 ϫ 10
12
͒͑3.7 ϫ 10
Ϫ5
͒
2.21 ϫ 10
Ϫ10
͑2.6 ϫ 10
Ϫ18
͒͑8.5 ϫ 10
7
͒
7.462 ϫ 10
Ϫ13
͑9.1 ϫ 10
Ϫ17
͒͑8.2 ϫ 10
3
͒
1.395 ϫ 10
3
͑3.1 ϫ 10
5
͒͑4.5 ϫ 10
Ϫ3
͒

1.67 ϫ 10
Ϫ24
2.319 ϫ 10
8
6.27 ϫ 10
Ϫ10
5.41 ϫ 10
Ϫ8
8.409 ϫ 10
11
3.496 ϫ 10
10
3.15 ϫ 10
Ϫ6
2.07 ϫ 10
7
4.1 ϫ 10
6
8.3 ϫ 10
Ϫ5
1.137 ϫ 10
12
1.6 ϫ 10
Ϫ5
0.000016 m
3
5.14 ϫ 10
Ϫ9
1.04 ϫ 10
Ϫ3

8.904 ϫ 10
9
2.346 ϫ 10
11
9.2 ϫ 10
Ϫ4
3.9 ϫ 10
Ϫ7
1.67 ϫ 10
6
4.05 ϫ 10
5
ͩ
125p
12
q
Ϫ14
r
22
25p
8
q
6
r
Ϫ15
ͪ
Ϫ4
ͩ
24a
10

b
Ϫ8
c
7
12a
6
b
Ϫ3
c
5
ͪ
Ϫ5
ͩ
3x
5
y
Ϫ8
z
Ϫ2
ͪ
4
ͩ
2x
Ϫ3
y
7
z
Ϫ1
ͪ
3

128n
7
͑4n
Ϫ1
͒
2
͑2n
3
͒
3
͑3m
4
͒
3
͑2m
Ϫ5
͒
4
x
15
z
6
Ϫ64
͑Ϫ4x
Ϫ5
z
Ϫ2
͒
Ϫ3
c

2
d
4
25
͑Ϫ5c
Ϫ1
d
Ϫ2
͒
Ϫ2
81x
8
͑Ϫ3x
2
͒
4
Ϫ32x
15
͑Ϫ2x
3
͒
5
16x
2
y
6
͑4xy
3
͒
2

8a
3
b
6
͑2ab
2
͒
3
Section R.2 • Integer Exponents, Scientific Notation, and Order of Operations 15
Ճ Answers to Exercises 39 and 41–44 can be found on p. IA-1.
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 15
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Calculate.
81. 2
82. 18
83. 2048
84. 2
85. 5
86. Ϫ5
Compound Interest. Use the compound interest
formula from Example 10 in Exercises 87– 90.
Round to the nearest cent.
87. Suppose that $2125 is invested at 6.2%, compounded
semiannually. How much is in the account at the end
of 5 yr? $2883.67
88. Suppose that $9550 is invested at 5.4%, compounded
semiannually. How much is in the account at the end
of 7 yr? $13,867.23
89. Suppose that $6700 is invested at 4.5%, compounded
quarterly. How much is in the account at the end

of 6 yr? $8763.54
90. Suppose that $4875 is invested at 5.8%, compounded
quarterly. How much is in the account at the end
of 9 yr? $8185.56
Collaborative Discussion and Writing
91. Are the parentheses necessary in the expression
? Why or why not?
92. Is for any negative value(s) of x? Why or
why not?
Synthesis
Savings Plan. The formula
gives the amount S accumulated in a savings plan when
a deposit of P dollars is made each month for t years in
an account with interest rate r, compounded monthly.
Use this formula for Exercises 93–96.
93. Marisol deposits $250 in a retirement account each
month beginning at age 40. If the investment earns
5% interest, compounded monthly, how much will
have accumulated in the account when she retires
27 yr later? $170,797.30
94. Gordon deposits $100 in a retirement account each
month beginning at age 25. If the investment earns
4% interest, compounded monthly, how much will
have accumulated in the account when Gordon
retires at age 65? $118,196.13
95. Gina wants to establish a college fund for her newborn
daughter that will have accumulated $120,000 at the
end of 18 yr. If she can count on an interest rate of
6%, compounded monthly, how much should she
deposit each month to accomplish this? $309.79

96. Liam wants to have $200,000 accumulated in a
retirement account by age 70. If he starts making
monthly deposits to the plan at age 30 and can count
on an interest rate of 4.5%, compounded monthly,
how much should he deposit each month in order
to accomplish this? $149.13
S ෇ P
ͫ
ͩ
1 ϩ
r
12
ͪ
12иt
Ϫ 1
r
12
ͬ
x
Ϫ2
Ͻ x
Ϫ1
4 и 25 Ϭ ͑10 Ϫ 5͒
͓4͑8 Ϫ 6͒
2
ϩ 4͔͑3 Ϫ 2 и 8͒
2
2
͑2
3

ϩ 5͒
4͑8 Ϫ 6͒
2
Ϫ 4 и 3 ϩ 2 и 8
3
1
ϩ 19
0
2
6
и 2
Ϫ3
Ϭ 2
10
Ϭ 2
Ϫ8
16 Ϭ 4 и 4 Ϭ 2 и 256
3͓͑2 ϩ 4 и 2
2
͒ Ϫ 6͑3 Ϫ 1͔͒
3 и 2 Ϫ 4 и 2
2
ϩ 6͑3 Ϫ 1͒
16 Chapter R • Basic Concepts of Algebra
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 16
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Simplify. Assume that all exponents are integers, all
denominators are nonzero, and zero is not raised to a
nonpositive power.
97. 98. 1

99. 100.
101. 102.
, or
x
6r
y
18t
x
6r
y
Ϫ18t
9x
2a
y
2b
ͫͩ
x
r
y
t
ͪ
2
ͩ
x
2r
y
4t
ͪ
Ϫ2
ͬ

Ϫ3
ͫ
͑3x
a
y
b
͒
3
͑Ϫ3x
a
y
b
͒
2
ͬ
2
m
x
2
n
x
2
͑m
xϪb
и n
xϩb
͒
x
͑m
b

n
Ϫb
͒
x
t
8x
͑t
aϩx
и t
xϪa
͒
4
͑x
y
и x
Ϫy
͒
3
x
8t
͑x
t
и x
3t
͒
2
17
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 17
Section R.2 • Integer Exponents, Scientific Notation, and Order of Operations
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley

Section R.3 • Addition, Subtraction, and Multiplication of Polynomials 17
R.3
Addition,
Subtraction, and
Multiplication of
Polynomials
•Identify the terms, coefficients, and degree of a polynomial.
•Add, subtract, and multiply polynomials.
Polynomials
Polynomials are a type of algebraic expression that you will often encounter
in your study of algebra. Some examples of polynomials are
,,,and .
All but the first are polynomials in one variable.
Polynomials in One Variable
A polynomial in one variable is any expression of the type
,
where n is a nonnegative integer and are real numbers,
called coefficients.The parts of a polynomial separated by plus
signs are called terms.The leading coefficient is , and the
constant term is . If the degree of the polynomial is n.
The polynomial is said to be written in descending order,because
the exponents decrease from left to right.
EXAMPLE 1 Identify the terms of the polynomial
.
Solution Writing plus signs between the terms, we have
,
so the terms are
,,x, and Ϫ12.
A polynomial, like 23, consisting of only a nonzero constant term has
degree 0. It is agreed that the polynomial consisting only of 0 has no degree.

Ϫ7.5x
3
2x
4
2x
4
Ϫ 7.5x
3
ϩ x Ϫ 12 ෇ 2x
4
ϩ ͑Ϫ7.5x
3
͒ ϩ x ϩ ͑Ϫ12͒
2x
4
Ϫ 7.5x
3
ϩ x Ϫ 12
a
n
 0,a
0
a
n
a
0
a
n
, ,
a

n
x
n
ϩ a
nϪ1
x
nϪ1
ϩ иии ϩ a
2
x
2
ϩ a
1
x ϩ a
0
z
6
Ϫ
͙
5Ϫ2.3a
4
5y
3
Ϫ
7
3
y
2
ϩ 3y Ϫ 23x Ϫ 4y
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 17

Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
EXAMPLE 2 Find the degree of each polynomial.
a) b) c) 7
Solution
POLYNOMIAL DEGREE
a) 3
b) 4
c) 0
Algebraic expressions like and
are polynomials in several variables.The degree of a term is the sum of
the exponents of the variables in that term. The degree of a polynomial is
the degree of the term of highest degree.
EXAMPLE 3 Find the degree of the polynomial
.
Solution The degrees of the terms of are 4, 6, and 0,
respectively, so the degree of the polynomial is 6.
A polynomial with just one term, like , is a monomial.Ifa poly-
nomial has two terms, like , it is a binomial.A polynomial with three
terms, like , is a trinomial.
Expressions like
,,and
are not polynomials, because they cannot be written in the form
,where the exponents are all nonnegative in-
tegers and the coefficients are all real numbers.
Addition and Subtraction
If two terms of an expression have the same variables raised to the same
powers, they are called like terms,or similar terms.We can combine,or
collect, like terms using the distributive property. For example, and
are like terms and
.

We add or subtract polynomials by combining like terms.
EXAMPLE 4 Add or subtract each of the following.
a)
b) ͑6x
2
y
3
Ϫ 9xy͒ Ϫ ͑5x
2
y
3
Ϫ 4xy͒
͑Ϫ5x
3
ϩ 3x
2
Ϫ x͒ ϩ ͑12x
3
Ϫ 7x
2
ϩ 3͒
෇ 8y
2
3 y
2
ϩ 5y
2
෇ ͑3 ϩ 5͒y
2
5y

2
3y
2
a
nϪ1
x
nϪ1
ϩ иии ϩ a
1
x ϩ a
0
a
n
x
n
ϩ
x ϩ 1
x
4
ϩ 5
9 Ϫ
͙
x2x
2
Ϫ 5x ϩ
3
x
4x
2
Ϫ 4xy ϩ 1

x
2
ϩ 4
Ϫ9y
6
7ab
3
Ϫ 11a
2
b
4
ϩ 8
7ab
3
Ϫ 11a
2
b
4
ϩ 8
5x
4
y
2
Ϫ 3x
3
y
8
ϩ 7xy
2
ϩ 63ab

3
Ϫ 8
7 ෇ 7x
0
y
2
Ϫ
3
2
ϩ 5y
4
෇ 5y
4
ϩ y
2
Ϫ
3
2
2x
3
Ϫ 9
y
2
Ϫ
3
2
ϩ 5y
4
2x
3

Ϫ 9
18 Chapter R • Basic Concepts of Algebra
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Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Solution
a)
Rearranging using
the commutative
and associative
properties
Using the distribu-
tive property
b) We can subtract by adding an opposite:
Adding the opposite of
. Combining like terms
Multiplication
Multiplication of polynomials is based on the distributive property—for
example,
Using the distributive property
Using the distributive property
two more times
. Combining like terms
In general, to multiply two polynomials, we multiply each term of one
by each term of the other and add the products.
EXAMPLE 5 Multiply: .
Solution We have
Using the distributive
property
Using the distributive
property three more times

. Combining like terms
We can also use columns to organize our work, aligning like terms under
each other in the products.
Multiplying by
Multiplying by 2y
Adding Ϫ12x
6
y
2
ϩ 29x
4
y
2
Ϫ 23x
2
y
2
ϩ 6y
2
8 x
4
y
2
Ϫ 14x
2
y
2
ϩ 6y
2
؊3x

2
y Ϫ12x
6
y
2
ϩ 21x
4
y
2
Ϫ 9x
2
y
2
2 y Ϫ 3x
2
y
4 x
4
y Ϫ 7x
2
y ϩ 3y
෇ 29x
4
y
2
Ϫ 12x
6
y
2
Ϫ 23x

2
y
2
ϩ 6y
2
෇ 8x
4
y
2
Ϫ 12x
6
y
2
Ϫ 14x
2
y
2
ϩ 21x
4
y
2
ϩ 6y
2
Ϫ 9x
2
y
2
෇ 4x
4
y͑2y Ϫ 3x

2
y͒ Ϫ 7x
2
y͑2y Ϫ 3x
2
y͒ ϩ 3y͑2y Ϫ 3x
2
y͒
͑4x
4
y Ϫ 7x
2
y ϩ 3y͒͑2y Ϫ 3x
2
y͒
͑4x
4
y Ϫ 7x
2
y ϩ 3y͒͑2y Ϫ 3x
2
y͒
෇ x
2
ϩ 7x ϩ 12
෇ x
2
ϩ 3x ϩ 4x ϩ 12
͑x ϩ 4͒͑x ϩ 3͒ ෇ x͑x ϩ 3͒ ϩ 4͑x ϩ 3͒
෇ x

2
y
3
Ϫ 5xy
෇ 6x
2
y
3
Ϫ 9xy Ϫ 5x
2
y
3
ϩ 4xy
5x
2
y
3
؊ 4xy
෇ ͑6x
2
y
3
Ϫ 9xy͒ ϩ ͑Ϫ5x
2
y
3
ϩ 4xy͒
͑6x
2
y

3
Ϫ 9xy͒ Ϫ ͑5x
2
y
3
Ϫ 4xy͒
෇ 7x
3
Ϫ 4x
2
Ϫ x ϩ 3
෇ ͑Ϫ5 ϩ 12͒x
3
ϩ ͑3 Ϫ 7͒x
2
Ϫ x ϩ 3
෇ ͑Ϫ5x
3
ϩ 12x
3
͒ ϩ ͑3x
2
Ϫ 7x
2
͒ Ϫ x ϩ 3
͑Ϫ5x
3
ϩ 3x
2
Ϫ x͒ ϩ ͑12x

3
Ϫ 7x
2
ϩ 3͒
Section R.3 • Addition, Subtraction, and Multiplication of Polynomials 19
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 19
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
20 Chapter R • Basic Concepts of Algebra
We can find the product of two binomials by multiplying the First
terms, then the Outer terms, then the Inner terms, then the Last terms. Then
we combine like terms, if possible. This procedure is sometimes called FOIL.
EXAMPLE 6 Multiply: .
Solution We have
FL
FOIL
I
O
We can use FOIL to find some special products.
Special Products of Binomials
Square of a sum
Square of a difference
Product of a sum and a difference
EXAMPLE 7 Multiply each of the following.
a) b) c)
Solution
a)
b)
c)
͑x
2

ϩ 3y͒͑x
2
Ϫ 3y͒ ෇ ͑x
2
͒
2
Ϫ ͑3y͒
2
෇ x
4
Ϫ 9y
2
͑3y
2
Ϫ 2͒
2
෇ ͑3y
2
͒
2
Ϫ 2 и 3y
2
и 2 ϩ 2
2
෇ 9y
4
Ϫ 12y
2
ϩ 4
͑4x ϩ 1͒

2
෇ ͑4x͒
2
ϩ 2 и 4x и 1 ϩ 1
2
෇ 16x
2
ϩ 8x ϩ 1
͑x
2
ϩ 3y͒͑x
2
Ϫ 3y͒͑3y
2
Ϫ 2͒
2
͑4x ϩ 1͒
2
͑A ϩ B͒͑A Ϫ B͒ ෇ A
2
Ϫ B
2
͑A Ϫ B͒
2
෇ A
2
Ϫ 2AB ϩ B
2
͑A ϩ B͒
2

෇ A
2
ϩ 2AB ϩ B
2
෇ 6x
2
Ϫ 13x Ϫ 28
͑2x Ϫ 7͒͑3x ϩ 4͒ ෇ 6x
2
ϩ 8x Ϫ 21x Ϫ 28
͑2x Ϫ 7͒͑3x ϩ 4͒
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 20
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
R.3
Exercise Set
Determine the terms and the degree of the polynomial.
1. ,,,
,;4
2. ,,Ϫ4m, 11; 3
3. ,,5ab,
;6
4. ,, ,
5; 6
Perform the operations indicated.
5.
6.
7.
3x ϩ 2y Ϫ 2z Ϫ 3͑Ϫ3x ϩ y Ϫ 2z Ϫ 4͒
͑2x ϩ 3y ϩ z Ϫ 7͒ ϩ ͑4x Ϫ 2y Ϫ z ϩ 8͒ ϩ
2x

2
y Ϫ 7xy
2
ϩ 8xy ϩ 5
͑Ϫ4x
2
y Ϫ 4xy
2
ϩ 3xy ϩ 8͒
͑6x
2
y Ϫ 3xy
2
ϩ 5xy Ϫ 3͒ ϩ
3x
2
y Ϫ 5xy
2
ϩ 7xy ϩ 2
͑Ϫ2x
2
y Ϫ 3xy
2
ϩ 4xy ϩ 7͒
͑5x
2
y Ϫ 2xy
2
ϩ 3xy Ϫ 5͒ ϩ
Ϫ3pq

2
Ϫp
2
q
4
6p
3
q
2
6p
3
q
2
Ϫ p
2
q
4
Ϫ 3pq
2
ϩ 5
Ϫ2
Ϫ7a
3
b
3
3a
4
b3a
4
b Ϫ 7a

3
b
3
ϩ 5ab Ϫ 2
Ϫm
2
2m
3
2m
3
Ϫ m
2
Ϫ 4m ϩ 11
Ϫ4Ϫy
7y
2
3y
3
Ϫ5y
4
Ϫ5y
4
ϩ 3y
3
ϩ 7y
2
Ϫ y Ϫ 4
20 Chapter R • Basic Concepts of Algebra
BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 20
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley

8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.

38.
39.
40.
41.
42.
Collaborative Discussion and Writing
43. Is the sum of two polynomials of degree n always a
polynomial of degree n? Why or why not?
44. Explain how you would convince a classmate that
.
Synthesis
Multiply. Assume that all exponents are natural
numbers.
45.
46.
47.
48.
49.
50.
51.
52.
53.
a
2
ϩ b
2
ϩ c
2
ϩ 2ab ϩ 2ac ϩ 2bc͑a ϩ b ϩ c͒
2

t
2m
2
ϩ2n
2
͑t
mϩn
͒
mϩn
и ͑t
mϪn
͒
mϪn
x
a
2
Ϫb
2
͑x
aϪb
͒
aϩb
16x
4
Ϫ 32x
3
ϩ 16x
2
͓͑2x Ϫ 1͒
2

Ϫ 1͔
2
x
6
Ϫ 1͑x Ϫ 1͒͑x
2
ϩ x ϩ 1͒͑x
3
ϩ 1͒
x
6m
Ϫ 2x
3m
t
5n
ϩ t
10n
͑x
3m
Ϫ t
5n
͒
2
a
2n
ϩ 2a
n
b
n
ϩ b

2n
͑a
n
ϩ b
n
͒
2
t
2a
Ϫ 3t
a
Ϫ 28͑t
a
ϩ 4͒͑t
a
Ϫ 7͒
a
2n
Ϫ b
2n
͑a
n
ϩ b
n
͒͑a
n
Ϫ b
n
͒
͑A ϩ B͒

2
 A
2
ϩ B
2
y
4
Ϫ 16͑y Ϫ 2͒͑y ϩ 2͒͑y
2
ϩ 4͒
x
4
Ϫ 1͑x ϩ 1͒͑x Ϫ 1͒͑x
2
ϩ 1͒
25x
2
ϩ 20xy ϩ 4y
2
Ϫ 9
͑5x ϩ 2y ϩ 3͒͑5x ϩ 2y Ϫ 3͒
4x
2
ϩ 12xy ϩ 9y
2
Ϫ 16
͑2x ϩ 3y ϩ 4͒͑2x ϩ 3y Ϫ 4͒
9x
2
Ϫ 25y

2
͑3x ϩ 5y͒͑3x Ϫ 5y͒
9x
2
Ϫ 4y
2
͑3x Ϫ 2y͒͑3x ϩ 2y͒
16y
2
Ϫ 1͑4y Ϫ 1͒͑4y ϩ 1͒
4x
2
Ϫ 25͑2x Ϫ 5͒͑2x ϩ 5͒
b
2
Ϫ 16͑b ϩ 4͒͑b Ϫ 4͒
a
2
Ϫ 9͑a ϩ 3͒͑a Ϫ 3͒
16x
4
Ϫ 40x
2
y ϩ 25y
2
͑4x
2
Ϫ 5y͒
2
4x

4
Ϫ 12x
2
y ϩ 9y
2
͑2x
2
Ϫ 3y͒
2
25x
2
ϩ 20xy ϩ 4y
2
͑5x ϩ 2y͒
2
4x
2
ϩ 12xy ϩ 9y
2
͑2x ϩ 3y͒
2
9x
2
Ϫ 12x ϩ 4͑3x Ϫ 2͒
2
25x
2
Ϫ 30x ϩ 9͑5x Ϫ 3͒
2
a

2
Ϫ 12a ϩ 36͑a Ϫ 6͒
2
x
2
Ϫ 8x ϩ 16͑x Ϫ 4͒
2
y
2
ϩ 14y ϩ 49͑y ϩ 7͒
2
y
2
ϩ 10y ϩ 25͑y ϩ 5͒
2
4a
2
Ϫ 8ab ϩ 3b
2
͑2a Ϫ 3b͒͑2a Ϫ b͒
4x
2
ϩ 8xy ϩ 3y
2
͑2x ϩ 3y͒͑2x ϩ y͒
3b
2
Ϫ 5b Ϫ 2͑3b ϩ 1͒͑b Ϫ 2͒
2a
2

ϩ 13a ϩ 15͑2a ϩ 3͒͑a ϩ 5͒
n
2
Ϫ 13n ϩ 40͑n Ϫ 5͒͑n Ϫ 8͒
x
2
ϩ 10x ϩ 24͑x ϩ 6͒͑x ϩ 4͒
y
2
Ϫ 3y Ϫ 4͑y Ϫ 4͒͑y ϩ 1͒
x
2
ϩ 2x Ϫ 15͑x ϩ 5͒͑x Ϫ 3͒
n
3
Ϫ 5n
2
Ϫ 10n Ϫ 4͑n ϩ 1͒͑n
2
Ϫ 6n Ϫ 4͒
2a
4
Ϫ 2a
3
b Ϫ a
2
b ϩ 4ab
2
Ϫ 3b
3

͑a Ϫ b͒͑2a
3
Ϫ ab ϩ 3b
2
͒
2x
4
Ϫ 5x
3
Ϫ 5x
2
ϩ 10x Ϫ 5
͑2x
4
Ϫ 3x
2
ϩ 7x͒ Ϫ ͑5x
3
ϩ 2x
2
Ϫ 3x ϩ 5͒
x
4
Ϫ 3x
3
Ϫ 4x
2
ϩ 9x Ϫ 3
͑x
4

Ϫ 3x
2
ϩ 4x͒ Ϫ ͑3x
3
ϩ x
2
Ϫ 5x ϩ 3͒
Ϫ4x
2
ϩ 8xy Ϫ 5y
2
ϩ 3
͑5x
2
ϩ 4xy Ϫ 3y
2
ϩ 2͒ Ϫ ͑9x
2
Ϫ 4xy ϩ 2y
2
Ϫ 1͒
Ϫ2x
2
ϩ 6x Ϫ 2
͑3x
2
Ϫ 2x Ϫ x
3
ϩ 2͒ Ϫ ͑5x
2

Ϫ 8x Ϫ x
3
ϩ 4͒
7x
2
ϩ 12xy Ϫ 2x Ϫ y Ϫ 9͑Ϫx
2
Ϫ y Ϫ 2͒
͑2x
2
ϩ 12xy Ϫ 11͒ ϩ ͑6x
2
Ϫ 2x ϩ 4͒ ϩ
Section R.3 • Addition, Subtraction, and Multiplication of Polynomials 21
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Factor polynomials by removing a common factor.
Factor polynomials by grouping.
Factor trinomials of the type .
Factor trinomials of the type , , using the FOIL
method and the grouping method.
Factor special products of polynomials.
To factor a polynomial, we do the reverse of multiplying; that is, we find an
equivalent expression that is written as a product.
Terms with Common Factors
When a polynomial is to be factored, we should always look first to factor
out a factor that is common to all the terms using the distributive property.
We usually look for the constant common factor with the largest absolute
value and for variables with the largest exponent common to all the terms.
In this sense, we factor out the “largest” common factor.

EXAMPLE 1 Factor each of the following.
a) b)
Solution
a)
We can always check a factorization by multiplying:
.
b) There are several factors common to the terms of , but
is the “largest” of these.
Factoring by Grouping
In some polynomials, pairs of terms have a common binomial factor that
can be removed in a process called factoring by grouping.
EXAMPLE 2 Factor: .
Solution We have
Grouping; each
group of terms has
a common factor.
Factoring a common
factor out of each
group
. Factoring out the
common binomial
factor
෇ ͑x ϩ 3͒ ͑x
2
Ϫ 5͒
෇ x
2
͑x ϩ 3͒ Ϫ 5͑x ϩ 3͒
x
3

ϩ 3x
2
Ϫ 5x Ϫ 15 ෇ ͑x
3
ϩ 3x
2
͒ ϩ ͑Ϫ5x Ϫ 15͒
x
3
ϩ 3x
2
Ϫ 5x Ϫ 15
෇ 4x
2
y͑3y Ϫ 5x͒
12x
2
y
2
Ϫ 20x
3
y ෇ 4x
2
y и 3y Ϫ 4x
2
y и 5x
4x
2
y
12x

2
y
2
Ϫ 20x
3
y
5͑3 ϩ 2x Ϫ x
2
͒ ෇ 15 ϩ 10x Ϫ 5x
2
15 ϩ 10x Ϫ 5x
2
෇ 5 и 3 ϩ 5 и 2x Ϫ 5 и x
2
෇ 5͑3 ϩ 2x Ϫ x
2
͒
12x
2
y
2
Ϫ 20x
3
y15 ϩ 10x Ϫ 5x
2
a  1ax
2
ϩ bx ϩ c
x
2

ϩ bx ϩ c
22 Chapter R • Basic Concepts of Algebra
R.4
Factoring
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