03 Algebra and Trigonometry 3rd Edition
Table of Contents
Chapter R 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
Chapter 1 Graphs, Functions, and Models
1.1 Introduction to Graphing
1.2 Functions and Graphs
1.3 Linear Functions, Slope, and Applications
1.4 Equations of Lines and Modeling
1.5 More on Functions
1.6 The Algebra of Functions
1.7 Symmetry and Transformations
Chapter 2 Functions, Equations, and Inequalities
2.1 Linear Equations, Functions, and Models
2.2 The Complex Numbers
2.3 Quadratic Equations, Functions, and Models
2.4 Analyzing Graphs of Quadratic Functions
2.5 More Equation Solving
2.6 Solving Linear Inequalities
Chapter 3 Polynomial And Rational Functions
3.1 Polynomial Functions and Models
3.2 Graphing Polynomial Functions
3.3 Polynomial Division; The Remainder and Factor Theorems
3.4 Theorems about Zeros of Polynomial Functions
3.5 Rational Functions
3.6 Polynomial and Rational Inequalities
3.7 Variation and Applications
Chapter 4 Exponential and Logarithmic Functions
4.1 Inverse Functions
4.2 Exponential Functions and Graphs
4.3 Logarithmic Functions and Graphs
4.4 Properties of Logarithmic Functions
4.5 Solving Exponential and Logarithmic Equations
4.6 Applications and Models: Growth and Decay, and Compound Interest
Chapter 5 The Trigonometric Functions
5.1 Trigonometric Functions of Acute Angles
5.2 Applications of Right Triangles
5.3 Trigonometric Functions of Any Angle
5.4 Radians, Arc Length, and Angular Speed
5.5 Circular Functions: Graphs and Properties
5.6 Graphs of Transformed Sine and Cosine Functions
Chapter 6 Trigonometric Identities, Inverse Functions, and Equations
6.1 Identities: Pythagorean and Sum and Difference
6.2 Identities: Cofunction, Double‐Angle, and Half‐Angle
6.3 Proving Trigonometric Identities
6.4 Inverses of the Trigonometric Functions
6.5 Solving Trigonometric Equations
Chapter 7 Applications of Trigonometry
7.1 The Law of Sines
7.2 The Law of Cosines
7.3 Complex Numbers: Trigonometric Form
7.4 Polar Coordinates and Graphs
7.5 Vectors and Applications
7.6 Vector Operations
Chapter 8 Systems of Equations and Matrices
8.1 Systems of Equations in Two Variables
8.2 Systems of Equations in Three Variables
8.3 Matrices and Systems of Equations
8.4 Matrix Operations
8.5 Inverses of Matrices
8.6 Determinants and Cramer's Rule
8.7 Systems of Inequalities and Linear Programming
8.8 Partial Fractions
Chapter 9 Analytic Geometry Topics
9.1 The Parabola
9.2 The Circle and the Eclipse
9.3 The Hyperbola
9.4 Nonlinear Systems of Equations and Inequalities
9.5 Rotation of Axes
9.6 Polar Equations of Conics
9.7 Parametric Equations
Chapter 10 Sequences, Series, and Combinatorics
10.1 Sequences and Series
10.2 Arithmetic Sequences and Series
10.3 Geometric Sequences and Series
10.4 Mathematical Induction
10.5 Combinatorics: Permutations
10.6 Combinatorics: Combinations
10.7 The Binomial Theorem
10.8 Probability
Basic Concepts
of Algebra
R.1
R.2
R.3
R.4
R.5
R.6
R.7
The Real-Number System
Integer Exponents, Scientific Notation,
and Order of Operations
Addition, Subtraction, and
Multiplication of Polynomials
Factoring
Rational Expressions
Radical Notation and Rational Exponents
The Basics of Equation Solving
R
SUMMARY AND REVIEW
TEST
A P P L I C A T I O N
G
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.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
2
Chapter R
• Basic Concepts of Algebra
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.
R.1
2.1
The Real-Number
Polynomial
Functions
System
and
Modeling
Real Numbers
In applications of algebraic concepts, we use real numbers to represent
quantities such as distance, time, speed, area, profit, loss, and temperature. Some frequently used sets of real numbers and the relationships
among them are shown below.
Natural numbers
(positive integers):
1, 2, 3, …
Whole numbers:
0, 1, 2, 3, …
Integers:
…, −3, −2, −1, 0,
1, 2, 3, …
Rational
numbers
Rational numbers
that are not integers:
Real
numbers
Negative integers:
−1, −2, −3, …
2 4 19 −7
−,
− −, −−, −−, 8.3,
3 5 −5 8
−
Irrational numbers:
5
Zero: 0
0.56, …
4
√2, p, −√3, √27,
−4.030030003…, …
Numbers that can be expressed in the form p兾q, where p and q are integers and q 苷 0, are rational numbers. Decimal notation for rational
numbers either terminates (ends) or repeats. Each of the following is a
rational number.
a) 0
b) ⫺7
1
苷 0.25
4
5
d) ⫺ 苷 ⫺0.45
11
c)
0
for any nonzero integer a
a
ⴚ7
7
ⴚ7 ⴝ
, or
ⴚ1
1
0ⴝ
Terminating decimal
Repeating decimal
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.1
• The Real-Number System
3
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) 苷 3.1415926535 . . .
There is no repeating block of digits.
22
共 7 and 3.14 are rational approximations of the irrational number . 兲
b) 兹2 苷 1.414213562 . . .
c) ⫺6.12122122212222 . . .
There is no repeating block of digits.
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.
⫺2.9
兹3
⫺E
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
*
p
3
4
5
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
共a ⬍ b兲. Similarly, a is greater than b 共a ⬎ b兲 if a is to the right of b on
the number line. For example, we see from the number line above that
17
⫺2.9 ⬍ ⫺ 53 , because ⫺2.9 is to the left of ⫺ 53 . Also, 4 ⬎ 兹3, because 174
is to the right of 兹3.
The statement a ⱕ b, read “a is less than or equal to b,” is true if either
a ⬍ b is true or a 苷 b is true.
The symbol 僆 is used to indicate that a member, or element, belongs to
a set. Thus if we let ⺡ represent the set of rational numbers, we can see from
the diagram on page 2 that 0.56 僆 ⺡. We can also write 兹2 僆 ⺡ to indicate that 兹2 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 債 is used to denote this.
For instance, if we let ⺢ represent the set of real numbers, we can see from
the diagram that ⺡ 債 ⺢ (read “⺡ is a subset of ⺢”).
Interval Notation
Sets of real numbers can be expressed using interval notation. For example,
for real numbers a and b such that a ⬍ b, the open interval 共a, b兲 is the set
of real numbers between, but not including, a and b. That is,
(
a
[
a
(a, b)
[a, ∞)
)
b
共a, b兲 苷 兵x 兩 a ⬍ x ⬍ b其.
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 关a, ⬁ 兲, for example, begins at a and extends to the right without
bound. That is,
关a, ⬁ 兲 苷 兵x 兩 x ⱖ a其.
The bracket indicates that a is included in the interval.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4
Chapter R
• Basic Concepts of Algebra
The various types of intervals are listed below.
Intervals: Types, Notation, and Graphs
TYPE
INTERVAL
NOTATION
SET
NOTATION
Open
共a, b兲
兵x 兩 a ⬍ x ⬍ b其
GRAPH
(
)
[
]
[
)
(
]
a
Closed
关a, b兴
兵x 兩 a ⱕ x ⱕ b其
b
a
Half-open
关a, b兲
兵x 兩 a ⱕ x ⬍ b其
b
a
Half-open
共a, b兴
兵x 兩 a ⬍ x ⱕ b其
b
a
Open
共a, ⬁ 兲
兵x 兩 x ⬎ a其
b
(
a
Half-open
关a, ⬁ 兲
兵x 兩 x ⱖ a其
[
a
Open
共⫺⬁, b兲
兵x 兩 x ⬍ b其
)
b
Half-open
共⫺⬁, b兴
兵x 兩 x ⱕ b其
]
b
The interval 共⫺⬁, ⬁ 兲, graphed below, names the set of all real numbers, ⺢.
EXAMPLE 1
Write interval notation for each set and graph the set.
a) 兵x 兩 ⫺4 ⬍ x ⬍ 5其
c) 兵x 兩 ⫺5 ⬍ x ⱕ ⫺2其
b) 兵x 兩 x ⱖ 1.7其
d) 兵 x 兩 x ⬍ 兹5 其
Solution
a) 兵x 兩 ⫺4 ⬍ x ⬍ 5其 苷 共⫺4, 5兲;
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
2
3
4
5
4
5
b) 兵x 兩 x ⱖ 1.7其 苷 关1.7, ⬁ 兲;
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
c) 兵x 兩 ⫺5 ⬍ x ⱕ ⫺2其 苷 共⫺5, ⫺2兴;
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
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4
5
Section R.1
• The Real-Number System
5
d) 兵 x 兩 x ⬍ 兹5 其 苷 共 ⫺⬁, 兹5 兲;
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
5
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:
a ⫹ b 苷 b ⫹ a and
ab 苷 ba
Commutative properties of
addition and multiplication
a ⫹ 共b ⫹ c兲 苷 共a ⫹ b兲 ⫹ c and
a共bc兲 苷 共ab兲c
Associative properties of
addition and multiplication
a⫹0苷0⫹a苷a
Additive identity property
⫺a ⫹ a 苷 a ⫹ 共⫺a兲 苷 0
Additive inverse property
a⭈1苷1⭈a苷a
Multiplicative identity property
a⭈
1
1
苷 ⭈ a 苷 1 共a 苷 0兲
a
a
a共b ⫹ c兲 苷 ab ⫹ ac
Multiplicative inverse property
Distributive property
Note that the distributive property is also true for subtraction since
a共b ⫺ c兲 苷 a关b ⫹ 共⫺c兲兴 苷 ab ⫹ a共⫺c兲 苷 ab ⫺ ac .
EXAMPLE 2
State the property being illustrated in each sentence.
a) 8 ⭈ 5 苷 5 ⭈ 8
c) 14 ⫹ 共⫺14兲 苷 0
e) 2共a ⫺ b兲 苷 2a ⫺ 2b
Solution
SENTENCE
b) 5 ⫹ 共m ⫹ n兲 苷 共5 ⫹ m兲 ⫹ n
d) 6 ⭈ 1 苷 1 ⭈ 6 苷 6
PROPERTY
a) 8 ⭈ 5 苷 5 ⭈ 8
b)
c)
d)
e)
Commutative property of multiplication:
ab 苷 ba
5 ⫹ 共m ⫹ n兲 苷 共5 ⫹ m兲 ⫹ n Associative property of addition:
a ⫹ 共b ⫹ c兲 苷 共a ⫹ b兲 ⫹ c
14 ⫹ 共⫺14兲 苷 0
Additive inverse property: a ⫹ 共⫺a兲 苷 0
6⭈1苷1⭈6苷6
Multiplicative identity property:
a⭈1苷1⭈a苷a
2共a ⫺ b兲 苷 2a ⫺ 2b
Distributive property:
a共b ⫹ c兲 苷 ab ⫹ ac
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
6
Chapter R
• Basic Concepts of Algebra
Absolute Value
The number line can be used to provide a geometric interpretation of
absolute value. The absolute value of a number a, denoted 兩a兩, is its distance from 0 on the number line. For example, 兩⫺5兩 苷 5, because the
3
3
3
distance of ⫺5 from 0 is 5. Similarly, 4 苷 4 , because the distance of 4
3
from 0 is 4 .
ⱍⱍ
Absolute Value
For any real number a,
兩a兩 苷
再
a, if a ⱖ 0,
⫺a, if a ⬍ 0.
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,
兩a兩 is never negative; that is, for any real number a, 兩a兩 ⱖ 0.
Absolute value can be used to find the distance between two points on
the number line.
a
Distance Between Two Points on the Number Line
For any real numbers a and b, the distance between a and b is
兩a ⫺ b兩, or equivalently, 兩b ⫺ a兩.
b
兩a ⫺ b兩 ⫽ 兩b ⫺ a兩
GCM
EXAMPLE 3
Solution
Find the distance between ⫺2 and 3.
The distance is
兩⫺2 ⫺ 3兩 苷 兩⫺5兩 苷 5, or equivalently,
兩3 ⫺ 共⫺2兲兩 苷 兩3 ⫹ 2兩 苷 兩5兩 苷 5.
We can also use the absolute-value operation on a graphing calculator to
find the distance between two points. On many graphing calculators, absolute value is denoted “abs” and is found in the MATH NUM menu and also
in the CATALOG.
abs (⫺2⫺3)
5
abs (3⫺(⫺2))
5
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.1
R.1
Exercise Set
In Exercises 1– 10, consider the numbers ⫺12, 兹7, 5.3,
25. 关x, x ⫹ h兴
[
5
3
⫺ 37 , 兹 8, 0, 5.242242224 . . . , ⫺兹14, 兹 5, ⫺1.96, 9,
3
4 32 ,
• The Real-Number System
5
7.
兹25, 兹 4,
3
1. Which are whole numbers? 兹 8, 0, 9, 兹25
(
⫺12, 兹 8, 0, 9, 兹25
兹7, 5.242242224 . . . ,
3
5
3. Which are irrational numbers? ⫺兹14, 兹
5, 兹 4
3
4. Which are natural numbers? 兹 8, 9, 兹25 7 3
⫺12, 5.3, ⫺ 3 , 兹 8, 0,
5. Which are rational numbers? ⫺1.96, 9, 4 2 , 兹25, 5
3
7
27. 共 p, ⬁ 兲
6. Which are real numbers?
28. 共⫺⬁, q兴
All of them 5.3, ⫺ 7 , ⫺1.96,
3
2 5
7. Which are rational numbers but not integers? 4 3 , 7
8. Which are integers but not whole numbers?
x⫹h
26. 共x, x ⫹ h兴
3
2. Which are integers?
]
x
]
x
x⫹h
(
p
]
q
⫺12
13. 兵x 兩 ⫺4 ⱕ x ⬍ ⫺1其 ⱓ
14. 兵x 兩 1 ⬍ x ⱕ 6其 ⱓ
In Exercises 29–46, the following notation is used:
⺞ 苷 the set of natural numbers, ⺧ 苷 the set of whole
numbers, ⺪ 苷 the set of integers, ⺡ 苷 the set of
rational numbers, ⺙ 苷 the set of irrational numbers, and
⺢ 苷 the set of real numbers. Classify the statement as
true or false.
29. 6 僆 ⺞ True
30. 0 僆 ⺞ True
15. 兵x 兩 x ⱕ ⫺2其 ⱓ
16. 兵x 兩 x ⬎ ⫺5其 ⱓ
18. 兵 x 兩 x ⱖ 兹3 其 ⱓ
31. 3.2 僆 ⺪
17. 兵x 兩 x ⬎ 3.8其 ⱓ
19. 兵x 兩 7 ⬍ x其 ⱓ
20. 兵x 兩 ⫺3 ⬎ x其 ⱓ
9. Which are integers but not natural numbers? ⫺12, 0
10. Which are real numbers but not integers? ⱓ
Write interval notation. Then graph the interval.
11. 兵x 兩 ⫺3 ⱕ x ⱕ 3其 ⱓ
12. 兵x 兩 ⫺4 ⬍ x ⬍ 4其 ⱓ
33. ⫺
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
)
5
37. 24 僆 ⺧
6
22. 关⫺1, 2兴
[
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
]
2
3
4
5
6
23. 关⫺9, ⫺4兲
[
)
⫺10 ⫺9 ⫺8 ⫺7 ⫺6 ⫺5
⫺4 ⫺3 ⫺2 ⫺1
0
1
2
]
⫺10 ⫺9 ⫺8 ⫺7 ⫺6 ⫺5
⫺4 ⫺3 ⫺2 ⫺1
0
1
2
False
False
32. ⫺10.1 僆 ⺢ True
34. ⫺兹6 僆 ⺡ False
36. ⫺1 僆 ⺧
38. 1 僆 ⺪
False
True
39. 1.089 僆 ⺙ True
40. ⺞ 債 ⺧
41. ⺧ 債 ⺪
True
42. ⺪ 債 ⺞ False
43. ⺡ 債 ⺢
True
44. ⺪ 債 ⺡ True
45. ⺢ 債 ⺪
False
46. ⺡ 債 ⺙ False
True
Name the property illustrated by the sentence.
47. 6 ⭈ x 苷 x ⭈ 6 Commutative property of
multiplication
48. 3 ⫹ 共x ⫹ y兲 苷 共3 ⫹ x兲 ⫹ y
of addition
24. 共⫺9, ⫺5兴
(
11
僆 ⺡ True
5
35. 兹11 僆 ⺢
Write interval notation for the graph.
21. 共0, 5兲
(
False
Associative property
49. ⫺3 ⭈ 1 苷 ⫺3
50. x ⫹ 4 苷 4 ⫹ x ⱓ
Multiplicative identity property
51. 5共ab兲 苷 共5a兲b ⱓ
52. 4共 y ⫺ z兲 苷 4y ⫺ 4z
Distributive property
ⱓ Answers to Exercises 10 – 20, 50, and 51 can be found on p. IA-1.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
7
8
Chapter R
• Basic Concepts of Algebra
53. 2共a ⫹ b兲 苷 共a ⫹ b兲2
Additive inverse property
54. ⫺7 ⫹ 7 苷 0
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?
1
苷 1 Multiplicative inverse property
8
80. Under what circumstances is 兹a a rational number?
Commutative property of multiplication
55. ⫺6共m ⫹ n兲 苷 ⫺6共n ⫹ m兲 Commutative property
of addition
56. t ⫹ 0 苷 t Additive identity property
57. 8 ⭈
58. 9x ⫹ 9y 苷 9共x ⫹ y兲 Distributive property
Synthesis
Simplify.
59. 兩⫺7.1兩 7.1
60. 兩⫺86.2兩 86.2
61. 兩347兩 347
62. 兩⫺54兩 54
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.
ⱍ
63. ⫺兹97
ⱍ
兹97
65. 兩0兩 0
67.
兩兩
5
4
64.
兩兩
12
19
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
12
19
66. 兩15兩 15
ⱍ
5
4
68. ⫺兹3
ⱍ
兹3
Find the distance between the given pair of points on
the number line.
69. ⫺5, 6 11
70. ⫺2.5, 0 2.5
15 23 1
,
8 12 24
71. ⫺8, ⫺2 6
72.
73. 6.7, 12.1 5.4
74. ⫺14, ⫺3 11
75. ⫺
3 15
,
4 8
77. ⫺7, 0 7
21
8
Answers may vary; 0.124124412444 . . .
82. A rational number between ⫺兹2.01 and ⫺兹2
Answers may vary; ⫺1.415
83. A rational number between ⫺
Answers may vary; ⫺0.00999
1
1
and ⫺
101
100
84. An irrational number between 兹5.99 and 兹6
Answers may vary; 兹5.995
85. The hypotenuse of an isosceles right triangle with
legs of length 1 unit can be used to “measure” a
value for 兹2 by using the Pythagorean theorem,
as shown.
76. ⫺3.4, 10.2 13.6
78. 3, 19 16
c
1
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
c 2 苷 12 ⫹ 12
c2 苷 2
c 苷 兹2
1
Draw a right triangle that could be used to
“measure” 兹10 units. ⱓ
ⱓ Answer to Exercise 85 can be found on p. IA-1.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2
9
Simplify expressions with integer exponents.
Solve problems using scientific notation.
Use the rules for order of operations.
R.2
Integer
Exponents,
Scientific
Notation, and
Order of
Operations
• Integer Exponents, Scientific Notation, and 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, 73 means 7 ⭈ 7 ⭈ 7 and 51
means 5.
For any positive integer n,
a n 苷 a ⭈ a ⭈ a ⭈ ⭈ ⭈ a,
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,
a0 苷 1 and a⫺m 苷
EXAMPLE 1
a) 6
1
.
am
Simplify each of the following.
b) 共⫺3.4兲0
0
Solution
a) 60 苷 1
b) 共⫺3.4兲0 苷 1
EXAMPLE 2
a) 4⫺5
Write each of the following with positive exponents.
b)
1
共0.82兲⫺7
Solution
a) 4⫺5 苷
1
45
1
苷 共0.82兲⫺共⫺7兲 苷 共0.82兲7
共0.82兲⫺7
x ⫺3
1
1
y8
c) ⫺8 苷 x ⫺3 ⭈ ⫺8 苷 3 ⭈ y 8 苷 3
y
y
x
x
b)
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
c)
x ⫺3
y ⫺8
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⫺m
bn
苷
.
b ⫺n
am
(A factor can be moved to the other side of the fraction bar if the
sign of the exponent is changed.)
Write an equivalent expression without negative exponents:
EXAMPLE 3
x ⫺3y ⫺8
.
z ⫺10
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:
x ⫺3y ⫺8
z 10
苷
.
z ⫺10
x 3y 8
The following properties of exponents can be used to simplify
expressions.
Proper ties of Exponents
For any real numbers a and b and any integers m and n, assuming 0 is
not raised to a nonpositive power:
am ⭈ an 苷 am⫹n
Product rule
m
a
苷 am⫺n 共a 苷 0兲
an
Quotient rule
共am 兲n 苷 amn
Power rule
m m
共ab兲 苷 a b
Raising a product to a power
am
共b 苷 0兲
bm
Raising a quotient to a power
m
冉冊
a
b
m
苷
Simplify each of the following.
EXAMPLE 4
a) y ⫺5 ⭈ y 3
c) 共t ⫺3 兲5
e)
冉
48x 12
16x 4
d) 共2s ⫺2 兲5
b)
冊
45x ⫺4y 2
9z ⫺8
⫺3
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2
• Integer Exponents, Scientific Notation, and Order of Operations
11
Solution
a) y ⫺5 ⭈ y 3 苷 y ⫺5⫹3 苷 y ⫺2, or
b)
1
y2
48x 12 48 12⫺4
苷 x
苷 3x 8
16x 4
16
c) 共t ⫺3 兲5 苷 t ⫺3⭈5 苷 t ⫺15, or
1
t 15
d) 共2s ⫺2 兲5 苷 25共s ⫺2 兲5 苷 32s ⫺10, or
e)
冉
冊 冉 冊
45x ⫺4y 2
9z ⫺8
⫺3
32
s 10
5x ⫺4y 2 ⫺3
z ⫺8
x 12
x 12
5⫺3x 12y ⫺6
苷 3 6 24 , or
苷
24
z
5y z
125y 6z 24
苷
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
N ⫻ 10 m,
where 1 ⱕ N ⬍ 10, 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 苷 1.6539 ⫻ 10 7
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 scientific 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.
0.00000000000000000000000000167 苷 1.67 ⫻ 10 ⫺27
EXAMPLE 7
Convert each of the following to decimal notation.
b) 9.4 ⫻ 10 5
a) 7.632 ⫻ 10 ⫺4
Solution
a) The exponent is negative, so the number is between 0 and 1. We move the
decimal point 4 places to the left.
7.632 ⫻ 10 ⫺4 苷 0.0007632
b) The exponent is positive, so the number is greater than 10. We move the
decimal point 5 places to the right.
9.4 ⫻ 10 5 苷 940,000
Most calculators make use of scientific notation. For example, the number 48,000,000,000,000 might be expressed in one of the ways shown below.
4.8E13
GCM
4.8 13
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 5.88 ⫻ 10 12 miles. How many miles is it
from Earth to Alpha Centauri C? Express your answer in scientific notation.
Solution
4.22 ⫻ 共5.88 ⫻ 10 12 兲 苷 共4.22 ⫻ 5.88兲 ⫻ 10 12
苷 24.8136 ⫻ 10 12
4.22ⴱ5.88E12
2.48136E13
This is not scientific
notation because
24.8136 w 10.
苷 共2.48136 ⫻ 10 1 兲 ⫻ 10 12
苷 2.48136 ⫻ 共10 1 ⫻ 10 12 兲
苷 2.48136 ⫻ 10 13 miles
Writing scientific
notation
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 3 ⫹ 4 ⭈ 5, first we multiply 4 and 5 to
get 20 and then add 3 to get 23. Mathematicians have agreed on the following 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.
GCM
EXAMPLE 9
Calculate each of the following.
a) 8共5 ⫺ 3兲3 ⫺ 20
b)
Solution
a) 8共5 ⫺ 3兲3 ⫺ 20 苷 8 ⭈ 23 ⫺ 20
苷 8 ⭈ 8 ⫺ 20
苷 64 ⫺ 20
苷 44
b)
10 ⫼ 共8 ⫺ 6兲 ⫹ 9 ⭈ 4
25 ⫹ 32
Doing the calculation within
parentheses
Evaluating the exponential expression
Multiplying
Subtracting
10 ⫼ 共8 ⫺ 6兲 ⫹ 9 ⭈ 4 10 ⫼ 2 ⫹ 9 ⭈ 4
苷
25 ⫹ 32
32 ⫹ 9
5 ⫹ 36 41
苷
苷1
苷
41
41
Note that fraction bars act as grouping symbols. That is, the given expression is equivalent to 关10 ⫼ 共8 ⫺ 6兲 ⫹ 9 ⭈ 4兴 ⫼ 共25 ⫹ 32 兲.
We can also enter these computations on a graphing calculator as shown
below.
8(5⫺3)ˆ3⫺20
44
(10/(8⫺6)⫹9ⴱ4)/(2ˆ5⫹32)
1
To confirm that it is essential to include parentheses around the numerator and around the denominator when the computation in Example 9(b) is
entered in a calculator, enter the computation without using these parentheses. What is the result?
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
14
Chapter R
• Basic Concepts of Algebra
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
冉 冊
A苷P 1⫹
r
n
nt
.
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 P 苷 1250, r 苷 4.6%, or 0.046, n 苷 4, and t 苷 8. Substituting, we find that the amount in the account at the end of 8 years is
given by
冉
A 苷 1250 1 ⫹
冊
0.046
4
4⭈8
.
Next, we evaluate this expression:
A 苷 1250共1 ⫹ 0.0115兲4⭈8
苷 1250共1.0115兲4⭈8
苷 1250共1.0115兲32
⬇ 1250共1.441811175兲
⬇ 1802.263969
⬇ 1802.26.
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.
ⱓ Answers to Exercises 15 – 20 can be found on p. IA-1.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
14
Chapter R
• Basic Concepts of Algebra
R.2
Exercise Set
Simplify.
1. 18 0 1
9
3. x ⭈ x
0
2. 共
x
7. m
⭈m
5
兲
0
9
4. a ⭈ a
5. 58 ⭈ 5⫺6 52, or 25
⫺5
19. 共6x ⫺3y 5 兲 共⫺7x 2y ⫺9 兲 ⱓ 20. 共8ab 7 兲 共⫺7a⫺5b 2 兲 ⱓ
0
⫺ 43
1
4
1
a
1
6⫺5, or 5
6
1
6. 62 ⭈ 6⫺7
9
8. n ⭈ n
4
⫺9
1
9. y 3 ⭈ y ⫺7 y ⫺4, or 4
y
1
11. 73 ⭈ 7⫺5 ⭈ 7 7⫺1, or
7
13. 2x 3 ⭈ 3x 2 6x 5
14. 3y 4 ⭈ 4y 3
15. 共⫺3a⫺5 兲 共5a⫺7 兲 ⱓ
16. 共⫺6b ⫺4 兲 共2b ⫺7 兲 ⱓ
17. 共5a2b兲 共3a⫺3b 4 兲 ⱓ
18. 共4xy 2 兲 共3x ⫺4y 5 兲 ⱓ
10. b ⫺4 ⭈ b 12
6
12. 3 ⭈ 3
⫺5
b8
⭈3
4
5
21. 共2x兲3共3x兲2 72x 5
22. 共4y兲2共3y兲3 432y 5
23. 共⫺2n兲3共5n兲2
24. 共2x兲5共3x兲2 288x 7
25.
b 40
b 37
27.
26.
a39
a7
a32
x ⫺5
1
x ⫺21, or 21
x
x 16
28.
y ⫺24
y ⫺21
29.
x 2y ⫺2
x3
x 3y ⫺3, or 3
⫺1
y
x y
30.
x 3y ⫺3
x4
4 ⫺5
⫺1 2 x y , or y 5
x y
31.
32x ⫺4y 3
8x
8xy ⫺5, or 5
⫺5 8
y
4x y
32.
20a5b ⫺2
4b
4a ⫺2b , or 2
7 ⫺3
a
5a b
3
12y 7
⫺200n 5
b3
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
y ⫺3, or
1
y3
Section R.2
• Integer Exponents, Scientific Notation, and Order of Operations
33. 共2ab 2 兲3 8a 3b 6
34. 共4xy 3 兲2 16x 2y 6
35. 共⫺2x 3 兲5 ⫺32x 15
c 2d 4
37. 共⫺5c ⫺1d ⫺2 兲⫺2
25
39. 共3m4 兲3共2m⫺5 兲4 ⱓ
36. 共⫺3x 2 兲4 81x 8
41.
43.
冉 冊
冉 冊
2x ⫺3y 7
z ⫺1
38. 共⫺4x
42.
ⱓ
24a10b ⫺8c 7
12a6b ⫺3c 5
⫺5
44.
ⱓ
x 15z 6
⫺64
兲
⫺5 ⫺2 ⫺3
z
40. 共4n ⫺1 兲2共2n3 兲3
3
69.
128n 7
冉 冊
冉
冊
3x 5y ⫺8
z ⫺2
4
ⱓ
125p12q ⫺14r 22
25p8q 6r ⫺15
⫺4
ⱓ
Convert to scientific notation.
45. 405,000 4.05 ⫻ 10 5
46. 1,670,000 1.67 ⫻ 10 6
47. 0.00000039 3.9 ⫻ 10 ⫺7 48. 0.00092 9.2 ⫻ 10 ⫺4
49. 234,600,000,000
50. 8,904,000,000
2.346 ⫻ 10 11
51. 0.00104 1.04 ⫻ 10 ⫺3
8.904 ⫻ 10 9
52. 0.00000000514
5.14 ⫻ 10 ⫺9
53. One cubic inch is approximately equal to
0.000016 m3. 1.6 ⫻ 10 ⫺5
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).
1.137 ⫻ 10
12
Convert to decimal notation.
55. 8.3 ⫻ 10 ⫺5 0.000083 56. 4.1 ⫻ 10 6 4,100,000
57. 2.07 ⫻ 10 7 20,700,000 58. 3.15 ⫻ 10 ⫺6
0.00000315
59. 3.496 ⫻ 10 10
60. 8.409 ⫻ 10 11
61. 5.41 ⫻ 10 ⫺8
62. 6.27 ⫻ 10 ⫺10
34,960,000,000
840,900,000,000
0.0000000541
0.000000000627
6.4 ⫻ 10 ⫺7
1.1 ⫻ 10 ⫺40
⫺14
8
⫻
10
70.
8.0 ⫻ 10 6
2.0 ⫻ 10 ⫺71
1.8 ⫻ 10 ⫺3
1.3 ⫻ 10 4
5
71.
72.
⫺9 2.5 ⫻ 10
7.2 ⫻ 10
5.2 ⫻ 10 10
15
5.5 ⫻ 10 30
2.5 ⫻ 10 ⫺7
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. 3.627 ⫻ 10 9 mi
74. Parsecs. One parsec is about 3.26 light-years and
1 light-year is about 5.88 ⫻ 10 12 mi. Find the
number of miles in 1 parsec. 1.91688 ⫻ 10 13 mi
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. 3.6 ⫻ 10 ⫺7 m
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? $3.4749 ⫻ 10 7
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. $1.19 ⫻ 10 7
64. The mass of a proton is about 1.67 ⫻ 10 ⫺24 g.
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. 4.3 ⫻ 10 ⫺5 sq mi
Compute. Write the answer using scientific notation.
65. 共3.1 ⫻ 10 5 兲 共4.5 ⫻ 10 ⫺3 兲 1.395 ⫻ 10 3
79. Nuclear Disintegration. One gram of radium
produces 37 billion disintegrations per second. How
many disintegrations are produced in 1 hr?
63. The amount of solid waste generated in the United
States in a recent year was 2.319 ⫻ 10 8 tons (Source:
Franklin Associates, Ltd.). 231,900,000
0.00000000000000000000000167
66. 共9.1 ⫻ 10
⫺17
兲 共8.2 ⫻ 10 兲 7.462 ⫻ 10
67. 共2.6 ⫻ 10
⫺18
兲 共8.5 ⫻ 10 7 兲 2.21 ⫻ 10 ⫺10
3
⫺13
68. 共6.4 ⫻ 10 12 兲 共3.7 ⫻ 10 ⫺5 兲 2.368 ⫻ 10 8
1.332 ⫻ 10 14 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.) 5.8 ⫻ 10 8 mi
ⱓ Answers to Exercises 39 and 41 – 44 can be found on p. IA-1.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
16
Chapter R
• Basic Concepts of Algebra
Calculate.
81. 3 ⭈ 2 ⫺ 4 ⭈ 22 ⫹ 6共3 ⫺ 1兲 2
82. 3关共2 ⫹ 4 ⭈ 22 兲 ⫺ 6共3 ⫺ 1兲兴 18
83. 16 ⫼ 4 ⭈ 4 ⫼ 2 ⭈ 256
6
⫺3
84. 2 ⭈ 2
10
⫼2
⫺8
⫼2
2048
2
2
85.
4共8 ⫺ 6兲 ⫺ 4 ⭈ 3 ⫹ 2 ⭈ 8
31 ⫹ 19 0
86.
关4共8 ⫺ 6兲2 ⫹ 4兴 共3 ⫺ 2 ⭈ 8兲
⫺5
22共23 ⫹ 5兲
5
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
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
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
4 ⭈ 25 ⫼ 共10 ⫺ 5兲? Why or why not?
92. Is x ⫺2 ⬍ x ⫺1 for any negative value(s) of x? Why or
why not?
Synthesis
Savings Plan.
冋
S苷P
The formula
冉 冊
1⫹
册
r 12⭈t
⫺1
12
r
12
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
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2
• Integer Exponents, Scientific Notation, and Order of Operations
Simplify. Assume that all exponents are integers, all
denominators are nonzero, and zero is not raised to a
nonpositive power.
97. 共x t ⭈ x 3t 兲2 x 8t
98. 共x y ⭈ x ⫺y 兲3 1
99. 共t a⫹x ⭈ t x⫺a 兲4
t 8x
101.
冋
册
共3x ay b 兲3
共⫺3x ay b 兲2
2a 2b
2
102.
冋冉 冊 冉 冊 册
9x y
100. 共mx⫺b ⭈ nx⫹b 兲x共mbn⫺b 兲x
2
mx nx
2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
xr
yt
2
x 2r
y 4t
17
⫺2 ⫺3
x 6ry ⫺18t , or
x 6r
y 18t
Section R.3
R.3
Addition,
Subtraction, and
Multiplication of
Polynomials
• Addition, Subtraction, and Multiplication of Polynomials
17
• 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
7
3x ⫺ 4y , 5y 3 ⫺ 3 y 2 ⫹ 3y ⫺ 2, ⫺2.3a4,
and z 6 ⫺ 兹5.
All but the first are polynomials in one variable.
Polynomials in One Variable
A polynomial in one variable is any expression of the type
an x n ⫹ an⫺1x n⫺1 ⫹ ⭈ ⭈ ⭈ ⫹ a2x 2 ⫹ a1x ⫹ a0,
where n is a nonnegative integer and an , . . . , a0 are real numbers,
called coefficients. The parts of a polynomial separated by plus
signs are called terms. The leading coefficient is an , and the
constant term is a 0. If an 苷 0, 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
4
2x ⫺ 7.5x 3 ⫹ x ⫺ 12.
Solution
Writing plus signs between the terms, we have
2x ⫺ 7.5x 3 ⫹ x ⫺ 12 苷 2x 4 ⫹ 共⫺7.5x 3 兲 ⫹ x ⫹ 共⫺12兲,
4
so the terms are
2x 4,
⫺7.5x 3,
x, and ⫺12.
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.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
18
Chapter R
• Basic Concepts of Algebra
EXAMPLE 2
Find the degree of each polynomial.
b) y 2 ⫺ 23 ⫹ 5y 4
3
a) 2x ⫺ 9
Solution
POLYNOMIAL
c) 7
DEGREE
3
a) 2x ⫺ 9
b) y 2 ⫺ 23 ⫹ 5y 4 苷 5y 4 ⫹ y 2 ⫺ 23
c) 7 苷 7x 0
3
4
0
Algebraic expressions like 3ab 3 ⫺ 8 and 5x 4y 2 ⫺ 3x 3y 8 ⫹ 7xy 2 ⫹ 6
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
7ab 3 ⫺ 11a2b 4 ⫹ 8.
Solution The degrees of the terms of 7ab 3 ⫺ 11a2b 4 ⫹ 8 are 4, 6, and 0,
respectively, so the degree of the polynomial is 6.
A polynomial with just one term, like ⫺9y 6, is a monomial. If a polynomial has two terms, like x 2 ⫹ 4, it is a binomial. A polynomial with three
terms, like 4x 2 ⫺ 4xy ⫹ 1, is a trinomial.
Expressions like
2x 2 ⫺ 5x ⫹
3
,
x
9 ⫺ 兹x , and
x⫹1
x4 ⫹ 5
are not polynomials, because they cannot be written in the form an x n ⫹
an⫺1x n⫺1 ⫹ ⭈ ⭈ ⭈ ⫹ a1x ⫹ a 0, where the exponents are all nonnegative integers 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, 3y 2 and
5y 2 are like terms and
3y 2 ⫹ 5y 2 苷 共3 ⫹ 5兲y 2
苷 8y 2.
We add or subtract polynomials by combining like terms.
EXAMPLE 4
Add or subtract each of the following.
a) 共⫺5x ⫹ 3x ⫺ x兲 ⫹ 共12x 3 ⫺ 7x 2 ⫹ 3兲
b) 共6x 2y 3 ⫺ 9xy兲 ⫺ 共5x 2y 3 ⫺ 4xy兲
3
2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.3
• Addition, Subtraction, and Multiplication of Polynomials
Solution
a) 共⫺5x 3 ⫹ 3x 2 ⫺ x兲 ⫹ 共12x 3 ⫺ 7x 2 ⫹ 3兲
苷 共⫺5x 3 ⫹ 12x 3 兲 ⫹ 共3x 2 ⫺ 7x 2 兲 ⫺ x ⫹ 3
苷 共⫺5 ⫹ 12兲x 3 ⫹ 共3 ⫺ 7兲x 2 ⫺ x ⫹ 3
苷 7x 3 ⫺ 4x 2 ⫺ x ⫹ 3
19
Rearranging using
the commutative
and associative
properties
Using the distributive property
b) We can subtract by adding an opposite:
共6x 2y 3 ⫺ 9xy兲 ⫺ 共5x 2y 3 ⫺ 4xy兲
苷 共6x 2y 3 ⫺ 9xy兲 ⫹ 共⫺5x 2y 3 ⫹ 4xy兲
Adding the opposite of
5x2y3 ⴚ 4xy
苷 6x 2y 3 ⫺ 9xy ⫺ 5x 2y 3 ⫹ 4xy
苷 x 2y 3 ⫺ 5xy .
Combining like terms
Multiplication
Multiplication of polynomials is based on the distributive property—for
example,
共x ⫹ 4兲 共x ⫹ 3兲 苷 x共x ⫹ 3兲 ⫹ 4共x ⫹ 3兲
苷 x 2 ⫹ 3x ⫹ 4x ⫹ 12
苷 x 2 ⫹ 7x ⫹ 12.
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
Solution
Multiply: 共4x 4y ⫺ 7x 2y ⫹ 3y兲 共2y ⫺ 3x 2y兲.
We have
共4x y ⫺ 7x 2y ⫹ 3y兲 共2y ⫺ 3x 2y兲
苷 4x 4y共2y ⫺ 3x 2y兲 ⫺ 7x 2y共2y ⫺ 3x 2y兲 ⫹ 3y共2y ⫺ 3x 2y兲
4
Using the distributive
property
苷 8x 4y 2 ⫺ 12x 6y 2 ⫺ 14x 2y 2 ⫹ 21x 4y 2 ⫹ 6y 2 ⫺ 9x 2y 2
苷 29x 4y 2 ⫺ 12x 6y 2 ⫺ 23x 2y 2 ⫹ 6y 2.
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.
4x 4y ⫺ 7x 2y ⫹ 3y
2y ⫺ 3x 2y
⫺12x 6y 2 ⫹ 21x 4y 2 ⫺ 9x 2y 2
8x 4y 2 ⫺ 14x 2y 2 ⫹ 6y 2
⫺12x 6y 2 ⫹ 29x 4y 2 ⫺ 23x 2y 2 ⫹ 6y 2
Multiplying by ⴚ3x2y
Multiplying by 2y
Adding
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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
Solution
Multiply: 共2x ⫺ 7兲 共3x ⫹ 4兲.
We have
F
L
F
O
I
L
共2x ⫺ 7兲 共3x ⫹ 4兲 苷 6x 2 ⫹ 8x ⫺ 21x ⫺ 28
苷 6x 2 ⫺ 13x ⫺ 28
I
O
We can use FOIL to find some special products.
Special Products of Binomials
共A ⫹ B兲2 苷 A2 ⫹ 2AB ⫹ B 2
Square of a sum
共A ⫺ B兲2 苷 A2 ⫺ 2AB ⫹ B 2
Square of a difference
共A ⫹ B兲 共A ⫺ B兲 苷 A ⫺ B
Product of a sum and a difference
2
EXAMPLE 7
a) 共4x ⫹ 1兲2
2
Multiply each of the following.
b) 共3y 2 ⫺ 2兲2
c) 共x 2 ⫹ 3y兲 共x 2 ⫺ 3y兲
Solution
a) 共4x ⫹ 1兲2 苷 共4x兲2 ⫹ 2 ⭈ 4x ⭈ 1 ⫹ 12 苷 16x 2 ⫹ 8x ⫹ 1
b) 共3y 2 ⫺ 2兲2 苷 共3y 2 兲2 ⫺ 2 ⭈ 3y 2 ⭈ 2 ⫹ 22 苷 9y 4 ⫺ 12y 2 ⫹ 4
c) 共x 2 ⫹ 3y兲 共x 2 ⫺ 3y兲 苷 共x 2 兲2 ⫺ 共3y兲2 苷 x 4 ⫺ 9y 2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
20
Chapter R
R.3
• Basic Concepts of Algebra
Exercise Set
Determine the terms and the degree of the polynomial.
1. ⫺5y 4 ⫹ 3y 3 ⫹ 7y 2 ⫺ y ⫺ 4 ⫺5y 4, 3y 3, 7y 2,
⫺y , ⫺4; 4
2. 2m3 ⫺ m2 ⫺ 4m ⫹ 11 2m3, ⫺m2, ⫺4m, 11; 3
3. 3a4b ⫺ 7a3b 3 ⫹ 5ab ⫺ 2
⫺2; 6
3a 4b, ⫺7a 3b 3, 5ab,
4. 6p3q 2 ⫺ p2q 4 ⫺ 3pq 2 ⫹ 5 6p 3q 2, ⫺p 2q 4, ⫺3pq 2,
Perform the operations indicated.
5. 共5x 2y ⫺ 2xy 2 ⫹ 3xy ⫺ 5兲 ⫹
共⫺2x 2y ⫺ 3xy 2 ⫹ 4xy ⫹ 7兲
3x 2y ⫺ 5xy 2 ⫹ 7xy ⫹ 2
6. 共6x 2y ⫺ 3xy 2 ⫹ 5xy ⫺ 3兲 ⫹
共⫺4x 2y ⫺ 4xy 2 ⫹ 3xy ⫹ 8兲
2x 2y ⫺ 7xy 2 ⫹ 8xy ⫹ 5
7. 共2x ⫹ 3y ⫹ z ⫺ 7兲 ⫹ 共4x ⫺ 2y ⫺ z ⫹ 8兲 ⫹
共⫺3x ⫹ y ⫺ 2z ⫺ 4兲 3x ⫹ 2y ⫺ 2z ⫺ 3
5; 6
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.3
• Addition, Subtraction, and Multiplication of Polynomials
34. 共b ⫹ 4兲 共b ⫺ 4兲 b 2 ⫺ 16
8. 共2x 2 ⫹ 12xy ⫺ 11兲 ⫹ 共6x 2 ⫺ 2x ⫹ 4兲 ⫹
共⫺x 2 ⫺ y ⫺ 2兲 7x 2 ⫹ 12xy ⫺ 2x ⫺ y ⫺ 9
35. 共2x ⫺ 5兲 共2x ⫹ 5兲 4x 2 ⫺ 25
9. 共3x ⫺ 2x ⫺ x ⫹ 2兲 ⫺ 共5x ⫺ 8x ⫺ x ⫹ 4兲
2
3
2
3
⫺2x 2 ⫹ 6x ⫺ 2
10. 共5x ⫹ 4xy ⫺ 3y ⫹ 2兲 ⫺ 共9x ⫺ 4xy ⫹ 2y ⫺ 1兲
2
2
2
⫺4x 2 ⫹ 8xy ⫺ 5y 2 ⫹ 3
2
11. 共x ⫺ 3x ⫹ 4x兲 ⫺ 共3x ⫹ x ⫺ 5x ⫹ 3兲
4
2
3
2
x 4 ⫺ 3x 3 ⫺ 4x 2 ⫹ 9x ⫺ 3
2
3
2
2x 4 ⫺ 5x 3 ⫺ 5x 2 ⫹ 10x ⫺ 5
13. 共a ⫺ b兲 共2a ⫺ ab ⫹ 3b 兲
3
3
37. 共3x ⫺ 2y兲 共3x ⫹ 2y兲 9x 2 ⫺ 4y 2
39. 共2x ⫹ 3y ⫹ 4兲 共2x ⫹ 3y ⫺ 4兲
4x 2 ⫹ 12xy ⫹ 9y 2 ⫺ 16
2
4
36. 共4y ⫺ 1兲 共4y ⫹ 1兲 16y 2 ⫺ 1
38. 共3x ⫹ 5y兲 共3x ⫺ 5y兲 9x 2 ⫺ 25y 2
12. 共2x ⫺ 3x ⫹ 7x兲 ⫺ 共5x ⫹ 2x ⫺ 3x ⫹ 5兲
4
2
2
2a ⫺ 2a b ⫺ a b ⫹ 4ab ⫺ 3b
3
40. 共5x ⫹ 2y ⫹ 3兲 共5x ⫹ 2y ⫺ 3兲
25x 2 ⫹ 20xy ⫹ 4y 2 ⫺ 9
14. 共n ⫹ 1兲 共n2 ⫺ 6n ⫺ 4兲 n 3 ⫺ 5n 2 ⫺ 10n ⫺ 4
41. 共x ⫹ 1兲 共x ⫺ 1兲 共x 2 ⫹ 1兲 x 4 ⫺ 1
15. 共x ⫹ 5兲 共x ⫺ 3兲 x 2 ⫹ 2x ⫺ 15
42. 共 y ⫺ 2兲 共 y ⫹ 2兲 共 y 2 ⫹ 4兲
16. 共 y ⫺ 4兲 共 y ⫹ 1兲
21
y 4 ⫺ 16
2
y ⫺ 3y ⫺ 4
17. 共x ⫹ 6兲 共x ⫹ 4兲 x 2 ⫹ 10x ⫹ 24
Collaborative Discussion and Writing
18. 共n ⫺ 5兲 共n ⫺ 8兲 n 2 ⫺ 13n ⫹ 40
43. Is the sum of two polynomials of degree n always a
polynomial of degree n? Why or why not?
19. 共2a ⫹ 3兲 共a ⫹ 5兲 2a 2 ⫹ 13a ⫹ 15
20. 共3b ⫹ 1兲 共b ⫺ 2兲 3b 2 ⫺ 5b ⫺ 2
44. Explain how you would convince a classmate that
共A ⫹ B兲2 苷 A2 ⫹ B 2.
21. 共2x ⫹ 3y兲 共2x ⫹ y兲 4x 2 ⫹ 8xy ⫹ 3y 2
22. 共2a ⫺ 3b兲 共2a ⫺ b兲 4a 2 ⫺ 8ab ⫹ 3b 2
Synthesis
23. 共 y ⫹ 5兲2
y 2 ⫹ 10y ⫹ 25
24. 共 y ⫹ 7兲2
y 2 ⫹ 14y ⫹ 49
Multiply. Assume that all exponents are natural
numbers.
45. 共an ⫹ b n 兲 共an ⫺ b n 兲 a 2n ⫺ b 2n
25. 共x ⫺ 4兲2 x 2 ⫺ 8x ⫹ 16
26. 共a ⫺ 6兲
2
46. 共t a ⫹ 4兲 共t a ⫺ 7兲 t 2a ⫺ 3t a ⫺ 28
2
a ⫺ 12a ⫹ 36
27. 共5x ⫺ 3兲
2
47. 共an ⫹ b n 兲2 a 2n ⫹ 2a nb n ⫹ b 2n
2
25x ⫺ 30x ⫹ 9
28. 共3x ⫺ 2兲
2
48. 共x 3m ⫺ t 5n 兲2
2
9x ⫺ 12x ⫹ 4
29. 共2x ⫹ 3y兲
2
2
4x ⫹ 12xy ⫹ 9y
30. 共5x ⫹ 2y兲
2
2
49. 共x ⫺ 1兲 共x 2 ⫹ x ⫹ 1兲 共x 3 ⫹ 1兲 x 6 ⫺ 1
2
25x ⫹ 20xy ⫹ 4y
50. 关共2x ⫺ 1兲2 ⫺ 1兴 2 16x 4 ⫺ 32x 3 ⫹ 16x 2
2
51. 共x a⫺b 兲a⫹b
31. 共2x ⫺ 3y兲
4x ⫺ 12x y ⫹ 9y
32. 共4x 2 ⫺ 5y兲2
16x 4 ⫺ 40x 2y ⫹ 25y 2
2
2
4
2
x 6m ⫺ 2x 3m t 5n ⫹ t 10n
2
xa
2⫺b2
52. 共t m⫹n 兲m⫹n ⭈ 共t m⫺n 兲m⫺n
t 2m
2⫹2n 2
53. 共a ⫹ b ⫹ c兲2 a 2 ⫹ b 2 ⫹ c 2 ⫹ 2ab ⫹ 2ac ⫹ 2bc
33. 共a ⫹ 3兲 共a ⫺ 3兲 a ⫺ 9
2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
22
Chapter R
• Basic Concepts of Algebra
Factor polynomials by removing a common factor.
Factor polynomials by grouping.
Factor trinomials of the type x 2 ⫹ bx ⫹ c .
Factor trinomials of the type ax 2 ⫹ bx ⫹ c , a 苷 1, using the FOIL
method and the grouping method.
Factor special products of polynomials.
R.4
Factoring
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) 15 ⫹ 10x ⫺ 5x 2
b) 12x 2y 2 ⫺ 20x 3y
Solution
a) 15 ⫹ 10x ⫺ 5x 2 苷 5 ⭈ 3 ⫹ 5 ⭈ 2x ⫺ 5 ⭈ x 2 苷 5共3 ⫹ 2x ⫺ x 2 兲
We can always check a factorization by multiplying:
5共3 ⫹ 2x ⫺ x 2 兲 苷 15 ⫹ 10x ⫺ 5x 2.
b) There are several factors common to the terms of 12x 2y 2 ⫺ 20x 3y , but
4x 2y is the “largest” of these.
12x 2y 2 ⫺ 20x 3y 苷 4x 2y ⭈ 3y ⫺ 4x 2y ⭈ 5x
苷 4x 2y共3y ⫺ 5x兲
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
Solution
Factor: x 3 ⫹ 3x 2 ⫺ 5x ⫺ 15.
We have
x ⫹ 3x ⫺ 5x ⫺ 15 苷 共x 3 ⫹ 3x 2 兲 ⫹ 共⫺5x ⫺ 15兲
3
2
苷 x 2共x ⫹ 3兲 ⫺ 5共x ⫹ 3兲
苷 共x ⫹ 3兲 共x 2 ⫺ 5兲.
Grouping; each
group of terms has
a common factor.
Factoring a common
factor out of each
group
Factoring out the
common binomial
factor
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R. 4
• Factoring
23
Trinomials of the Type x 2 ⴙ bx ⴙ c
Some trinomials can be factored into the product of two binomials. To
factor a trinomial of the form x 2 ⫹ bx ⫹ c , we look for binomial factors
of the form
共x ⫹ p兲 共x ⫹ q兲,
where p ⭈ q 苷 c and p ⫹ q 苷 b. That is, we look for two numbers p and q
whose sum is the coefficient of the middle term of the polynomial, b, and
whose product is the constant term, c.
When we factor any polynomial, we should always check first to determine whether there is a factor common to all the terms. If there is, we factor
it out first.
EXAMPLE 3
Factor: x 2 ⫹ 5x ⫹ 6.
Solution First, we look for a common factor. There is none. Next, we look
for two numbers whose product is 6 and whose sum is 5. Since the constant
term, 6, and the coefficient of the middle term, 5, are both positive, we look
for a factorization of 6 in which both factors are positive.
PAIRS OF FACTORS
SUMS OF FACTORS
1, 6
7
2, 3
5
The numbers we
need are 2 and 3.
The factorization is 共x ⫹ 2兲 共x ⫹ 3兲. We have
x 2 ⫹ 5x ⫹ 6 苷 共x ⫹ 2兲 共x ⫹ 3兲.
We can check this by multiplying:
共x ⫹ 2兲 共x ⫹ 3兲 苷 x 2 ⫹ 3x ⫹ 2x ⫹ 6 苷 x 2 ⫹ 5x ⫹ 6.
EXAMPLE 4
Factor: 2y 2 ⫺ 14y ⫹ 24.
Solution First, we look for a common factor. Each term has a factor of 2,
so we factor it out first:
2y 2 ⫺ 14y ⫹ 24 苷 2共 y 2 ⫺ 7y ⫹ 12兲.
Now we consider the trinomial y 2 ⫺ 7y ⫹ 12. We look for two numbers
whose product is 12 and whose sum is
Chapter R 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
Chapter 1 Graphs, Functions, and Models
1.1 Introduction to Graphing
1.2 Functions and Graphs
1.3 Linear Functions, Slope, and Applications
1.4 Equations of Lines and Modeling
1.5 More on Functions
1.6 The Algebra of Functions
1.7 Symmetry and Transformations
Chapter 2 Functions, Equations, and Inequalities
2.1 Linear Equations, Functions, and Models
2.2 The Complex Numbers
2.3 Quadratic Equations, Functions, and Models
2.4 Analyzing Graphs of Quadratic Functions
2.5 More Equation Solving
2.6 Solving Linear Inequalities
Chapter 3 Polynomial And Rational Functions
3.1 Polynomial Functions and Models
3.2 Graphing Polynomial Functions
3.3 Polynomial Division; The Remainder and Factor Theorems
3.4 Theorems about Zeros of Polynomial Functions
3.5 Rational Functions
3.6 Polynomial and Rational Inequalities
3.7 Variation and Applications
Chapter 4 Exponential and Logarithmic Functions
4.1 Inverse Functions
4.2 Exponential Functions and Graphs
4.3 Logarithmic Functions and Graphs
4.4 Properties of Logarithmic Functions
4.5 Solving Exponential and Logarithmic Equations
4.6 Applications and Models: Growth and Decay, and Compound Interest
Chapter 5 The Trigonometric Functions
5.1 Trigonometric Functions of Acute Angles
5.2 Applications of Right Triangles
5.3 Trigonometric Functions of Any Angle
5.4 Radians, Arc Length, and Angular Speed
5.5 Circular Functions: Graphs and Properties
5.6 Graphs of Transformed Sine and Cosine Functions
Chapter 6 Trigonometric Identities, Inverse Functions, and Equations
6.1 Identities: Pythagorean and Sum and Difference
6.2 Identities: Cofunction, Double‐Angle, and Half‐Angle
6.3 Proving Trigonometric Identities
6.4 Inverses of the Trigonometric Functions
6.5 Solving Trigonometric Equations
Chapter 7 Applications of Trigonometry
7.1 The Law of Sines
7.2 The Law of Cosines
7.3 Complex Numbers: Trigonometric Form
7.4 Polar Coordinates and Graphs
7.5 Vectors and Applications
7.6 Vector Operations
Chapter 8 Systems of Equations and Matrices
8.1 Systems of Equations in Two Variables
8.2 Systems of Equations in Three Variables
8.3 Matrices and Systems of Equations
8.4 Matrix Operations
8.5 Inverses of Matrices
8.6 Determinants and Cramer's Rule
8.7 Systems of Inequalities and Linear Programming
8.8 Partial Fractions
Chapter 9 Analytic Geometry Topics
9.1 The Parabola
9.2 The Circle and the Eclipse
9.3 The Hyperbola
9.4 Nonlinear Systems of Equations and Inequalities
9.5 Rotation of Axes
9.6 Polar Equations of Conics
9.7 Parametric Equations
Chapter 10 Sequences, Series, and Combinatorics
10.1 Sequences and Series
10.2 Arithmetic Sequences and Series
10.3 Geometric Sequences and Series
10.4 Mathematical Induction
10.5 Combinatorics: Permutations
10.6 Combinatorics: Combinations
10.7 The Binomial Theorem
10.8 Probability
Basic Concepts
of Algebra
R.1
R.2
R.3
R.4
R.5
R.6
R.7
The Real-Number System
Integer Exponents, Scientific Notation,
and Order of Operations
Addition, Subtraction, and
Multiplication of Polynomials
Factoring
Rational Expressions
Radical Notation and Rational Exponents
The Basics of Equation Solving
R
SUMMARY AND REVIEW
TEST
A P P L I C A T I O N
G
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.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
2
Chapter R
• Basic Concepts of Algebra
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.
R.1
2.1
The Real-Number
Polynomial
Functions
System
and
Modeling
Real Numbers
In applications of algebraic concepts, we use real numbers to represent
quantities such as distance, time, speed, area, profit, loss, and temperature. Some frequently used sets of real numbers and the relationships
among them are shown below.
Natural numbers
(positive integers):
1, 2, 3, …
Whole numbers:
0, 1, 2, 3, …
Integers:
…, −3, −2, −1, 0,
1, 2, 3, …
Rational
numbers
Rational numbers
that are not integers:
Real
numbers
Negative integers:
−1, −2, −3, …
2 4 19 −7
−,
− −, −−, −−, 8.3,
3 5 −5 8
−
Irrational numbers:
5
Zero: 0
0.56, …
4
√2, p, −√3, √27,
−4.030030003…, …
Numbers that can be expressed in the form p兾q, where p and q are integers and q 苷 0, are rational numbers. Decimal notation for rational
numbers either terminates (ends) or repeats. Each of the following is a
rational number.
a) 0
b) ⫺7
1
苷 0.25
4
5
d) ⫺ 苷 ⫺0.45
11
c)
0
for any nonzero integer a
a
ⴚ7
7
ⴚ7 ⴝ
, or
ⴚ1
1
0ⴝ
Terminating decimal
Repeating decimal
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.1
• The Real-Number System
3
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) 苷 3.1415926535 . . .
There is no repeating block of digits.
22
共 7 and 3.14 are rational approximations of the irrational number . 兲
b) 兹2 苷 1.414213562 . . .
c) ⫺6.12122122212222 . . .
There is no repeating block of digits.
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.
⫺2.9
兹3
⫺E
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
*
p
3
4
5
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
共a ⬍ b兲. Similarly, a is greater than b 共a ⬎ b兲 if a is to the right of b on
the number line. For example, we see from the number line above that
17
⫺2.9 ⬍ ⫺ 53 , because ⫺2.9 is to the left of ⫺ 53 . Also, 4 ⬎ 兹3, because 174
is to the right of 兹3.
The statement a ⱕ b, read “a is less than or equal to b,” is true if either
a ⬍ b is true or a 苷 b is true.
The symbol 僆 is used to indicate that a member, or element, belongs to
a set. Thus if we let ⺡ represent the set of rational numbers, we can see from
the diagram on page 2 that 0.56 僆 ⺡. We can also write 兹2 僆 ⺡ to indicate that 兹2 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 債 is used to denote this.
For instance, if we let ⺢ represent the set of real numbers, we can see from
the diagram that ⺡ 債 ⺢ (read “⺡ is a subset of ⺢”).
Interval Notation
Sets of real numbers can be expressed using interval notation. For example,
for real numbers a and b such that a ⬍ b, the open interval 共a, b兲 is the set
of real numbers between, but not including, a and b. That is,
(
a
[
a
(a, b)
[a, ∞)
)
b
共a, b兲 苷 兵x 兩 a ⬍ x ⬍ b其.
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 关a, ⬁ 兲, for example, begins at a and extends to the right without
bound. That is,
关a, ⬁ 兲 苷 兵x 兩 x ⱖ a其.
The bracket indicates that a is included in the interval.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4
Chapter R
• Basic Concepts of Algebra
The various types of intervals are listed below.
Intervals: Types, Notation, and Graphs
TYPE
INTERVAL
NOTATION
SET
NOTATION
Open
共a, b兲
兵x 兩 a ⬍ x ⬍ b其
GRAPH
(
)
[
]
[
)
(
]
a
Closed
关a, b兴
兵x 兩 a ⱕ x ⱕ b其
b
a
Half-open
关a, b兲
兵x 兩 a ⱕ x ⬍ b其
b
a
Half-open
共a, b兴
兵x 兩 a ⬍ x ⱕ b其
b
a
Open
共a, ⬁ 兲
兵x 兩 x ⬎ a其
b
(
a
Half-open
关a, ⬁ 兲
兵x 兩 x ⱖ a其
[
a
Open
共⫺⬁, b兲
兵x 兩 x ⬍ b其
)
b
Half-open
共⫺⬁, b兴
兵x 兩 x ⱕ b其
]
b
The interval 共⫺⬁, ⬁ 兲, graphed below, names the set of all real numbers, ⺢.
EXAMPLE 1
Write interval notation for each set and graph the set.
a) 兵x 兩 ⫺4 ⬍ x ⬍ 5其
c) 兵x 兩 ⫺5 ⬍ x ⱕ ⫺2其
b) 兵x 兩 x ⱖ 1.7其
d) 兵 x 兩 x ⬍ 兹5 其
Solution
a) 兵x 兩 ⫺4 ⬍ x ⬍ 5其 苷 共⫺4, 5兲;
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
2
3
4
5
4
5
b) 兵x 兩 x ⱖ 1.7其 苷 关1.7, ⬁ 兲;
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
c) 兵x 兩 ⫺5 ⬍ x ⱕ ⫺2其 苷 共⫺5, ⫺2兴;
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
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4
5
Section R.1
• The Real-Number System
5
d) 兵 x 兩 x ⬍ 兹5 其 苷 共 ⫺⬁, 兹5 兲;
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
5
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:
a ⫹ b 苷 b ⫹ a and
ab 苷 ba
Commutative properties of
addition and multiplication
a ⫹ 共b ⫹ c兲 苷 共a ⫹ b兲 ⫹ c and
a共bc兲 苷 共ab兲c
Associative properties of
addition and multiplication
a⫹0苷0⫹a苷a
Additive identity property
⫺a ⫹ a 苷 a ⫹ 共⫺a兲 苷 0
Additive inverse property
a⭈1苷1⭈a苷a
Multiplicative identity property
a⭈
1
1
苷 ⭈ a 苷 1 共a 苷 0兲
a
a
a共b ⫹ c兲 苷 ab ⫹ ac
Multiplicative inverse property
Distributive property
Note that the distributive property is also true for subtraction since
a共b ⫺ c兲 苷 a关b ⫹ 共⫺c兲兴 苷 ab ⫹ a共⫺c兲 苷 ab ⫺ ac .
EXAMPLE 2
State the property being illustrated in each sentence.
a) 8 ⭈ 5 苷 5 ⭈ 8
c) 14 ⫹ 共⫺14兲 苷 0
e) 2共a ⫺ b兲 苷 2a ⫺ 2b
Solution
SENTENCE
b) 5 ⫹ 共m ⫹ n兲 苷 共5 ⫹ m兲 ⫹ n
d) 6 ⭈ 1 苷 1 ⭈ 6 苷 6
PROPERTY
a) 8 ⭈ 5 苷 5 ⭈ 8
b)
c)
d)
e)
Commutative property of multiplication:
ab 苷 ba
5 ⫹ 共m ⫹ n兲 苷 共5 ⫹ m兲 ⫹ n Associative property of addition:
a ⫹ 共b ⫹ c兲 苷 共a ⫹ b兲 ⫹ c
14 ⫹ 共⫺14兲 苷 0
Additive inverse property: a ⫹ 共⫺a兲 苷 0
6⭈1苷1⭈6苷6
Multiplicative identity property:
a⭈1苷1⭈a苷a
2共a ⫺ b兲 苷 2a ⫺ 2b
Distributive property:
a共b ⫹ c兲 苷 ab ⫹ ac
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
6
Chapter R
• Basic Concepts of Algebra
Absolute Value
The number line can be used to provide a geometric interpretation of
absolute value. The absolute value of a number a, denoted 兩a兩, is its distance from 0 on the number line. For example, 兩⫺5兩 苷 5, because the
3
3
3
distance of ⫺5 from 0 is 5. Similarly, 4 苷 4 , because the distance of 4
3
from 0 is 4 .
ⱍⱍ
Absolute Value
For any real number a,
兩a兩 苷
再
a, if a ⱖ 0,
⫺a, if a ⬍ 0.
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,
兩a兩 is never negative; that is, for any real number a, 兩a兩 ⱖ 0.
Absolute value can be used to find the distance between two points on
the number line.
a
Distance Between Two Points on the Number Line
For any real numbers a and b, the distance between a and b is
兩a ⫺ b兩, or equivalently, 兩b ⫺ a兩.
b
兩a ⫺ b兩 ⫽ 兩b ⫺ a兩
GCM
EXAMPLE 3
Solution
Find the distance between ⫺2 and 3.
The distance is
兩⫺2 ⫺ 3兩 苷 兩⫺5兩 苷 5, or equivalently,
兩3 ⫺ 共⫺2兲兩 苷 兩3 ⫹ 2兩 苷 兩5兩 苷 5.
We can also use the absolute-value operation on a graphing calculator to
find the distance between two points. On many graphing calculators, absolute value is denoted “abs” and is found in the MATH NUM menu and also
in the CATALOG.
abs (⫺2⫺3)
5
abs (3⫺(⫺2))
5
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.1
R.1
Exercise Set
In Exercises 1– 10, consider the numbers ⫺12, 兹7, 5.3,
25. 关x, x ⫹ h兴
[
5
3
⫺ 37 , 兹 8, 0, 5.242242224 . . . , ⫺兹14, 兹 5, ⫺1.96, 9,
3
4 32 ,
• The Real-Number System
5
7.
兹25, 兹 4,
3
1. Which are whole numbers? 兹 8, 0, 9, 兹25
(
⫺12, 兹 8, 0, 9, 兹25
兹7, 5.242242224 . . . ,
3
5
3. Which are irrational numbers? ⫺兹14, 兹
5, 兹 4
3
4. Which are natural numbers? 兹 8, 9, 兹25 7 3
⫺12, 5.3, ⫺ 3 , 兹 8, 0,
5. Which are rational numbers? ⫺1.96, 9, 4 2 , 兹25, 5
3
7
27. 共 p, ⬁ 兲
6. Which are real numbers?
28. 共⫺⬁, q兴
All of them 5.3, ⫺ 7 , ⫺1.96,
3
2 5
7. Which are rational numbers but not integers? 4 3 , 7
8. Which are integers but not whole numbers?
x⫹h
26. 共x, x ⫹ h兴
3
2. Which are integers?
]
x
]
x
x⫹h
(
p
]
q
⫺12
13. 兵x 兩 ⫺4 ⱕ x ⬍ ⫺1其 ⱓ
14. 兵x 兩 1 ⬍ x ⱕ 6其 ⱓ
In Exercises 29–46, the following notation is used:
⺞ 苷 the set of natural numbers, ⺧ 苷 the set of whole
numbers, ⺪ 苷 the set of integers, ⺡ 苷 the set of
rational numbers, ⺙ 苷 the set of irrational numbers, and
⺢ 苷 the set of real numbers. Classify the statement as
true or false.
29. 6 僆 ⺞ True
30. 0 僆 ⺞ True
15. 兵x 兩 x ⱕ ⫺2其 ⱓ
16. 兵x 兩 x ⬎ ⫺5其 ⱓ
18. 兵 x 兩 x ⱖ 兹3 其 ⱓ
31. 3.2 僆 ⺪
17. 兵x 兩 x ⬎ 3.8其 ⱓ
19. 兵x 兩 7 ⬍ x其 ⱓ
20. 兵x 兩 ⫺3 ⬎ x其 ⱓ
9. Which are integers but not natural numbers? ⫺12, 0
10. Which are real numbers but not integers? ⱓ
Write interval notation. Then graph the interval.
11. 兵x 兩 ⫺3 ⱕ x ⱕ 3其 ⱓ
12. 兵x 兩 ⫺4 ⬍ x ⬍ 4其 ⱓ
33. ⫺
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
)
5
37. 24 僆 ⺧
6
22. 关⫺1, 2兴
[
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
]
2
3
4
5
6
23. 关⫺9, ⫺4兲
[
)
⫺10 ⫺9 ⫺8 ⫺7 ⫺6 ⫺5
⫺4 ⫺3 ⫺2 ⫺1
0
1
2
]
⫺10 ⫺9 ⫺8 ⫺7 ⫺6 ⫺5
⫺4 ⫺3 ⫺2 ⫺1
0
1
2
False
False
32. ⫺10.1 僆 ⺢ True
34. ⫺兹6 僆 ⺡ False
36. ⫺1 僆 ⺧
38. 1 僆 ⺪
False
True
39. 1.089 僆 ⺙ True
40. ⺞ 債 ⺧
41. ⺧ 債 ⺪
True
42. ⺪ 債 ⺞ False
43. ⺡ 債 ⺢
True
44. ⺪ 債 ⺡ True
45. ⺢ 債 ⺪
False
46. ⺡ 債 ⺙ False
True
Name the property illustrated by the sentence.
47. 6 ⭈ x 苷 x ⭈ 6 Commutative property of
multiplication
48. 3 ⫹ 共x ⫹ y兲 苷 共3 ⫹ x兲 ⫹ y
of addition
24. 共⫺9, ⫺5兴
(
11
僆 ⺡ True
5
35. 兹11 僆 ⺢
Write interval notation for the graph.
21. 共0, 5兲
(
False
Associative property
49. ⫺3 ⭈ 1 苷 ⫺3
50. x ⫹ 4 苷 4 ⫹ x ⱓ
Multiplicative identity property
51. 5共ab兲 苷 共5a兲b ⱓ
52. 4共 y ⫺ z兲 苷 4y ⫺ 4z
Distributive property
ⱓ Answers to Exercises 10 – 20, 50, and 51 can be found on p. IA-1.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
7
8
Chapter R
• Basic Concepts of Algebra
53. 2共a ⫹ b兲 苷 共a ⫹ b兲2
Additive inverse property
54. ⫺7 ⫹ 7 苷 0
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?
1
苷 1 Multiplicative inverse property
8
80. Under what circumstances is 兹a a rational number?
Commutative property of multiplication
55. ⫺6共m ⫹ n兲 苷 ⫺6共n ⫹ m兲 Commutative property
of addition
56. t ⫹ 0 苷 t Additive identity property
57. 8 ⭈
58. 9x ⫹ 9y 苷 9共x ⫹ y兲 Distributive property
Synthesis
Simplify.
59. 兩⫺7.1兩 7.1
60. 兩⫺86.2兩 86.2
61. 兩347兩 347
62. 兩⫺54兩 54
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.
ⱍ
63. ⫺兹97
ⱍ
兹97
65. 兩0兩 0
67.
兩兩
5
4
64.
兩兩
12
19
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
12
19
66. 兩15兩 15
ⱍ
5
4
68. ⫺兹3
ⱍ
兹3
Find the distance between the given pair of points on
the number line.
69. ⫺5, 6 11
70. ⫺2.5, 0 2.5
15 23 1
,
8 12 24
71. ⫺8, ⫺2 6
72.
73. 6.7, 12.1 5.4
74. ⫺14, ⫺3 11
75. ⫺
3 15
,
4 8
77. ⫺7, 0 7
21
8
Answers may vary; 0.124124412444 . . .
82. A rational number between ⫺兹2.01 and ⫺兹2
Answers may vary; ⫺1.415
83. A rational number between ⫺
Answers may vary; ⫺0.00999
1
1
and ⫺
101
100
84. An irrational number between 兹5.99 and 兹6
Answers may vary; 兹5.995
85. The hypotenuse of an isosceles right triangle with
legs of length 1 unit can be used to “measure” a
value for 兹2 by using the Pythagorean theorem,
as shown.
76. ⫺3.4, 10.2 13.6
78. 3, 19 16
c
1
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
c 2 苷 12 ⫹ 12
c2 苷 2
c 苷 兹2
1
Draw a right triangle that could be used to
“measure” 兹10 units. ⱓ
ⱓ Answer to Exercise 85 can be found on p. IA-1.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2
9
Simplify expressions with integer exponents.
Solve problems using scientific notation.
Use the rules for order of operations.
R.2
Integer
Exponents,
Scientific
Notation, and
Order of
Operations
• Integer Exponents, Scientific Notation, and 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, 73 means 7 ⭈ 7 ⭈ 7 and 51
means 5.
For any positive integer n,
a n 苷 a ⭈ a ⭈ a ⭈ ⭈ ⭈ a,
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,
a0 苷 1 and a⫺m 苷
EXAMPLE 1
a) 6
1
.
am
Simplify each of the following.
b) 共⫺3.4兲0
0
Solution
a) 60 苷 1
b) 共⫺3.4兲0 苷 1
EXAMPLE 2
a) 4⫺5
Write each of the following with positive exponents.
b)
1
共0.82兲⫺7
Solution
a) 4⫺5 苷
1
45
1
苷 共0.82兲⫺共⫺7兲 苷 共0.82兲7
共0.82兲⫺7
x ⫺3
1
1
y8
c) ⫺8 苷 x ⫺3 ⭈ ⫺8 苷 3 ⭈ y 8 苷 3
y
y
x
x
b)
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
c)
x ⫺3
y ⫺8
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⫺m
bn
苷
.
b ⫺n
am
(A factor can be moved to the other side of the fraction bar if the
sign of the exponent is changed.)
Write an equivalent expression without negative exponents:
EXAMPLE 3
x ⫺3y ⫺8
.
z ⫺10
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:
x ⫺3y ⫺8
z 10
苷
.
z ⫺10
x 3y 8
The following properties of exponents can be used to simplify
expressions.
Proper ties of Exponents
For any real numbers a and b and any integers m and n, assuming 0 is
not raised to a nonpositive power:
am ⭈ an 苷 am⫹n
Product rule
m
a
苷 am⫺n 共a 苷 0兲
an
Quotient rule
共am 兲n 苷 amn
Power rule
m m
共ab兲 苷 a b
Raising a product to a power
am
共b 苷 0兲
bm
Raising a quotient to a power
m
冉冊
a
b
m
苷
Simplify each of the following.
EXAMPLE 4
a) y ⫺5 ⭈ y 3
c) 共t ⫺3 兲5
e)
冉
48x 12
16x 4
d) 共2s ⫺2 兲5
b)
冊
45x ⫺4y 2
9z ⫺8
⫺3
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2
• Integer Exponents, Scientific Notation, and Order of Operations
11
Solution
a) y ⫺5 ⭈ y 3 苷 y ⫺5⫹3 苷 y ⫺2, or
b)
1
y2
48x 12 48 12⫺4
苷 x
苷 3x 8
16x 4
16
c) 共t ⫺3 兲5 苷 t ⫺3⭈5 苷 t ⫺15, or
1
t 15
d) 共2s ⫺2 兲5 苷 25共s ⫺2 兲5 苷 32s ⫺10, or
e)
冉
冊 冉 冊
45x ⫺4y 2
9z ⫺8
⫺3
32
s 10
5x ⫺4y 2 ⫺3
z ⫺8
x 12
x 12
5⫺3x 12y ⫺6
苷 3 6 24 , or
苷
24
z
5y z
125y 6z 24
苷
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
N ⫻ 10 m,
where 1 ⱕ N ⬍ 10, 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 苷 1.6539 ⫻ 10 7
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 scientific 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.
0.00000000000000000000000000167 苷 1.67 ⫻ 10 ⫺27
EXAMPLE 7
Convert each of the following to decimal notation.
b) 9.4 ⫻ 10 5
a) 7.632 ⫻ 10 ⫺4
Solution
a) The exponent is negative, so the number is between 0 and 1. We move the
decimal point 4 places to the left.
7.632 ⫻ 10 ⫺4 苷 0.0007632
b) The exponent is positive, so the number is greater than 10. We move the
decimal point 5 places to the right.
9.4 ⫻ 10 5 苷 940,000
Most calculators make use of scientific notation. For example, the number 48,000,000,000,000 might be expressed in one of the ways shown below.
4.8E13
GCM
4.8 13
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 5.88 ⫻ 10 12 miles. How many miles is it
from Earth to Alpha Centauri C? Express your answer in scientific notation.
Solution
4.22 ⫻ 共5.88 ⫻ 10 12 兲 苷 共4.22 ⫻ 5.88兲 ⫻ 10 12
苷 24.8136 ⫻ 10 12
4.22ⴱ5.88E12
2.48136E13
This is not scientific
notation because
24.8136 w 10.
苷 共2.48136 ⫻ 10 1 兲 ⫻ 10 12
苷 2.48136 ⫻ 共10 1 ⫻ 10 12 兲
苷 2.48136 ⫻ 10 13 miles
Writing scientific
notation
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 3 ⫹ 4 ⭈ 5, first we multiply 4 and 5 to
get 20 and then add 3 to get 23. Mathematicians have agreed on the following 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.
GCM
EXAMPLE 9
Calculate each of the following.
a) 8共5 ⫺ 3兲3 ⫺ 20
b)
Solution
a) 8共5 ⫺ 3兲3 ⫺ 20 苷 8 ⭈ 23 ⫺ 20
苷 8 ⭈ 8 ⫺ 20
苷 64 ⫺ 20
苷 44
b)
10 ⫼ 共8 ⫺ 6兲 ⫹ 9 ⭈ 4
25 ⫹ 32
Doing the calculation within
parentheses
Evaluating the exponential expression
Multiplying
Subtracting
10 ⫼ 共8 ⫺ 6兲 ⫹ 9 ⭈ 4 10 ⫼ 2 ⫹ 9 ⭈ 4
苷
25 ⫹ 32
32 ⫹ 9
5 ⫹ 36 41
苷
苷1
苷
41
41
Note that fraction bars act as grouping symbols. That is, the given expression is equivalent to 关10 ⫼ 共8 ⫺ 6兲 ⫹ 9 ⭈ 4兴 ⫼ 共25 ⫹ 32 兲.
We can also enter these computations on a graphing calculator as shown
below.
8(5⫺3)ˆ3⫺20
44
(10/(8⫺6)⫹9ⴱ4)/(2ˆ5⫹32)
1
To confirm that it is essential to include parentheses around the numerator and around the denominator when the computation in Example 9(b) is
entered in a calculator, enter the computation without using these parentheses. What is the result?
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
14
Chapter R
• Basic Concepts of Algebra
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
冉 冊
A苷P 1⫹
r
n
nt
.
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 P 苷 1250, r 苷 4.6%, or 0.046, n 苷 4, and t 苷 8. Substituting, we find that the amount in the account at the end of 8 years is
given by
冉
A 苷 1250 1 ⫹
冊
0.046
4
4⭈8
.
Next, we evaluate this expression:
A 苷 1250共1 ⫹ 0.0115兲4⭈8
苷 1250共1.0115兲4⭈8
苷 1250共1.0115兲32
⬇ 1250共1.441811175兲
⬇ 1802.263969
⬇ 1802.26.
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.
ⱓ Answers to Exercises 15 – 20 can be found on p. IA-1.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
14
Chapter R
• Basic Concepts of Algebra
R.2
Exercise Set
Simplify.
1. 18 0 1
9
3. x ⭈ x
0
2. 共
x
7. m
⭈m
5
兲
0
9
4. a ⭈ a
5. 58 ⭈ 5⫺6 52, or 25
⫺5
19. 共6x ⫺3y 5 兲 共⫺7x 2y ⫺9 兲 ⱓ 20. 共8ab 7 兲 共⫺7a⫺5b 2 兲 ⱓ
0
⫺ 43
1
4
1
a
1
6⫺5, or 5
6
1
6. 62 ⭈ 6⫺7
9
8. n ⭈ n
4
⫺9
1
9. y 3 ⭈ y ⫺7 y ⫺4, or 4
y
1
11. 73 ⭈ 7⫺5 ⭈ 7 7⫺1, or
7
13. 2x 3 ⭈ 3x 2 6x 5
14. 3y 4 ⭈ 4y 3
15. 共⫺3a⫺5 兲 共5a⫺7 兲 ⱓ
16. 共⫺6b ⫺4 兲 共2b ⫺7 兲 ⱓ
17. 共5a2b兲 共3a⫺3b 4 兲 ⱓ
18. 共4xy 2 兲 共3x ⫺4y 5 兲 ⱓ
10. b ⫺4 ⭈ b 12
6
12. 3 ⭈ 3
⫺5
b8
⭈3
4
5
21. 共2x兲3共3x兲2 72x 5
22. 共4y兲2共3y兲3 432y 5
23. 共⫺2n兲3共5n兲2
24. 共2x兲5共3x兲2 288x 7
25.
b 40
b 37
27.
26.
a39
a7
a32
x ⫺5
1
x ⫺21, or 21
x
x 16
28.
y ⫺24
y ⫺21
29.
x 2y ⫺2
x3
x 3y ⫺3, or 3
⫺1
y
x y
30.
x 3y ⫺3
x4
4 ⫺5
⫺1 2 x y , or y 5
x y
31.
32x ⫺4y 3
8x
8xy ⫺5, or 5
⫺5 8
y
4x y
32.
20a5b ⫺2
4b
4a ⫺2b , or 2
7 ⫺3
a
5a b
3
12y 7
⫺200n 5
b3
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
y ⫺3, or
1
y3
Section R.2
• Integer Exponents, Scientific Notation, and Order of Operations
33. 共2ab 2 兲3 8a 3b 6
34. 共4xy 3 兲2 16x 2y 6
35. 共⫺2x 3 兲5 ⫺32x 15
c 2d 4
37. 共⫺5c ⫺1d ⫺2 兲⫺2
25
39. 共3m4 兲3共2m⫺5 兲4 ⱓ
36. 共⫺3x 2 兲4 81x 8
41.
43.
冉 冊
冉 冊
2x ⫺3y 7
z ⫺1
38. 共⫺4x
42.
ⱓ
24a10b ⫺8c 7
12a6b ⫺3c 5
⫺5
44.
ⱓ
x 15z 6
⫺64
兲
⫺5 ⫺2 ⫺3
z
40. 共4n ⫺1 兲2共2n3 兲3
3
69.
128n 7
冉 冊
冉
冊
3x 5y ⫺8
z ⫺2
4
ⱓ
125p12q ⫺14r 22
25p8q 6r ⫺15
⫺4
ⱓ
Convert to scientific notation.
45. 405,000 4.05 ⫻ 10 5
46. 1,670,000 1.67 ⫻ 10 6
47. 0.00000039 3.9 ⫻ 10 ⫺7 48. 0.00092 9.2 ⫻ 10 ⫺4
49. 234,600,000,000
50. 8,904,000,000
2.346 ⫻ 10 11
51. 0.00104 1.04 ⫻ 10 ⫺3
8.904 ⫻ 10 9
52. 0.00000000514
5.14 ⫻ 10 ⫺9
53. One cubic inch is approximately equal to
0.000016 m3. 1.6 ⫻ 10 ⫺5
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).
1.137 ⫻ 10
12
Convert to decimal notation.
55. 8.3 ⫻ 10 ⫺5 0.000083 56. 4.1 ⫻ 10 6 4,100,000
57. 2.07 ⫻ 10 7 20,700,000 58. 3.15 ⫻ 10 ⫺6
0.00000315
59. 3.496 ⫻ 10 10
60. 8.409 ⫻ 10 11
61. 5.41 ⫻ 10 ⫺8
62. 6.27 ⫻ 10 ⫺10
34,960,000,000
840,900,000,000
0.0000000541
0.000000000627
6.4 ⫻ 10 ⫺7
1.1 ⫻ 10 ⫺40
⫺14
8
⫻
10
70.
8.0 ⫻ 10 6
2.0 ⫻ 10 ⫺71
1.8 ⫻ 10 ⫺3
1.3 ⫻ 10 4
5
71.
72.
⫺9 2.5 ⫻ 10
7.2 ⫻ 10
5.2 ⫻ 10 10
15
5.5 ⫻ 10 30
2.5 ⫻ 10 ⫺7
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. 3.627 ⫻ 10 9 mi
74. Parsecs. One parsec is about 3.26 light-years and
1 light-year is about 5.88 ⫻ 10 12 mi. Find the
number of miles in 1 parsec. 1.91688 ⫻ 10 13 mi
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. 3.6 ⫻ 10 ⫺7 m
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? $3.4749 ⫻ 10 7
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. $1.19 ⫻ 10 7
64. The mass of a proton is about 1.67 ⫻ 10 ⫺24 g.
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. 4.3 ⫻ 10 ⫺5 sq mi
Compute. Write the answer using scientific notation.
65. 共3.1 ⫻ 10 5 兲 共4.5 ⫻ 10 ⫺3 兲 1.395 ⫻ 10 3
79. Nuclear Disintegration. One gram of radium
produces 37 billion disintegrations per second. How
many disintegrations are produced in 1 hr?
63. The amount of solid waste generated in the United
States in a recent year was 2.319 ⫻ 10 8 tons (Source:
Franklin Associates, Ltd.). 231,900,000
0.00000000000000000000000167
66. 共9.1 ⫻ 10
⫺17
兲 共8.2 ⫻ 10 兲 7.462 ⫻ 10
67. 共2.6 ⫻ 10
⫺18
兲 共8.5 ⫻ 10 7 兲 2.21 ⫻ 10 ⫺10
3
⫺13
68. 共6.4 ⫻ 10 12 兲 共3.7 ⫻ 10 ⫺5 兲 2.368 ⫻ 10 8
1.332 ⫻ 10 14 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.) 5.8 ⫻ 10 8 mi
ⱓ Answers to Exercises 39 and 41 – 44 can be found on p. IA-1.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
16
Chapter R
• Basic Concepts of Algebra
Calculate.
81. 3 ⭈ 2 ⫺ 4 ⭈ 22 ⫹ 6共3 ⫺ 1兲 2
82. 3关共2 ⫹ 4 ⭈ 22 兲 ⫺ 6共3 ⫺ 1兲兴 18
83. 16 ⫼ 4 ⭈ 4 ⫼ 2 ⭈ 256
6
⫺3
84. 2 ⭈ 2
10
⫼2
⫺8
⫼2
2048
2
2
85.
4共8 ⫺ 6兲 ⫺ 4 ⭈ 3 ⫹ 2 ⭈ 8
31 ⫹ 19 0
86.
关4共8 ⫺ 6兲2 ⫹ 4兴 共3 ⫺ 2 ⭈ 8兲
⫺5
22共23 ⫹ 5兲
5
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
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
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
4 ⭈ 25 ⫼ 共10 ⫺ 5兲? Why or why not?
92. Is x ⫺2 ⬍ x ⫺1 for any negative value(s) of x? Why or
why not?
Synthesis
Savings Plan.
冋
S苷P
The formula
冉 冊
1⫹
册
r 12⭈t
⫺1
12
r
12
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
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.2
• Integer Exponents, Scientific Notation, and Order of Operations
Simplify. Assume that all exponents are integers, all
denominators are nonzero, and zero is not raised to a
nonpositive power.
97. 共x t ⭈ x 3t 兲2 x 8t
98. 共x y ⭈ x ⫺y 兲3 1
99. 共t a⫹x ⭈ t x⫺a 兲4
t 8x
101.
冋
册
共3x ay b 兲3
共⫺3x ay b 兲2
2a 2b
2
102.
冋冉 冊 冉 冊 册
9x y
100. 共mx⫺b ⭈ nx⫹b 兲x共mbn⫺b 兲x
2
mx nx
2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
xr
yt
2
x 2r
y 4t
17
⫺2 ⫺3
x 6ry ⫺18t , or
x 6r
y 18t
Section R.3
R.3
Addition,
Subtraction, and
Multiplication of
Polynomials
• Addition, Subtraction, and Multiplication of Polynomials
17
• 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
7
3x ⫺ 4y , 5y 3 ⫺ 3 y 2 ⫹ 3y ⫺ 2, ⫺2.3a4,
and z 6 ⫺ 兹5.
All but the first are polynomials in one variable.
Polynomials in One Variable
A polynomial in one variable is any expression of the type
an x n ⫹ an⫺1x n⫺1 ⫹ ⭈ ⭈ ⭈ ⫹ a2x 2 ⫹ a1x ⫹ a0,
where n is a nonnegative integer and an , . . . , a0 are real numbers,
called coefficients. The parts of a polynomial separated by plus
signs are called terms. The leading coefficient is an , and the
constant term is a 0. If an 苷 0, 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
4
2x ⫺ 7.5x 3 ⫹ x ⫺ 12.
Solution
Writing plus signs between the terms, we have
2x ⫺ 7.5x 3 ⫹ x ⫺ 12 苷 2x 4 ⫹ 共⫺7.5x 3 兲 ⫹ x ⫹ 共⫺12兲,
4
so the terms are
2x 4,
⫺7.5x 3,
x, and ⫺12.
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.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
18
Chapter R
• Basic Concepts of Algebra
EXAMPLE 2
Find the degree of each polynomial.
b) y 2 ⫺ 23 ⫹ 5y 4
3
a) 2x ⫺ 9
Solution
POLYNOMIAL
c) 7
DEGREE
3
a) 2x ⫺ 9
b) y 2 ⫺ 23 ⫹ 5y 4 苷 5y 4 ⫹ y 2 ⫺ 23
c) 7 苷 7x 0
3
4
0
Algebraic expressions like 3ab 3 ⫺ 8 and 5x 4y 2 ⫺ 3x 3y 8 ⫹ 7xy 2 ⫹ 6
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
7ab 3 ⫺ 11a2b 4 ⫹ 8.
Solution The degrees of the terms of 7ab 3 ⫺ 11a2b 4 ⫹ 8 are 4, 6, and 0,
respectively, so the degree of the polynomial is 6.
A polynomial with just one term, like ⫺9y 6, is a monomial. If a polynomial has two terms, like x 2 ⫹ 4, it is a binomial. A polynomial with three
terms, like 4x 2 ⫺ 4xy ⫹ 1, is a trinomial.
Expressions like
2x 2 ⫺ 5x ⫹
3
,
x
9 ⫺ 兹x , and
x⫹1
x4 ⫹ 5
are not polynomials, because they cannot be written in the form an x n ⫹
an⫺1x n⫺1 ⫹ ⭈ ⭈ ⭈ ⫹ a1x ⫹ a 0, where the exponents are all nonnegative integers 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, 3y 2 and
5y 2 are like terms and
3y 2 ⫹ 5y 2 苷 共3 ⫹ 5兲y 2
苷 8y 2.
We add or subtract polynomials by combining like terms.
EXAMPLE 4
Add or subtract each of the following.
a) 共⫺5x ⫹ 3x ⫺ x兲 ⫹ 共12x 3 ⫺ 7x 2 ⫹ 3兲
b) 共6x 2y 3 ⫺ 9xy兲 ⫺ 共5x 2y 3 ⫺ 4xy兲
3
2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.3
• Addition, Subtraction, and Multiplication of Polynomials
Solution
a) 共⫺5x 3 ⫹ 3x 2 ⫺ x兲 ⫹ 共12x 3 ⫺ 7x 2 ⫹ 3兲
苷 共⫺5x 3 ⫹ 12x 3 兲 ⫹ 共3x 2 ⫺ 7x 2 兲 ⫺ x ⫹ 3
苷 共⫺5 ⫹ 12兲x 3 ⫹ 共3 ⫺ 7兲x 2 ⫺ x ⫹ 3
苷 7x 3 ⫺ 4x 2 ⫺ x ⫹ 3
19
Rearranging using
the commutative
and associative
properties
Using the distributive property
b) We can subtract by adding an opposite:
共6x 2y 3 ⫺ 9xy兲 ⫺ 共5x 2y 3 ⫺ 4xy兲
苷 共6x 2y 3 ⫺ 9xy兲 ⫹ 共⫺5x 2y 3 ⫹ 4xy兲
Adding the opposite of
5x2y3 ⴚ 4xy
苷 6x 2y 3 ⫺ 9xy ⫺ 5x 2y 3 ⫹ 4xy
苷 x 2y 3 ⫺ 5xy .
Combining like terms
Multiplication
Multiplication of polynomials is based on the distributive property—for
example,
共x ⫹ 4兲 共x ⫹ 3兲 苷 x共x ⫹ 3兲 ⫹ 4共x ⫹ 3兲
苷 x 2 ⫹ 3x ⫹ 4x ⫹ 12
苷 x 2 ⫹ 7x ⫹ 12.
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
Solution
Multiply: 共4x 4y ⫺ 7x 2y ⫹ 3y兲 共2y ⫺ 3x 2y兲.
We have
共4x y ⫺ 7x 2y ⫹ 3y兲 共2y ⫺ 3x 2y兲
苷 4x 4y共2y ⫺ 3x 2y兲 ⫺ 7x 2y共2y ⫺ 3x 2y兲 ⫹ 3y共2y ⫺ 3x 2y兲
4
Using the distributive
property
苷 8x 4y 2 ⫺ 12x 6y 2 ⫺ 14x 2y 2 ⫹ 21x 4y 2 ⫹ 6y 2 ⫺ 9x 2y 2
苷 29x 4y 2 ⫺ 12x 6y 2 ⫺ 23x 2y 2 ⫹ 6y 2.
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.
4x 4y ⫺ 7x 2y ⫹ 3y
2y ⫺ 3x 2y
⫺12x 6y 2 ⫹ 21x 4y 2 ⫺ 9x 2y 2
8x 4y 2 ⫺ 14x 2y 2 ⫹ 6y 2
⫺12x 6y 2 ⫹ 29x 4y 2 ⫺ 23x 2y 2 ⫹ 6y 2
Multiplying by ⴚ3x2y
Multiplying by 2y
Adding
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
Solution
Multiply: 共2x ⫺ 7兲 共3x ⫹ 4兲.
We have
F
L
F
O
I
L
共2x ⫺ 7兲 共3x ⫹ 4兲 苷 6x 2 ⫹ 8x ⫺ 21x ⫺ 28
苷 6x 2 ⫺ 13x ⫺ 28
I
O
We can use FOIL to find some special products.
Special Products of Binomials
共A ⫹ B兲2 苷 A2 ⫹ 2AB ⫹ B 2
Square of a sum
共A ⫺ B兲2 苷 A2 ⫺ 2AB ⫹ B 2
Square of a difference
共A ⫹ B兲 共A ⫺ B兲 苷 A ⫺ B
Product of a sum and a difference
2
EXAMPLE 7
a) 共4x ⫹ 1兲2
2
Multiply each of the following.
b) 共3y 2 ⫺ 2兲2
c) 共x 2 ⫹ 3y兲 共x 2 ⫺ 3y兲
Solution
a) 共4x ⫹ 1兲2 苷 共4x兲2 ⫹ 2 ⭈ 4x ⭈ 1 ⫹ 12 苷 16x 2 ⫹ 8x ⫹ 1
b) 共3y 2 ⫺ 2兲2 苷 共3y 2 兲2 ⫺ 2 ⭈ 3y 2 ⭈ 2 ⫹ 22 苷 9y 4 ⫺ 12y 2 ⫹ 4
c) 共x 2 ⫹ 3y兲 共x 2 ⫺ 3y兲 苷 共x 2 兲2 ⫺ 共3y兲2 苷 x 4 ⫺ 9y 2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
20
Chapter R
R.3
• Basic Concepts of Algebra
Exercise Set
Determine the terms and the degree of the polynomial.
1. ⫺5y 4 ⫹ 3y 3 ⫹ 7y 2 ⫺ y ⫺ 4 ⫺5y 4, 3y 3, 7y 2,
⫺y , ⫺4; 4
2. 2m3 ⫺ m2 ⫺ 4m ⫹ 11 2m3, ⫺m2, ⫺4m, 11; 3
3. 3a4b ⫺ 7a3b 3 ⫹ 5ab ⫺ 2
⫺2; 6
3a 4b, ⫺7a 3b 3, 5ab,
4. 6p3q 2 ⫺ p2q 4 ⫺ 3pq 2 ⫹ 5 6p 3q 2, ⫺p 2q 4, ⫺3pq 2,
Perform the operations indicated.
5. 共5x 2y ⫺ 2xy 2 ⫹ 3xy ⫺ 5兲 ⫹
共⫺2x 2y ⫺ 3xy 2 ⫹ 4xy ⫹ 7兲
3x 2y ⫺ 5xy 2 ⫹ 7xy ⫹ 2
6. 共6x 2y ⫺ 3xy 2 ⫹ 5xy ⫺ 3兲 ⫹
共⫺4x 2y ⫺ 4xy 2 ⫹ 3xy ⫹ 8兲
2x 2y ⫺ 7xy 2 ⫹ 8xy ⫹ 5
7. 共2x ⫹ 3y ⫹ z ⫺ 7兲 ⫹ 共4x ⫺ 2y ⫺ z ⫹ 8兲 ⫹
共⫺3x ⫹ y ⫺ 2z ⫺ 4兲 3x ⫹ 2y ⫺ 2z ⫺ 3
5; 6
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R.3
• Addition, Subtraction, and Multiplication of Polynomials
34. 共b ⫹ 4兲 共b ⫺ 4兲 b 2 ⫺ 16
8. 共2x 2 ⫹ 12xy ⫺ 11兲 ⫹ 共6x 2 ⫺ 2x ⫹ 4兲 ⫹
共⫺x 2 ⫺ y ⫺ 2兲 7x 2 ⫹ 12xy ⫺ 2x ⫺ y ⫺ 9
35. 共2x ⫺ 5兲 共2x ⫹ 5兲 4x 2 ⫺ 25
9. 共3x ⫺ 2x ⫺ x ⫹ 2兲 ⫺ 共5x ⫺ 8x ⫺ x ⫹ 4兲
2
3
2
3
⫺2x 2 ⫹ 6x ⫺ 2
10. 共5x ⫹ 4xy ⫺ 3y ⫹ 2兲 ⫺ 共9x ⫺ 4xy ⫹ 2y ⫺ 1兲
2
2
2
⫺4x 2 ⫹ 8xy ⫺ 5y 2 ⫹ 3
2
11. 共x ⫺ 3x ⫹ 4x兲 ⫺ 共3x ⫹ x ⫺ 5x ⫹ 3兲
4
2
3
2
x 4 ⫺ 3x 3 ⫺ 4x 2 ⫹ 9x ⫺ 3
2
3
2
2x 4 ⫺ 5x 3 ⫺ 5x 2 ⫹ 10x ⫺ 5
13. 共a ⫺ b兲 共2a ⫺ ab ⫹ 3b 兲
3
3
37. 共3x ⫺ 2y兲 共3x ⫹ 2y兲 9x 2 ⫺ 4y 2
39. 共2x ⫹ 3y ⫹ 4兲 共2x ⫹ 3y ⫺ 4兲
4x 2 ⫹ 12xy ⫹ 9y 2 ⫺ 16
2
4
36. 共4y ⫺ 1兲 共4y ⫹ 1兲 16y 2 ⫺ 1
38. 共3x ⫹ 5y兲 共3x ⫺ 5y兲 9x 2 ⫺ 25y 2
12. 共2x ⫺ 3x ⫹ 7x兲 ⫺ 共5x ⫹ 2x ⫺ 3x ⫹ 5兲
4
2
2
2a ⫺ 2a b ⫺ a b ⫹ 4ab ⫺ 3b
3
40. 共5x ⫹ 2y ⫹ 3兲 共5x ⫹ 2y ⫺ 3兲
25x 2 ⫹ 20xy ⫹ 4y 2 ⫺ 9
14. 共n ⫹ 1兲 共n2 ⫺ 6n ⫺ 4兲 n 3 ⫺ 5n 2 ⫺ 10n ⫺ 4
41. 共x ⫹ 1兲 共x ⫺ 1兲 共x 2 ⫹ 1兲 x 4 ⫺ 1
15. 共x ⫹ 5兲 共x ⫺ 3兲 x 2 ⫹ 2x ⫺ 15
42. 共 y ⫺ 2兲 共 y ⫹ 2兲 共 y 2 ⫹ 4兲
16. 共 y ⫺ 4兲 共 y ⫹ 1兲
21
y 4 ⫺ 16
2
y ⫺ 3y ⫺ 4
17. 共x ⫹ 6兲 共x ⫹ 4兲 x 2 ⫹ 10x ⫹ 24
Collaborative Discussion and Writing
18. 共n ⫺ 5兲 共n ⫺ 8兲 n 2 ⫺ 13n ⫹ 40
43. Is the sum of two polynomials of degree n always a
polynomial of degree n? Why or why not?
19. 共2a ⫹ 3兲 共a ⫹ 5兲 2a 2 ⫹ 13a ⫹ 15
20. 共3b ⫹ 1兲 共b ⫺ 2兲 3b 2 ⫺ 5b ⫺ 2
44. Explain how you would convince a classmate that
共A ⫹ B兲2 苷 A2 ⫹ B 2.
21. 共2x ⫹ 3y兲 共2x ⫹ y兲 4x 2 ⫹ 8xy ⫹ 3y 2
22. 共2a ⫺ 3b兲 共2a ⫺ b兲 4a 2 ⫺ 8ab ⫹ 3b 2
Synthesis
23. 共 y ⫹ 5兲2
y 2 ⫹ 10y ⫹ 25
24. 共 y ⫹ 7兲2
y 2 ⫹ 14y ⫹ 49
Multiply. Assume that all exponents are natural
numbers.
45. 共an ⫹ b n 兲 共an ⫺ b n 兲 a 2n ⫺ b 2n
25. 共x ⫺ 4兲2 x 2 ⫺ 8x ⫹ 16
26. 共a ⫺ 6兲
2
46. 共t a ⫹ 4兲 共t a ⫺ 7兲 t 2a ⫺ 3t a ⫺ 28
2
a ⫺ 12a ⫹ 36
27. 共5x ⫺ 3兲
2
47. 共an ⫹ b n 兲2 a 2n ⫹ 2a nb n ⫹ b 2n
2
25x ⫺ 30x ⫹ 9
28. 共3x ⫺ 2兲
2
48. 共x 3m ⫺ t 5n 兲2
2
9x ⫺ 12x ⫹ 4
29. 共2x ⫹ 3y兲
2
2
4x ⫹ 12xy ⫹ 9y
30. 共5x ⫹ 2y兲
2
2
49. 共x ⫺ 1兲 共x 2 ⫹ x ⫹ 1兲 共x 3 ⫹ 1兲 x 6 ⫺ 1
2
25x ⫹ 20xy ⫹ 4y
50. 关共2x ⫺ 1兲2 ⫺ 1兴 2 16x 4 ⫺ 32x 3 ⫹ 16x 2
2
51. 共x a⫺b 兲a⫹b
31. 共2x ⫺ 3y兲
4x ⫺ 12x y ⫹ 9y
32. 共4x 2 ⫺ 5y兲2
16x 4 ⫺ 40x 2y ⫹ 25y 2
2
2
4
2
x 6m ⫺ 2x 3m t 5n ⫹ t 10n
2
xa
2⫺b2
52. 共t m⫹n 兲m⫹n ⭈ 共t m⫺n 兲m⫺n
t 2m
2⫹2n 2
53. 共a ⫹ b ⫹ c兲2 a 2 ⫹ b 2 ⫹ c 2 ⫹ 2ab ⫹ 2ac ⫹ 2bc
33. 共a ⫹ 3兲 共a ⫺ 3兲 a ⫺ 9
2
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
22
Chapter R
• Basic Concepts of Algebra
Factor polynomials by removing a common factor.
Factor polynomials by grouping.
Factor trinomials of the type x 2 ⫹ bx ⫹ c .
Factor trinomials of the type ax 2 ⫹ bx ⫹ c , a 苷 1, using the FOIL
method and the grouping method.
Factor special products of polynomials.
R.4
Factoring
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) 15 ⫹ 10x ⫺ 5x 2
b) 12x 2y 2 ⫺ 20x 3y
Solution
a) 15 ⫹ 10x ⫺ 5x 2 苷 5 ⭈ 3 ⫹ 5 ⭈ 2x ⫺ 5 ⭈ x 2 苷 5共3 ⫹ 2x ⫺ x 2 兲
We can always check a factorization by multiplying:
5共3 ⫹ 2x ⫺ x 2 兲 苷 15 ⫹ 10x ⫺ 5x 2.
b) There are several factors common to the terms of 12x 2y 2 ⫺ 20x 3y , but
4x 2y is the “largest” of these.
12x 2y 2 ⫺ 20x 3y 苷 4x 2y ⭈ 3y ⫺ 4x 2y ⭈ 5x
苷 4x 2y共3y ⫺ 5x兲
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
Solution
Factor: x 3 ⫹ 3x 2 ⫺ 5x ⫺ 15.
We have
x ⫹ 3x ⫺ 5x ⫺ 15 苷 共x 3 ⫹ 3x 2 兲 ⫹ 共⫺5x ⫺ 15兲
3
2
苷 x 2共x ⫹ 3兲 ⫺ 5共x ⫹ 3兲
苷 共x ⫹ 3兲 共x 2 ⫺ 5兲.
Grouping; each
group of terms has
a common factor.
Factoring a common
factor out of each
group
Factoring out the
common binomial
factor
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Section R. 4
• Factoring
23
Trinomials of the Type x 2 ⴙ bx ⴙ c
Some trinomials can be factored into the product of two binomials. To
factor a trinomial of the form x 2 ⫹ bx ⫹ c , we look for binomial factors
of the form
共x ⫹ p兲 共x ⫹ q兲,
where p ⭈ q 苷 c and p ⫹ q 苷 b. That is, we look for two numbers p and q
whose sum is the coefficient of the middle term of the polynomial, b, and
whose product is the constant term, c.
When we factor any polynomial, we should always check first to determine whether there is a factor common to all the terms. If there is, we factor
it out first.
EXAMPLE 3
Factor: x 2 ⫹ 5x ⫹ 6.
Solution First, we look for a common factor. There is none. Next, we look
for two numbers whose product is 6 and whose sum is 5. Since the constant
term, 6, and the coefficient of the middle term, 5, are both positive, we look
for a factorization of 6 in which both factors are positive.
PAIRS OF FACTORS
SUMS OF FACTORS
1, 6
7
2, 3
5
The numbers we
need are 2 and 3.
The factorization is 共x ⫹ 2兲 共x ⫹ 3兲. We have
x 2 ⫹ 5x ⫹ 6 苷 共x ⫹ 2兲 共x ⫹ 3兲.
We can check this by multiplying:
共x ⫹ 2兲 共x ⫹ 3兲 苷 x 2 ⫹ 3x ⫹ 2x ⫹ 6 苷 x 2 ⫹ 5x ⫹ 6.
EXAMPLE 4
Factor: 2y 2 ⫺ 14y ⫹ 24.
Solution First, we look for a common factor. Each term has a factor of 2,
so we factor it out first:
2y 2 ⫺ 14y ⫹ 24 苷 2共 y 2 ⫺ 7y ⫹ 12兲.
Now we consider the trinomial y 2 ⫺ 7y ⫹ 12. We look for two numbers
whose product is 12 and whose sum is