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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.

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• 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 p͞q, where p and q are integers and q 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) Ϫ7
1
෇ 0.25
4
5
d) Ϫ ෇ Ϫ0.45
11
c)

0
for any nonzero integer a
a
؊7
7
؊7 ‫؍‬
, or
؊1
1

0‫؍‬

Terminating decimal
Repeating decimal

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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.
There is no repeating block of digits.
a) ␲ ෇ 3.1415926535 . . .
22
͑ 7 and 3.14 are rational approximations of the irrational number ␲. ͒

b) ͙2 ෇ 1.414213562 . . .
c) Ϫ6.12122122212222 . . .

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.
Ϫ2.9

ϪE

Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

0

͙3
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
͑a Ͻ b͒. Similarly, a is greater than b ͑a Ͼ b͒ if a is to the right of b on
the number line. For example, we see from the number line above that
17

Ϫ2.9 Ͻ Ϫ 35 , because Ϫ2.9 is to the left of Ϫ 35 . Also, 4 Ͼ ͙3, because 174
is to the right of ͙3.
The statement a Յ b, read “a is less than or equal to b,” is true if either
a Ͻ b is true or a ෇ b is true.
The symbol ʦ is used to indicate that a member, or element, belongs to
a set. Thus if we let ‫ ޑ‬represent the set of rational numbers, we can see from
the diagram on page 2 that 0.56 ʦ ‫ޑ‬. We can also write ͙2 ‫ ޑ‬to indicate that ͙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 ʕ is used to denote this.
For instance, if we let ‫ ޒ‬represent the set of real numbers, we can see from
the diagram that ‫ ޑ‬ʕ ‫( ޒ‬read “‫ ޑ‬is a subset of ‫)”ޒ‬.

Interval Notation
Sets of real numbers can be expressed using interval notation. For example,
for real numbers a and b such that a Ͻ b, the open interval ͑a, b͒ is the set
of real numbers between, but not including, a and b. That is,

(
a

[
a

)
(a, b)

[a, ∞)

b


͑a, b͒ ෇ ͕x ͉ a Ͻ x Ͻ b͖.
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 ͓a, ϱ ͒, for example, begins at a and extends to the right without
bound. That is,
͓a, ϱ ͒ ෇ ͕x ͉ x Ն a͖.
The bracket indicates that a is included in the interval.

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• Basic Concepts of Algebra

The various types of intervals are listed below.

Intervals: Types, Notation, and Graphs
TYPE


INTERVAL
NOTATION

SET
NOTATION

Open

͑a, b͒

͕x ͉ a Ͻ x Ͻ b͖

Closed
Half-open
Half-open
Open

͓a, b͔
͓a, b͒
͑a, b͔
͑a, ϱ ͒

GRAPH

͕x ͉ a Յ x Յ b͖
͕x ͉ a Յ x Ͻ b͖
͕x ͉ a Ͻ x Յ b͖
͕x ͉ x Ͼ a͖


(

)

a

b

[

]

a

b

[

)

a

b

(

]

a


b

(
a

Half-open

͓a, ϱ ͒

͕x ͉ x Ն a͖

[
a

Open

͑Ϫϱ, b͒

͕x ͉ x Ͻ b͖

)
b

Half-open

͑Ϫϱ, b͔

͕x ͉ x Յ b͖

]

b

The interval ͑Ϫϱ, ϱ ͒, graphed below, names the set of all real numbers, ‫ޒ‬.

EXAMPLE 1

Write interval notation for each set and graph the set.

a) ͕x ͉ Ϫ4 Ͻ x Ͻ 5͖
c) ͕x ͉ Ϫ5 Ͻ x Յ Ϫ2͖

b) ͕x ͉ x Ն 1.7͖
d) ͕ x ͉ x Ͻ ͙5 ͖

Solution
a) ͕x ͉ Ϫ4 Ͻ x Ͻ 5͖ ෇ ͑Ϫ4, 5͒;
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

0

1

2

3

2

3


4

5

4

5

b) ͕x ͉ x Ն 1.7͖ ෇ ͓1.7, ϱ ͒;
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

0

1

c) ͕x ͉ Ϫ5 Ͻ x Յ Ϫ2͖ ෇ ͑Ϫ5, Ϫ2͔;
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

0

1

2

3

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Section R.1

d) ͕ x ͉ x Ͻ ͙5 ͖ ෇ ͑ Ϫϱ, ͙5 ͒;

• The Real-Number System

Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

0

1

2

3

4

5


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 ϩ b ෇ b ϩ a and
ab ෇ ba

Commutative properties of
addition and multiplication

a ϩ ͑b ϩ c͒ ෇ ͑a ϩ b͒ ϩ c and
a͑bc͒ ෇ ͑ab͒c

Associative properties of
addition and multiplication

aϩ0෇0ϩa෇a

Additive identity property

Ϫa ϩ a ෇ a ϩ ͑Ϫa͒ ෇ 0

Additive inverse property

aи1෇1иa෇a


Multiplicative identity property

aи

1
1
෇ и a ෇ 1 ͑a
a
a

a͑b ϩ c͒ ෇ ab ϩ ac

0͒

Multiplicative inverse property
Distributive property

Note that the distributive property is also true for subtraction since
a͑b Ϫ c͒ ෇ a͓b ϩ ͑Ϫc͔͒ ෇ ab ϩ a͑Ϫc͒ ෇ ab Ϫ ac .
EXAMPLE 2

State the property being illustrated in each sentence.

a) 8 и 5 ෇ 5 и 8
c) 14 ϩ ͑Ϫ14͒ ෇ 0
e) 2͑a Ϫ b͒ ෇ 2a Ϫ 2b
Solution
SENTENCE


b) 5 ϩ ͑m ϩ n͒ ෇ ͑5 ϩ m͒ ϩ n
d) 6 и 1 ෇ 1 и 6 ෇ 6

PROPERTY

a) 8 и 5 ෇ 5 и 8
b)
c)
d)
e)

Commutative property of multiplication:
ab ෇ ba
5 ϩ ͑m ϩ n͒ ෇ ͑5 ϩ m͒ ϩ n Associative property of addition:
a ϩ ͑b ϩ c͒ ෇ ͑a ϩ b͒ ϩ c
14 ϩ ͑Ϫ14͒ ෇ 0
Additive inverse property: a ϩ ͑Ϫa͒ ෇ 0
6и1෇1и6෇6
Multiplicative identity property:
aи1෇1иa෇a
Distributive property:
2͑a Ϫ b͒ ෇ 2a Ϫ 2b
a͑b ϩ c͒ ෇ ab ϩ ac

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• 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 ͉a͉, is its distance from 0 on the number line. For example, ͉Ϫ5͉ ෇ 5, because the
3
3
3
distance of Ϫ5 from 0 is 5. Similarly, 4 ෇ 4 , because the distance of 4
3
from 0 is 4 .

ԽԽ

Absolute Value
For any real number a,
͉a͉ ෇

ͭ

a, if a Ն 0,

Ϫa, if a Ͻ 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,
͉a͉ is never negative; that is, for any real number a, ͉a͉ Ն 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
͉a Ϫ b͉, or equivalently, ͉b Ϫ a͉.

b

͉a Ϫ b͉ ϭ ͉b Ϫ a͉

GCM

EXAMPLE 3
Solution

Find the distance between Ϫ2 and 3.

The distance is

͉Ϫ2 Ϫ 3͉ ෇ ͉Ϫ5͉ ෇ 5, or equivalently,
͉3 Ϫ ͑Ϫ2͉͒ ෇ ͉3 ϩ 2͉ ෇ ͉5͉ ෇ 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 (Ϫ2Ϫ3)
abs (3Ϫ(Ϫ2))

5
5

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Section R.1

R.1

The Real-Number System

Exercise Set

In Exercises 1– 10, consider the numbers Ϫ12, ͙7, 5.3,
3


25. ͓x, x ϩ h͔

5

[

Ϫ 73 , ͙ 8, 0, 5.242242224 . . . , Ϫ͙14, ͙ 5, Ϫ1.96, 9,
3

4 23 ,

•

5
7.

͙25, ͙ 4,
3
1. Which are whole numbers? ͙ 8, 0, 9, ͙25

26. ͑x, x ϩ h͔

3

(

Ϫ12, ͙ 8, 0, 9, ͙25
͙7, 5.242242224 . . . ,
5

3
3. Which are irrational numbers? Ϫ͙14, ͙
5, ͙ 4
3
4. Which are natural numbers? ͙ 8, 9, ͙25 7 3
Ϫ12, 5.3, Ϫ 3 , ͙ 8, 0,
5. Which are rational numbers? Ϫ1.96, 9, 4 2 , ͙25, 5
3
7

27. ͑ p, ϱ ͒

6. Which are real numbers?

28. ͑Ϫϱ, q͔

2. Which are integers?

All of them 5.3, Ϫ 7 , Ϫ1.96,
3
2 5
7. Which are rational numbers but not integers? 4 3 , 7

8. Which are integers but not whole numbers?

]

xϩh

x


]

xϩh

x

(
p

]
q

Ϫ12

13. ͕x ͉ Ϫ4 Յ x Ͻ Ϫ1͖ Ճ

14. ͕x ͉ 1 Ͻ x Յ 6͖ Ճ

In Exercises 29–46, the following notation is used:
‫ ෇ ގ‬the set of natural numbers, ‫ ෇ ޗ‬the set of whole
numbers, ‫ ෇ ޚ‬the set of integers, ‫ ෇ ޑ‬the set of
rational numbers, ‫ ෇ މ‬the set of irrational numbers, and
‫ ෇ ޒ‬the set of real numbers. Classify the statement as
true or false.
29. 6 ʦ ‫ ގ‬True
30. 0 ‫ ގ‬True

15. ͕x ͉ x Յ Ϫ2͖ Ճ


16. ͕x ͉ x Ͼ Ϫ5͖ Ճ
18. ͕ x ͉ x Ն ͙3 ͖ Ճ

31. 3.2 ʦ ‫ޚ‬

17. ͕x ͉ x Ͼ 3.8͖ Ճ
19. ͕x ͉ 7 Ͻ x͖ Ճ

20. ͕x ͉ Ϫ3 Ͼ x͖ Ճ

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. ͕x ͉ Ϫ3 Յ x Յ 3͖ Ճ
12. ͕x ͉ Ϫ4 Ͻ x Ͻ 4͖ Ճ

33. Ϫ

Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

(

0

1

2

3


4

)

5

37. 24
6

22. ͓Ϫ1, 2͔

[

Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

0

1

]

2

3

4

5


6

23. ͓Ϫ9, Ϫ4͒

[

Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5

)

Ϫ4 Ϫ3 Ϫ2 Ϫ1

0

1

2

‫ ޒ‬False
‫ޗ‬

39. 1.089

]

Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5

Ϫ4 Ϫ3 Ϫ2 Ϫ1

0


1

2

False

‫ މ‬True

32. Ϫ10.1 ʦ ‫ ޒ‬True
34. Ϫ͙6 ʦ ‫ ޑ‬False
36. Ϫ1 ʦ ‫ ޗ‬False
38. 1 ʦ ‫ޚ‬

True

40. ‫ ގ‬ʕ ‫ ޗ‬True

41. ‫ ޗ‬ʕ ‫ ޚ‬True

42. ‫ ޚ‬ʕ ‫ ގ‬False

43. ‫ ޑ‬ʕ ‫ ޒ‬True

44. ‫ ޚ‬ʕ ‫ ޑ‬True

45. ‫ ޒ‬ʕ ‫ ޚ‬False

46. ‫ ޑ‬ʕ ‫ މ‬False


Name the property illustrated by the sentence.
47. 6 и x ෇ x и 6 Commutative property of

multiplication

48. 3 ϩ ͑x ϩ y͒ ෇ ͑3 ϩ x͒ ϩ y

of addition

24. ͑Ϫ9, Ϫ5͔

(

11
ʦ ‫ ޑ‬True
5

35. ͙11

Write interval notation for the graph.
21. ͑0, 5͒

False

Associative property

49. Ϫ3 и 1 ෇ Ϫ3
50. x ϩ 4 ෇ 4 ϩ x Ճ
Multiplicative identity property
51. 5͑ab͒ ෇ ͑5a͒b Ճ

52. 4͑ y Ϫ z͒ ෇ 4y Ϫ 4z
Distributive property

Ճ Answers to Exercises 10 – 20, 50, and 51 can be found on p. IA-1.

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Basic Concepts of Algebra

53. 2͑a ϩ b͒ ෇ ͑a ϩ b͒2

Additive inverse property


54. Ϫ7 ϩ 7 ෇ 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
෇ 1 Multiplicative inverse property
8

80. Under what circumstances is ͙a a rational number?

Commutative property of multiplication
55. Ϫ6͑m ϩ n͒ ෇ Ϫ6͑n ϩ m͒ Commutative property
of addition
56. t ϩ 0 ෇ t Additive identity property

57. 8 и

58. 9x ϩ 9y ෇ 9͑x ϩ y͒ Distributive property

Synthesis

Simplify.
59. ͉Ϫ7.1͉ 7.1

60. ͉Ϫ86.2͉ 86.2

61. ͉347͉ 347


62. ͉Ϫ54͉ 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.

Խ

63. Ϫ͙97

Խ

͙97

65. ͉0͉ 0
67.

͉͉
5
4

64.

͉͉
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. ͉15͉ 15
5
4

Խ

68. Ϫ͙3

Խ

͙3

Find the distance between the given pair of points on
the number line.
69. Ϫ5, 6 11
70. Ϫ2.5, 0 2.5
15 23 1
,
8 12 24

71. Ϫ8, Ϫ2 6

72.


73. 6.7, 12.1 5.4

74. Ϫ14, Ϫ3 11

75. Ϫ

3 15 21
,
4 8 8

77. Ϫ7, 0 7

Answers may vary; 0.124124412444 . . .

82. A rational number between Ϫ͙2.01 and Ϫ͙2

Answers may vary; Ϫ1.415

83. A rational number between Ϫ

Answers may vary; Ϫ0.00999

1
1
and Ϫ
101
100

84. An irrational number between ͙5.99 and ͙6


Answers may vary; ͙5.995

85. The hypotenuse of an isosceles right triangle with
legs of length 1 unit can be used to “measure” a
value for ͙2 by using the Pythagorean theorem,
as shown.

76. Ϫ3.4, 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 ෇ 12 ϩ 12
c2 ෇ 2
c ෇ ͙2

1

Draw a right triangle that could be used to

“measure” ͙10 units. Ճ

Ճ Answer to Exercise 85 can be found on p. IA-1.

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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 и 7 и 7 and 51
means 5.

For any positive integer n,








a n ෇ a и a и a и и и 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 ෇ 1 and aϪm ෇

EXAMPLE 1
a) 6

1

.
am

Simplify each of the following.
b) ͑Ϫ3.4͒0

0

Solution
a) 60 ෇ 1

b) ͑Ϫ3.4͒0 ෇ 1

EXAMPLE 2
a) 4Ϫ5

Write each of the following with positive exponents.
b)

1
͑0.82͒Ϫ7

Solution
a) 4Ϫ5 ෇

1
45

1
෇ ͑0.82͒Ϫ͑Ϫ7͒ ෇ ͑0.82͒7

͑0.82͒Ϫ7
x Ϫ3
1
1
y8
c) Ϫ8 ෇ x Ϫ3 и Ϫ8 ෇ 3 и y 8 ෇ 3
y
y
x
x
b)

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c)

x Ϫ3
y Ϫ8


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• 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,
aϪm
bn
.
෇
b Ϫn
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 Ϫ3y Ϫ8
.
z Ϫ10
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 Ϫ3y Ϫ8
z 10
෇
.
z Ϫ10
x 3y 8

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:
am и an ෇ amϩn

Product rule

m

a
෇ amϪn ͑a
an

Quotient rule

͑am ͒n ෇ amn

Power rule

͑ab͒ ෇ a b

Raising a product to a power

m

ͩͪ
a

b

m

m m

෇

am
͑b
bm

a) y Ϫ5 и y 3

Raising a quotient to a power

48x 12
16x 4
d) ͑2s Ϫ2 ͒5

b)

c) ͑t Ϫ3 ͒5

ͩ

0͒

Simplify each of the following.


EXAMPLE 4

e)

0͒

ͪ

45x Ϫ4y 2
9z Ϫ8

Ϫ3

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Section R.2

• Integer Exponents, Scientific Notation, and Order of Operations

11


Solution
a) y Ϫ5 и y 3 ෇ y Ϫ5ϩ3 ෇ y Ϫ2, or
b)

1
y2

48x 12 48 12Ϫ4
෇ x
෇ 3x 8
16x 4
16

c) ͑t Ϫ3 ͒5 ෇ t Ϫ3и5 ෇ t Ϫ15, or

1
t 15

d) ͑2s Ϫ2 ͒5 ෇ 25͑s Ϫ2 ͒5 ෇ 32s Ϫ10, or
e)

ͩ

ͪ ͩ ͪ

45x Ϫ4y 2
9z Ϫ8

Ϫ3


32
s 10

5x Ϫ4y 2 Ϫ3
z Ϫ8
x 12
5Ϫ3x 12y Ϫ6
x 12
෇
෇ 3 6 24 , or
24
z
5y z
125y 6z 24
෇

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 ϫ 10 m,
where 1 Յ N Ͻ 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 ෇ 1.6539 ϫ 10 7

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• 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 ෇ 1.67 ϫ 10 Ϫ27

EXAMPLE 7

Convert each of the following to decimal notation.

a) 7.632 ϫ 10 Ϫ4

b) 9.4 ϫ 10 5

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 ϫ 10 Ϫ4 ෇ 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 ϫ 10 5 ෇ 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 ϫ 10 12 miles. How many miles is it
from Earth to Alpha Centauri C? Express your answer in scientific notation.
Solution
4.22 ϫ ͑5.88 ϫ 10 12 ͒ ෇ ͑4.22 ϫ 5.88͒ ϫ 10 12
෇ 24.8136 ϫ 10 12


4.22‫ء‬5.88E12
2.48136E13

This is not scientific
notation because
24.8136 w 10.

෇ ͑2.48136 ϫ 10 1 ͒ ϫ 10 12
෇ 2.48136 ϫ ͑10 1 ϫ 10 12 ͒
෇ 2.48136 ϫ 10 13 miles
Writing scientific
notation

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Section R.2

• Integer Exponents, Scientific Notation, and Order of Operations

13


Order of Operations
Recall that to simplify the expression 3 ϩ 4 и 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) 8͑5 Ϫ 3͒3 Ϫ 20

b)

Solution
a) 8͑5 Ϫ 3͒3 Ϫ 20 ෇ 8 и 23 Ϫ 20
෇ 8 и 8 Ϫ 20
෇ 64 Ϫ 20
෇ 44
b)

10 Ϭ ͑8 Ϫ 6͒ ϩ 9 и 4

25 ϩ 32

Doing the calculation within
parentheses
Evaluating the exponential expression
Multiplying
Subtracting

10 Ϭ ͑8 Ϫ 6͒ ϩ 9 и 4 10 Ϭ 2 ϩ 9 и 4
෇
25 ϩ 32
32 ϩ 9
5 ϩ 36 41
෇
෇1
෇
41
41
Note that fraction bars act as grouping symbols. That is, the given expression is equivalent to ͓10 Ϭ ͑8 Ϫ 6͒ ϩ 9 и 4͔ Ϭ ͑25 ϩ 32 ͒.

We can also enter these computations on a graphing calculator as shown
below.
8(5Ϫ3)ˆ3Ϫ20
44
(10/(8Ϫ6)ϩ9‫ء‬4)/(2ˆ5ϩ32)
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?


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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

ͩ ͪ

A෇P 1ϩ

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 ෇ 1250, r ෇ 4.6%, or 0.046, n ෇ 4, and t ෇ 8. Substituting, we find that the amount in the account at the end of 8 years is
given by

ͩ

A ෇ 1250 1 ϩ

ͪ

0.046
4

4и8

.

Next, we evaluate this expression:
A ෇ 1250͑1 ϩ 0.0115͒4и8
෇ 1250͑1.0115͒4и8
෇ 1250͑1.0115͒32
Ϸ 1250͑1.441811175͒
Ϸ 1802.263969
Ϸ 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.

Ճ Answers to Exercises 15 – 20 can be found on p. IA-1.

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Basic Concepts of Algebra


R.2
Exercise Set
Simplify.
1. 18 0 1
3. x и x
9

0

2. ͑
x

4. a и a
0

9

5. 58 и 5Ϫ6 52, or 25
7. m

Ϫ5

иm

5

͒

0
Ϫ 43


1

4

19. ͑6x Ϫ3y 5 ͒ ͑Ϫ7x 2y Ϫ9 ͒ Ճ 20. ͑8ab 7 ͒ ͑Ϫ7aϪ5b 2 ͒ Ճ
1
a

4

1
6Ϫ5, or 5
6
1

6. 62 и 6Ϫ7
8. n и n
9

Ϫ9

1
9. y 3 и y Ϫ7 y Ϫ4, or 4
y
1
11. 73 и 7Ϫ5 и 7 7Ϫ1, or
7
13. 2x 3 и 3x 2 6x 5


14. 3y 4 и 4y 3 12y 7

15. ͑Ϫ3aϪ5 ͒ ͑5aϪ7 ͒ Ճ

16. ͑Ϫ6b Ϫ4 ͒ ͑2b Ϫ7 ͒ Ճ

17. ͑5a2b͒ ͑3aϪ3b 4 ͒ Ճ

18. ͑4xy 2 ͒ ͑3x Ϫ4y 5 ͒ Ճ

10. b Ϫ4 и b 12
Ϫ5

12. 3 и 3
6

b8

и3

4

5

21. ͑2x͒3͑3x͒2 72x 5

22. ͑4y͒2͑3y͒3 432y 5

23. ͑Ϫ2n͒3͑5n͒2


24. ͑2x͒5͑3x͒2 288x 7

25.

b 40
b 37

27.

Ϫ200n 5

26.

a39
a7
a32

x Ϫ5
1
x Ϫ21, or 21
x
x 16

28.

y Ϫ24
y Ϫ21

29.


x 2y Ϫ2
x3
x 3y Ϫ3, or 3
Ϫ1
y
x y

30.

x 3y Ϫ3
x4
4 Ϫ5
Ϫ1 2 x y , or y 5
x y

31.

32x Ϫ4y 3
4x Ϫ5y 8

32.

20a5b Ϫ2
4b
4a Ϫ2b , or 2
7 Ϫ3
a
5a b

3


b3

8xy Ϫ5, or

8x
y5

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Ճ Answers to Exercises 15 – 20 can be found on p. IA-1.

y Ϫ3, or

1
y3


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Section R.2

33. ͑2ab 2 ͒3
37. ͑Ϫ5c


36. ͑Ϫ3x 2 ͒4

Ϫ32x 15

Ϫ1 Ϫ2 Ϫ2

͒

d

c 2d 4
25

38. ͑Ϫ4x

39. ͑3m4 ͒3͑2mϪ5 ͒4 Ճ
41.
43.

ͩ ͪ
ͩ ͪ
2x Ϫ3y 7
z Ϫ1

Ճ
Ϫ5

69.


81x 8

x 15z 6
Ϫ64

Ϫ5 Ϫ2 Ϫ3

͒

z

40. ͑4nϪ1 ͒2͑2n3 ͒3 128n 7

3

24a10b Ϫ8c 7
12a6b Ϫ3c 5

Integer Exponents, Scientific Notation, and Order of Operations

34. ͑4xy 3 ͒2 16x 2y 6

8a 3b 6

35. ͑Ϫ2x 3 ͒5

•

42.
Ճ


44.

ͩ ͪ
ͩ
ͪ
3x 5y Ϫ8
z Ϫ2

4

Ճ

125p12q Ϫ14r 22
25p8q 6r Ϫ15

Ϫ4

Ճ

Convert to scientific notation.
45. 405,000 4.05 ϫ 10 5
46. 1,670,000 1.67 ϫ 10 6
47. 0.00000039 3.9 ϫ 10 Ϫ7 48. 0.00092 9.2 ϫ 10 Ϫ4
49. 234,600,000,000

50. 8,904,000,000

2.346 ϫ 10 11
51. 0.00104 1.04 ϫ 10 Ϫ3


8.904 ϫ 10 9

52. 0.00000000514
5.14 ϫ 10 Ϫ9

53. One cubic inch is approximately equal to
0.000016 m3. 1.6 ϫ 10 Ϫ5

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 ϫ 10

12

Convert to decimal notation.
55. 8.3 ϫ 10 Ϫ5 0.000083 56. 4.1 ϫ 10 6 4,100,000
57. 2.07 ϫ 10 7
59. 3.496 ϫ 10 10

20,700,000 58. 3.15 ϫ 10 Ϫ6
0.00000315

60. 8.409 ϫ 10 11

34,960,000,000

840,900,000,000


61. 5.41 ϫ 10 Ϫ8

62. 6.27 ϫ 10 Ϫ10

0.0000000541

0.000000000627

6.4 ϫ 10 Ϫ7
1.1 ϫ 10 Ϫ40
Ϫ14
8
ϫ
10
70.
8.0 ϫ 10 6
2.0 ϫ 10 Ϫ71

1.8 ϫ 10 Ϫ3
1.3 ϫ 10 4
5
71.
72.
Ϫ9 2.5 ϫ 10
7.2 ϫ 10
5.2 ϫ 10 10

15

5.5 ϫ 10 30


2.5 ϫ 10 Ϫ7

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 ϫ 10 9 mi
74. Parsecs. One parsec is about 3.26 light-years and
1 light-year is about 5.88 ϫ 10 12 mi. Find the
number of miles in 1 parsec. 1.91688 ϫ 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 ϫ 10 Ϫ7 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 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 in 2050.
Find the number of square miles of land per person
in 2050. 4.3 ϫ 10 Ϫ5 sq mi

Compute. Write the answer using scientific notation.
65. ͑3.1 ϫ 10 5 ͒ ͑4.5 ϫ 10 Ϫ3 ͒ 1.395 ϫ 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 ϫ 10 8 tons (Source:
Franklin Associates, Ltd.). 231,900,000
0.00000000000000000000000167

66. ͑9.1 ϫ 10

Ϫ17

͒ ͑8.2 ϫ 10 ͒ 7.462 ϫ 10

67. ͑2.6 ϫ 10

Ϫ18

͒ ͑8.5 ϫ 10 7 ͒ 2.21 ϫ 10 Ϫ10


3

Ϫ13

68. ͑6.4 ϫ 10 12 ͒ ͑3.7 ϫ 10 Ϫ5 ͒ 2.368 ϫ 10 8

1.332 ϫ 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 ϫ 10 8 mi

Ճ Answers to Exercises 39 and 41 – 44 can be found on p. IA-1.

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Basic Concepts of Algebra

Calculate.
81. 3 и 2 Ϫ 4 и 22 ϩ 6͑3 Ϫ 1͒ 2
82. 3͓͑2 ϩ 4 и 22 ͒ Ϫ 6͑3 Ϫ 1͔͒ 18
83. 16 Ϭ 4 и 4 Ϭ 2 и 256
84. 2 и 2
6

Ϫ3

Ϭ2

10

Ϫ8

Ϭ2

2048
2

85.

4͑8 Ϫ 6͒ Ϫ 4 и 3 ϩ 2 и 8
5
31 ϩ 19 0


86.

͓4͑8 Ϫ 6͒2 ϩ 4͔ ͑3 Ϫ 2 и 8͒
Ϫ5
22͑23 ϩ 5͒

2

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 и 25 Ϭ ͑10 Ϫ 5͒? Why or why not?
92. Is x Ϫ2 Ͻ x Ϫ1 for any negative value(s) of x? Why or
why not?

Synthesis
Savings Plan.

ͫ

S෇P

The formula


ͩ ͪ
1ϩ

ͬ

r 12иt
Ϫ1
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

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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. ͑x t и x 3t ͒2 x 8t
98. ͑x y и x Ϫy ͒3 1
99. ͑t aϩx и t xϪa ͒4

t 8x

101.

ͫ

ͬ

͑3x ay b ͒3
͑Ϫ3x ay b ͒2
2a 2b

9x y

2

102.


ͫͩ ͪ ͩ ͪ ͬ

100. ͑mxϪb и nxϩb ͒x͑mbnϪb ͒x
2

mx nx

2

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xr
yt

2

x 2r
y 4t

17

Ϫ2 Ϫ3

x 6ry Ϫ18t , or

x 6r
y 18t



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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
3x Ϫ 4y , 5y 3 Ϫ 73 y 2 ϩ 3y Ϫ 2, Ϫ2.3a4,


and z 6 Ϫ ͙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 ϩ anϪ1x nϪ1 ϩ и и и ϩ a2x 2 ϩ a1x ϩ 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 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

2x Ϫ 7.5x 3 ϩ x Ϫ 12.
4

Solution

Writing plus signs between the terms, we have

2x Ϫ 7.5x 3 ϩ x Ϫ 12 ෇ 2x 4 ϩ ͑Ϫ7.5x 3 ͒ ϩ x ϩ ͑Ϫ12͒,
4

so the terms are
2x 4,


Ϫ7.5x 3,

x, and Ϫ12.

A polynomial, like 23, consisting of only a nonzero constant term has
degree 0. It is agreed that the polynomial consisting only of 0 has no degree.

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• Basic Concepts of Algebra

EXAMPLE 2

Find the degree of each polynomial.
3
b) y 2 Ϫ 2 ϩ 5y 4


a) 2x Ϫ 9
3

Solution
POLYNOMIAL

c) 7
DEGREE

a) 2x Ϫ 9
3
3
b) y 2 Ϫ 2 ϩ 5y 4 ෇ 5y 4 ϩ y 2 Ϫ 2
c) 7 ෇ 7x 0
3

3
4
0

Algebraic expressions like 3ab 3 Ϫ 8 and 5x 4y 2 Ϫ 3x 3y 8 ϩ 7xy 2 ϩ 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 Ϫ 11a2b 4 ϩ 8.
Solution The degrees of the terms of 7ab 3 Ϫ 11a2b 4 ϩ 8 are 4, 6, and 0,

respectively, so the degree of the polynomial is 6.
A polynomial with just one term, like Ϫ9y 6, is a monomial. If a polynomial has two terms, like x 2 ϩ 4, it is a binomial. A polynomial with three
terms, like 4x 2 Ϫ 4xy ϩ 1, is a trinomial.
Expressions like
2x 2 Ϫ 5x ϩ

3
,
x

9 Ϫ ͙x , and

xϩ1
x4 ϩ 5

are not polynomials, because they cannot be written in the form an x n ϩ
anϪ1x nϪ1 ϩ и и и ϩ a1x ϩ 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 ϩ 5y 2 ෇ ͑3 ϩ 5͒y 2
෇ 8y 2.
We add or subtract polynomials by combining like terms.
EXAMPLE 4

Add or subtract each of the following.


a) ͑Ϫ5x ϩ 3x Ϫ x͒ ϩ ͑12x 3 Ϫ 7x 2 ϩ 3͒
b) ͑6x 2y 3 Ϫ 9xy͒ Ϫ ͑5x 2y 3 Ϫ 4xy͒
3

2

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Section R.3

• Addition, Subtraction, and Multiplication of Polynomials

Solution
a) ͑Ϫ5x 3 ϩ 3x 2 Ϫ x͒ ϩ ͑12x 3 Ϫ 7x 2 ϩ 3͒
෇ ͑Ϫ5x 3 ϩ 12x 3 ͒ ϩ ͑3x 2 Ϫ 7x 2 ͒ Ϫ x ϩ 3

෇ ͑Ϫ5 ϩ 12͒x 3 ϩ ͑3 Ϫ 7͒x 2 Ϫ x ϩ 3
෇ 7x 3 Ϫ 4x 2 Ϫ x ϩ 3

19


Rearranging using
the commutative
and associative
properties
Using the distributive property

b) We can subtract by adding an opposite:
͑6x 2y 3 Ϫ 9xy͒ Ϫ ͑5x 2y 3 Ϫ 4xy͒
෇ ͑6x 2y 3 Ϫ 9xy͒ ϩ ͑Ϫ5x 2y 3 ϩ 4xy͒

Adding the opposite of
5x2y3 ؊ 4xy

෇ 6x 2y 3 Ϫ 9xy Ϫ 5x 2y 3 ϩ 4xy
෇ x 2y 3 Ϫ 5xy .

Combining like terms

Multiplication
Multiplication of polynomials is based on the distributive property—for
example,
͑x ϩ 4͒ ͑x ϩ 3͒ ෇ x͑x ϩ 3͒ ϩ 4͑x ϩ 3͒
෇ x 2 ϩ 3x ϩ 4x ϩ 12
෇ x 2 ϩ 7x ϩ 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: ͑4x 4y Ϫ 7x 2y ϩ 3y͒ ͑2y Ϫ 3x 2y͒.

We have

͑4x y Ϫ 7x 2y ϩ 3y͒ ͑2y Ϫ 3x 2y͒
෇ 4x 4y͑2y Ϫ 3x 2y͒ Ϫ 7x 2y͑2y Ϫ 3x 2y͒ ϩ 3y͑2y Ϫ 3x 2y͒
4

Using the distributive
property

෇ 8x 4y 2 Ϫ 12x 6y 2 Ϫ 14x 2y 2 ϩ 21x 4y 2 ϩ 6y 2 Ϫ 9x 2y 2
෇ 29x 4y 2 Ϫ 12x 6y 2 Ϫ 23x 2y 2 ϩ 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 Ϫ 7x 2y ϩ 3y
2y Ϫ 3x 2y
Ϫ12x 6y 2 ϩ 21x 4y 2 Ϫ 9x 2y 2
8x 4y 2 Ϫ 14x 2y 2 ϩ 6y 2
Ϫ12x 6y 2 ϩ 29x 4y 2 Ϫ 23x 2y 2 ϩ 6y 2


Multiplying by ؊3x2y
Multiplying by 2y
Adding

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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: ͑2x Ϫ 7͒ ͑3x ϩ 4͒.

We have
F

L

F

O

I

L

͑2x Ϫ 7͒ ͑3x ϩ 4͒ ෇ 6x ϩ 8x Ϫ 21x Ϫ 28
෇ 6x 2 Ϫ 13x Ϫ 28
2

I

O

We can use FOIL to find some special products.

Special Products of Binomials
͑A ϩ B͒2 ෇ A2 ϩ 2AB ϩ B 2

Square of a sum


͑A Ϫ B͒2 ෇ A2 Ϫ 2AB ϩ B 2

Square of a difference

͑A ϩ B͒ ͑A Ϫ B͒ ෇ A Ϫ B

Product of a sum and a difference

2

EXAMPLE 7
a) ͑4x ϩ 1͒2

2

Multiply each of the following.
b) ͑3y 2 Ϫ 2͒2

c) ͑x 2 ϩ 3y͒ ͑x 2 Ϫ 3y͒

Solution
a) ͑4x ϩ 1͒2 ෇ ͑4x͒2 ϩ 2 и 4x и 1 ϩ 12 ෇ 16x 2 ϩ 8x ϩ 1
b) ͑3y 2 Ϫ 2͒2 ෇ ͑3y 2 ͒2 Ϫ 2 и 3y 2 и 2 ϩ 22 ෇ 9y 4 Ϫ 12y 2 ϩ 4
c) ͑x 2 ϩ 3y͒ ͑x 2 Ϫ 3y͒ ෇ ͑x 2 ͒2 Ϫ ͑3y͒2 ෇ x 4 Ϫ 9y 2

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R.3

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Basic Concepts of Algebra

Exercise Set

Determine the terms and the degree of the polynomial.
1. Ϫ5y 4 ϩ 3y 3 ϩ 7y 2 Ϫ y Ϫ 4 Ϫ5y 4, 3y 3, 7y 2,
Ϫy , Ϫ4; 4

2. 2m3 Ϫ m2 Ϫ 4m ϩ 11 2m3, Ϫm2, Ϫ4m, 11; 3
3. 3a4b Ϫ 7a3b 3 ϩ 5ab Ϫ 2 3a 4b, Ϫ7a 3b 3, 5ab,
Ϫ2; 6

4. 6p3q 2 Ϫ p2q 4 Ϫ 3pq 2 ϩ 5 6p 3q 2, Ϫp 2q 4, Ϫ3pq 2,
5; 6

Perform the operations indicated.

5. ͑5x 2y Ϫ 2xy 2 ϩ 3xy Ϫ 5͒ ϩ
͑Ϫ2x 2y Ϫ 3xy 2 ϩ 4xy ϩ 7͒

3x 2y Ϫ 5xy 2 ϩ 7xy ϩ 2

6. ͑6x 2y Ϫ 3xy 2 ϩ 5xy Ϫ 3͒ ϩ
͑Ϫ4x 2y Ϫ 4xy 2 ϩ 3xy ϩ 8͒

2x 2y Ϫ 7xy 2 ϩ 8xy ϩ 5

7. ͑2x ϩ 3y ϩ z Ϫ 7͒ ϩ ͑4x Ϫ 2y Ϫ z ϩ 8͒ ϩ
͑Ϫ3x ϩ y Ϫ 2z Ϫ 4͒ 3x ϩ 2y Ϫ 2z Ϫ 3

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Section R.3

•

Addition, Subtraction, and Multiplication of Polynomials


34. ͑b ϩ 4͒ ͑b Ϫ 4͒ b 2 Ϫ 16

8. ͑2x 2 ϩ 12xy Ϫ 11͒ ϩ ͑6x 2 Ϫ 2x ϩ 4͒ ϩ
͑Ϫx 2 Ϫ y Ϫ 2͒ 7x 2 ϩ 12xy Ϫ 2x Ϫ y Ϫ 9
9. ͑3x Ϫ 2x Ϫ x ϩ 2͒ Ϫ ͑5x Ϫ 8x Ϫ x ϩ 4͒
2

3

2

3

Ϫ2x 2 ϩ 6x Ϫ 2

10. ͑5x ϩ 4xy Ϫ 3y ϩ 2͒ Ϫ ͑9x Ϫ 4xy ϩ 2y Ϫ 1͒
2

2

2

Ϫ4x 2 ϩ 8xy Ϫ 5y 2 ϩ 3

2

11. ͑x Ϫ 3x ϩ 4x͒ Ϫ ͑3x ϩ x Ϫ 5x ϩ 3͒
4

2


3

2

x 4 Ϫ 3x 3 Ϫ 4x 2 ϩ 9x Ϫ 3

12. ͑2x Ϫ 3x ϩ 7x͒ Ϫ ͑5x ϩ 2x Ϫ 3x ϩ 5͒
4

2

3

2

2x 4 Ϫ 5x 3 Ϫ 5x 2 ϩ 10x Ϫ 5

13. ͑a Ϫ b͒ ͑2a Ϫ ab ϩ 3b ͒
3

2a Ϫ 2a b Ϫ a b ϩ 4ab Ϫ 3b
3

35. ͑2x Ϫ 5͒ ͑2x ϩ 5͒ 4x 2 Ϫ 25
36. ͑4y Ϫ 1͒ ͑4y ϩ 1͒ 16y 2 Ϫ 1
37. ͑3x Ϫ 2y͒ ͑3x ϩ 2y͒ 9x 2 Ϫ 4y 2
38. ͑3x ϩ 5y͒ ͑3x Ϫ 5y͒ 9x 2 Ϫ 25y 2
39. ͑2x ϩ 3y ϩ 4͒ ͑2x ϩ 3y Ϫ 4͒
4x 2 ϩ 12xy ϩ 9y 2 Ϫ 16


2

4

2

2

3

40. ͑5x ϩ 2y ϩ 3͒ ͑5x ϩ 2y Ϫ 3͒
25x 2 ϩ 20xy ϩ 4y 2 Ϫ 9

14. ͑n ϩ 1͒ ͑n 2 Ϫ 6n Ϫ 4͒ n 3 Ϫ 5n 2 Ϫ 10n Ϫ 4

41. ͑x ϩ 1͒ ͑x Ϫ 1͒ ͑x 2 ϩ 1͒ x 4 Ϫ 1

15. ͑x ϩ 5͒ ͑x Ϫ 3͒ x 2 ϩ 2x Ϫ 15

42. ͑ y Ϫ 2͒ ͑ y ϩ 2͒ ͑ y 2 ϩ 4͒

16. ͑ y Ϫ 4͒ ͑ y ϩ 1͒

21

y 4 Ϫ 16

y Ϫ 3y Ϫ 4
2


17. ͑x ϩ 6͒ ͑x ϩ 4͒ x 2 ϩ 10x ϩ 24

Collaborative Discussion and Writing

18. ͑n Ϫ 5͒ ͑n Ϫ 8͒ n 2 Ϫ 13n ϩ 40

43. Is the sum of two polynomials of degree n always a
polynomial of degree n? Why or why not?

19. ͑2a ϩ 3͒ ͑a ϩ 5͒ 2a 2 ϩ 13a ϩ 15
20. ͑3b ϩ 1͒ ͑b Ϫ 2͒ 3b 2 Ϫ 5b Ϫ 2

44. Explain how you would convince a classmate that
͑A ϩ B͒2 A2 ϩ B 2.

21. ͑2x ϩ 3y͒ ͑2x ϩ y͒ 4x 2 ϩ 8xy ϩ 3y 2
22. ͑2a Ϫ 3b͒ ͑2a Ϫ b͒ 4a 2 Ϫ 8ab ϩ 3b 2

Synthesis

23. ͑ y ϩ 5͒2

y 2 ϩ 10y ϩ 25

24. ͑ y ϩ 7͒2

y 2 ϩ 14y ϩ 49

Multiply. Assume that all exponents are natural

numbers.
45. ͑an ϩ b n ͒ ͑an Ϫ b n ͒ a 2n Ϫ b 2n

25. ͑x Ϫ 4͒2 x 2 Ϫ 8x ϩ 16
26. ͑a Ϫ 6͒

2

46. ͑t a ϩ 4͒ ͑t a Ϫ 7͒ t 2a Ϫ 3t a Ϫ 28

a Ϫ 12a ϩ 36
2

27. ͑5x Ϫ 3͒

25x Ϫ 30x ϩ 9

28. ͑3x Ϫ 2͒

9x Ϫ 12x ϩ 4

2

47. ͑an ϩ b n ͒2 a 2n ϩ 2a nb n ϩ b 2n

2

2

48. ͑x 3m Ϫ t 5n ͒2 x 6m Ϫ 2x 3mt 5n ϩ t 10n


2

29. ͑2x ϩ 3y͒

4x ϩ 12xy ϩ 9y

30. ͑5x ϩ 2y͒

25x ϩ 20xy ϩ 4y

2

2

2

2

49. ͑x Ϫ 1͒ ͑x 2 ϩ x ϩ 1͒ ͑x 3 ϩ 1͒ x 6 Ϫ 1

2

50. ͓͑2x Ϫ 1͒2 Ϫ 1͔ 2

2

31. ͑2x Ϫ 3y͒

4x Ϫ 12x y ϩ 9y


32. ͑4x 2 Ϫ 5y͒2

16x 4 Ϫ 40x 2y ϩ 25y 2

2

2

4

2

33. ͑a ϩ 3͒ ͑a Ϫ 3͒ a Ϫ 9
2

2

51. ͑x aϪb ͒aϩb

xa

16x 4 Ϫ 32x 3 ϩ 16x 2

2Ϫb2

52. ͑t mϩn ͒mϩn и ͑t mϪn ͒mϪn

t 2m


2ϩ2n 2

53. ͑a ϩ b ϩ c͒2 a 2 ϩ b 2 ϩ c 2 ϩ 2ab ϩ 2ac ϩ 2bc

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• Basic Concepts of Algebra

Factor polynomials by removing a common factor.
Factor polynomials by grouping.
Factor trinomials of the type x 2 ϩ bx ϩ c .
Factor trinomials of the type ax 2 ϩ bx ϩ c , a 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 ϩ 10x Ϫ 5x 2

b) 12x 2y 2 Ϫ 20x 3y

Solution
a) 15 ϩ 10x Ϫ 5x 2 ෇ 5 и 3 ϩ 5 и 2x Ϫ 5 и x 2 ෇ 5͑3 ϩ 2x Ϫ x 2 ͒
We can always check a factorization by multiplying:
5͑3 ϩ 2x Ϫ x 2 ͒ ෇ 15 ϩ 10x Ϫ 5x 2.
b) There are several factors common to the terms of 12x 2y 2 Ϫ 20x 3y , but
4x 2y is the “largest” of these.
12x 2y 2 Ϫ 20x 3y ෇ 4x 2y и 3y Ϫ 4x 2y и 5x
෇ 4x 2y͑3y Ϫ 5x͒

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 ϩ 3x 2 Ϫ 5x Ϫ 15.

We have

x ϩ 3x Ϫ 5x Ϫ 15 ෇ ͑x 3 ϩ 3x 2 ͒ ϩ ͑Ϫ5x Ϫ 15͒
3

2

෇ x 2͑x ϩ 3͒ Ϫ 5͑x ϩ 3͒
෇ ͑x ϩ 3͒ ͑x 2 Ϫ 5͒.

Grouping; each
group of terms has
a common factor.
Factoring a common
factor out of each
group
Factoring out the
common binomial
factor

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