Thông tin tài liệu
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, ∞)
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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
͒
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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.
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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
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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
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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
[...]... 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... 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 BBEPMC0R_0312279093.QXP 14 Chapter R • 12/2/04 2:42 PM Page 14 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... 13-week period? $3.4749 ϫ 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 ϫ 10 7 64 The mass of a proton is about 1.67 ϫ 10 Ϫ24 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... 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... 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... checked by multiplying Some trials show that the desired factorization is ͑3x ϩ 2͒ ͑x Ϫ 4͒ The Grouping Method The second method for factoring trinomials of the type ax 2 ϩ bx ϩ c , a 1, is known as the grouping method, or the ac-method Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 26 Chapter R 12/2/04 2:42 PM Page 26 • Basic Concepts of Algebra. .. FACTORS Ϫ1, Ϫ12 Ϫ13 Ϫ2, Ϫ6 Ϫ8 Ϫ3, Ϫ4 Ϫ7 The numbers we need are ؊3 and ؊4 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 24 Chapter R 12/2/04 2:42 PM Page 24 • Basic Concepts of Algebra The factorization of y 2 Ϫ 7y ϩ 12 is ͑ y Ϫ 3͒ ͑ y Ϫ 4͒ We must also include the common factor that we factored out earlier Thus we have 2y 2 Ϫ 14y ϩ 24 2͑ y Ϫ 3͒... Ϫ1, SUMS OF FACTORS 8 7 1, Ϫ8 Ϫ7 Ϫ2, 4 2 2, Ϫ4 Ϫ2 The numbers we need are 2 and ؊4 We might have observed at the outset that since the sum of the factors is Ϫ2, a negative number, we need consider only pairs of factors for which the negative factor has the greater absolute value Thus only the pairs 1, Ϫ8 and 2, Ϫ4 need have been considered Using the pair of factors 2 and Ϫ4, we see that the factorization... Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 20 Chapter R • 12/2/04 2:42 PM Page 20 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... 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 . 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. 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. 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
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