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CAREERS AND MATHEMATICS:
Pharmacist
Pharmacists distribute prescription
drugs to individuals. They also advise
patients, physicians, and other healthcare workers on the selection, dosages,
interactions, and side effects of medications. They also monitor patients to
ensure that they are using their medications safely and effectively. Some pharmacists specialize in oncology, nuclear
pharmacy, geriatric pharmacy, or psychiatric pharmacy.
© Istockphoto.com/Lucas Rucchin
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
A Review of Basic Algebra
Education and Mathematics Required
•
•
A.1
Sets of Real Numbers
A.2
Integer Exponents and
Notation
A.3
Rational Exponents and
Radicals
A.4
Polynomials
A.5
Factoring Polynomials
A.6
Rational Expressions
Appendix Review
Appendix Test
Pharmacists are required to possess a Pharm.D. degree from an accredited college or school of pharmacy. This degree generally takes four years to complete.
To be admitted to a Pharm.D. program, at least two years of college must be
completed, which includes courses in the natural sciences, mathematics, humanities, and the social sciences. A series of examinations must also be passed to
obtain a license to practice pharmacy.
College Algebra, Trigonometry, Statistics, and Calculus I are courses required
for admission to a Pharm.D. program.
How Pharmacists Use Math and Who Employs Them
•
•
Pharmacists use math throughout their work to calculate dosages of various
drugs. These dosages are based on weight and whether the medication is given
in pill form, by infusion, or intravenously.
Most pharmacists work in a community setting, such as a retail drugstore, or in
a healthcare facility, such as a hospital.
Career Outlook and Salary
•
•
Employment of pharmacists is expected to grow by 17 percent between 2008
and 2018, which is faster than the average for all occupations.
Median annual wages of wage and salary pharmacists is approximately
$106,410.
In this appendix, we review many
concepts and skills learned in
previous algebra courses.
For more information see: www.bls.gov/oco
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A1
11/16/11 10:08 AM
A2
Appendix
Not For Sale
A Review of Basic Algebra
A.1 Sets of Real Numbers
In this section, we will learn to
5 3
7
6
1 9 5
9 8
6
8
6
4
8
3
7
2
6
2 8
4 1 9
8
7
SUDOKU
3
1
6
5
9
Sudoku, a game that involves number placement, is very popular. The objective
is to fill a 9 by 9 grid so that each column, each row and each of the 3 by 3 blocks
contains the numbers from 1 to 9. A partially completed Sudoku grid is shown in
the margin.
To solve Sudoku puzzles, logic and the set of numbers, 5 1, 2, 3, 4, 5, 6, 7, 8, 9 6
are used. Sets of numbers are important in mathematics, and we begin our study of
algebra with this topic.
A set is a collection of objects, such as a set of dishes or a set of golf clubs. The
set of vowels in the English language can be denoted as 5 a, e, i, o, u 6 , where the
braces 5 6 are read as “the set of.”
If every member of one set B is also a member of another set A, we say that B
is a subset of A. We can denote this by writing B ( A, where the symbol ( is read
as “is a subset of.” (See Figure A.1 below.) If set B equals set A, we can write B 8 A.
If A and B are two sets, we can form a new set consisting of all members that are
in set A or set B or both. This set is called the union of A and B. We can denote this
set by writing, A c B where the symbol c is read as “union.” (See Figure A.1 below.)
We can also form the set consisting of all members that are in both set A and
set B. This set is called the intersection of A and B. We can denote this set by writing
A d B, where the symbol d is read as “intersection.” (See Figure A.1 below.)
A
B
A
BA
B
A
AB
B
AB
FIGURE A-1
EXAMPLE 1Understanding Subsets and Finding the Union and Intersection of Two Sets
Let A 5 5 a, e, i 6 , B 5 5 c, d, e 6 , and V 5 5 a, e, i, o, u 6 .
a. Is A ( V? b. Find A c B. c. Find A d B.
SOLUTION a. Since each member of set A is also a member of set V, A ( V.
b. The union of set A and set B contains the members of set A, set B, or both. Thus,
A c B 5 5 a, c, d, e, i 6 .
c. The intersection of set A and set B contains the members that are in both set A
and set B. Thus, A d B 5 5 e 6 .
Self Check 1 a. Is B ( V? b. Find B c V c. Find A d V Now Try Exercise 33.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
1.Identify sets of real numbers.
2.Identify properties of real numbers.
3.Graph subsets of real numbers on the number line.
4.Graph intervals on the number line.
5.Define absolute value.
6.Find distances on the number line.
Section
.1 Sets of Real Numbers A3
1. Identify Sets of Real Numbers
There are several sets of numbers that we use in everyday life.
Basic Sets of Numbers
Natural numbers
The numbers that we use for counting: 5 1, 2, 3, 4, 5, 6, % 6
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Whole numbers
The set of natural numbers including 0: 5 0, 1, 2, 3, 4, 5, 6, % 6
Integers
The set of whole numbers and their negatives:
5 % , 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, % 6
In the definitions above, each group of three dots (called an ellipsis) indicates
that the numbers continue forever in the indicated direction.
Two important subsets of the natural numbers are the prime and composite
numbers. A prime number is a natural number greater than 1 that is divisible only
by itself and 1. A composite number is a natural number greater than 1 that is not
prime.
• The set of prime numbers: 5 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, % 6
• The set of composite numbers: 5 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, % 6
Two important subsets of the set of integers are the even and odd integers. The
even integers are the integers that are exactly divisible by 2. The odd integers are the
integers that are not exactly divisible by 2.
• The set of even integers: 5 % , 210, 28, 26, 24, 22, 0, 2, 4, 6, 8, 10, % 6 • The set of odd integers: 5 % , 29, 27, 25, 23, 21, 1, 3, 5, 7, 9, % 6
So far, we have listed numbers inside braces to specify sets. This method is
called the roster method. When we give a rule to determine which numbers are in a
set, we are using set-builder notation. To use set-builder notation to denote the set
of prime numbers, we write
5 x 0 x is a prime number 6 Read as “the set of all numbers x such that x is a prime
▲ ▲
Caution
variable such
that
Remember that the denominator of
a fraction can never be 0.
▲
number.” Recall that when a letter stands for a number, it is
called a variable.
rule that determines
membership in the set
The fractions of arithmetic are called rational numbers.
Rational Numbers
Rational numbers are fractions that have an integer numerator and a nonzero
integer denominator. Using set-builder notation, the rational numbers are
e
a
0 a is an integer and b is a nonzero integer f
b
Rational numbers can be written as fractions or decimals. Some examples of
rational numbers are
5
3
5 0.75, 5 5 , 1
4
1
5
5 20.333 % , 2 5 20.454545 % The = sign indicates
3
11
that two quantities
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2
are equal.
A4
Appendix
Not For Sale
A Review of Basic Algebra
These examples suggest that the decimal forms of all rational numbers are
either terminating decimals or repeating decimals.
EXAMPLE 2Determining whether the Decimal Form of a Fraction Terminates or
Repeats
Determine whether the decimal form of each fraction terminates or repeats:
a.
7
65
b. 16
99
7
7
a. To change 16
to a decimal, we perform a long division to get 16
5 0.4375. Since
7
0.4375 terminates, we can write 16
as a terminating decimal.
65
b. To change 65
99 to a decimal, we perform a long division to get 99 5 0.656565 % .
65
Since 0.656565 ... repeats, we can write 99 as a repeating decimal.
We can write repeating decimals in compact form by using an overbar. For
example, 0.656565 % 5 0.65.
Self Check 2 Determine whether the decimal form of each fraction terminates or repeats:
a.
38
99
7
b. 8
Now Try Exercise 35.
Some numbers have decimal forms that neither terminate nor repeat. These
nonterminating, nonrepeating decimals are called irrational numbers. Three examples of irrational numbers are
1.010010001000010 % "2 5 1.414213562 % and p 5 3.141592654 %
The union of the set of rational numbers (the terminating and repeating decimals) and the set of irrational numbers (the nonterminating, nonrepeating decimals) is the set of real numbers (the set of all decimals).
Real Numbers
A real number is any number that is rational or irrational. Using set-builder
notation, the set of real numbers is
5 x 0 x is a rational or an irrational number 6
EXAMPLE 3 Classifying Real Numbers
In the set 5 23, 22, 0, 12, 1, "5, 2, 4, 5, 6 6 , list all
a. even integers b. prime numbers c. rational numbers
SOLUTION We will check to see whether each number is a member of the set of even integers,
the set of prime numbers, and the set of rational numbers. a. even integers: –2, 0, 2, 4, 6
b. prime numbers: 2, 5
c. rational numbers: –3, –2, 0, 12, 1, 2, 4, 5, 6
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
SOLUTION In each case, we perform a long division and write the quotient as a decimal.
Section
.1 Sets of Real Numbers A5
Self Check 3 In the set in Example 3, list all a. odd integers b. composite numbers
c. irrational numbers. Now Try Exercise 43.
Figure A-2 shows how the previous sets of numbers are related.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Real numbers
8
11
−3, – – , −√2, 0, –– , π, 9.9
5
16
Rational numbers
−6, –1.25, 0, 2– , 5 3– , 80
3 4
Noninteger rational numbers
111
– 13
––, – 0.1, 2–, –––, 3.17
7
5 53
Irrational numbers
−√5, π, √21, 2√101
Integers
−4, −1, 0, 21
Negative integers
−47, −17, −5, −1
Whole numbers
0, 1, 4, 8, 10, 53, 101
Zero
0
Natural numbers
(positive integers)
1, 12, 38, 990
FIGURE A-2
2. Identify Properties of Real Numbers
When we work with real numbers, we will use the following properties.
Properties of Real Numbers
If a, b, and c are real numbers,
The Commutative Properties for Addition and Multiplication
a 1 b 5 b 1 a
ab 5 ba
The Associative Properties for Addition and Multiplication
1a 1 b2 1 c 5 a 1 1b 1 c2 1ab2 c 5 a 1bc2
The Distributive Property of Multiplication over Addition or Subtraction
a 1b 1 c2 5 ab 1 ac or a 1b 2 c2 5 ab 2 ac
The Double Negative Rule
2 12a2 5 a
Caution
When the Associative Property is used, the order of the real numbers does not change. The
real numbers that occur within the parentheses change.
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The Distributive Property also applies when more than two terms are within
parentheses.
Appendix A
A6
Not For Sale
A Review of Basic Algebra
EXAMPLE 4
SOLUTION
Identifying Properties of Real Numbers
a. 19 1 22 1 3 5 9 1 12 1 32
b. 3 1x 1 y 1 22 5 3x 1 3y 1 3 ? 2
We will compare the form of each statement to the forms listed in the properties of
real numbers box.
a. This form matches the Associative Property of Addition.
Self Check 4
a. mn 5 nm b. 1xy2 z 5 x 1yz2
c. p 1 q 5 q 1 p
Now Try Exercise 17.
3. Graph Subsets of Real Numbers on the Number Line
We can graph subsets of real numbers on the number line. The number line shown
in Figure A-3 continues forever in both directions. The positive numbers are represented by the points to the right of 0, and the negative numbers are represented by
the points to the left of 0.
Negative numbers
Comment
–5
–4
–3
–2
Positive numbers
–1
Zero is neither positive nor negative.
0
1
2
3
4
5
FIGURE A-3
Figure A-4(a) shows the graph of the natural numbers from 1 to 5. The point
associated with each number is called the graph of the number, and the number is
called the coordinate of its point.
Figure A-4(b) shows the graph of the prime numbers that are less than 10.
Figure A-4(c) shows the graph of the integers from 24 to 3.
7
3
Figure A-4(d) shows the graph of the real numbers 23, 24, 0.3, and "2.
–1
Comment
0
1
2
3
4
5
6
0
1
2
3
4
(a)
square with sides of length 1.
–7/3
–5 –4 –3 –2 –1
2
1
6
7
8
9 10
(b)
"2 can be shown as the diagonal of a
1
5
0
1
2
3
4
–3
–2
–3/4
–1
–
0.3
0
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
b. This form matches the Distributive Property.
2
1
2
(d)
(c)
FIGURE A-4
1
The graphs in Figure A-4 suggest that there is a one-to-one correspondence between the set of real numbers and the points on a number line. This means that to
each real number there corresponds exactly one point on the number line, and to each
point on the number line there corresponds exactly one real-number coordinate.
1
EXAMPLE 5
Graphing a Set of Numbers on a Number Line
Graph the set U23, 243, 0, "5V.
11/16/11 10:08 AM
Section
.1 Sets of Real Numbers A7
SOLUTION We will mark (plot) each number on the number line. To the nearest tenth, "5 5 2.2.
5
– 4/3
–3
–2
–1
0
1
2
3
Self Check 5 Graph the set U22, 34, "3V. (Hint: To the nearest tenth, "3 5 1.7.)
Now Try Exercise 51.
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4. Graph Intervals on the Number Line
To show that two quantities are not equal, we can use an inequality symbol.
Symbol Read as
Examples
2
“is not equal to”
528
and
0.25 2 13
,
“is less than”
12 , 20 and
0.17 , 1.1
.
“is greater than”
15 . 9 and
1
2
#
“is less than or equal to”
25 # 25 and
1.7 # 2.3
$
“is greater than or equal to”
19 $ 19 and
15.2 $ 13.7
<
“is approximately equal to”
"2 < 1.414 and
"3 < 1.732
. 0.2
It is possible to write an inequality with the inequality symbol pointing in the
opposite direction. For example,
• 12 , 20 is equivalent to 20 . 12
• 2.3 $ 21.7 is equivalent to 21.7 # 2.3
In Figure A-3, the coordinates of points get larger as we move from left to right
on a number line. Thus, if a and b are the coordinates of two points, the one to the
right is the greater. This suggests the following facts:
• If a . b, point a lies to the right of point b on a number line.
• If a , b, point a lies to the left of point b on a number line.
Figure A-5(a) shows the graph of the inequality x . 22 (or 22 , x). This
graph includes all real numbers x that are greater than 22. The parenthesis at 22
indicates that 22 is not included in the graph. Figure A-5(b) shows the graph of
x # 5 (or 5 $ x). The bracket at 5 indicates that 5 is included in the graph.
]
(
−2
5
(a)
(b)
FIGURE A-5
Sometimes two inequalities can be written as a single expression called a
compound inequality. For example, the compound inequality
5 , x , 12
is a combination of the inequalities 5 , x and x , 12. It is read as “5 is less than x,
and x is less than 12,” and it means that x is between 5 and 12. Its graph is shown
in Figure A-6.
(
)
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5
12
FIGURE A-6
A8
Appendix
Not For Sale
A Review of Basic Algebra
The graphs shown in Figures A-5 and A-6 are portions of a number line
called intervals. The interval shown in Figure A-7(a) is denoted by the inequality
22 , x , 4, or in interval notation as 122, 42 . The parentheses indicate that the
endpoints are not included. The interval shown in Figure A-7(b) is denoted by the
inequality x . 1, or as 11, ` 2 in interval notation.
(
)
4
1
(1, ∞)
x>1
x is greater than 1.
(−2, 4)
−2 < x < 4
x is between −2 and 4.
(a)
(b)
FIGURE A-7
Caution
The symbol ` (infinity) is not a real number. It is used to indicate that the graph in Figure
A-7(b) extendsinfinitely far to
right.
A compound inequality such as 22 , x , 4 can be written as two separate
inequalities:
x . 22 and x , 4
This expression represents the intersection of two intervals. In interval notation, this expression can be written as
122, ` 2 d 12`, 42 Read the symbol d as “intersection.”
Since the graph of 22 , x , 4 will include all points whose coordinates satisfy
both x . 22 and x , 4 at the same time, its graph will include all points that are
larger than 22 but less than 4. This is the interval 122, 42 , whose graph is shown in
Figure A-7(a).
EXAMPLE 6 Writing an Inequality in Interval Notation and Graphing the Inequality
Write the inequality 23 , x , 5 in interval notation and graph it.
SOLUTION This is the interval 123, 52 . Its graph includes all real numbers between 23 and 5,
as shown in Figure A-8.
(
−3
)
5
FIGURE A-8
Self Check 6 Write the inequality 22 , x # 5 in interval notation and graph it. Now Try Exercise 63.
If an interval extends forever in one direction, it is called an unbounded interval. © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
(
−2
Section
.1 Sets of Real Numbers A9
Unbounded Intervals
Interval
Inequality
1a, ` 2 x . a
12, ` 2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
3 a, ` 2 (
)
−3
2
(−3, 2)
−3 < x < 2
(a)
[
)
−2
3
[−2, 3)
−2 ≤ x < 3
(b)
FIGURE A-9
[
−2
3 2, ` 2 12`, a2 x . 2
x $ 2
–2 –1
12`, 2 4 x # 2
(
0
1
2
3
4
5
6
0
1
2
3
4
5
6
– 4 –3 –2 –1
0
1
3
4
0
1
3
4
[
a
–2 –1
[
(
x , a
x , 2
12`, ` 2 (
a
x $ a
12`, 22 12`, a 4 Graph
a
)
2
[
x # a
a
–4 –3 –2 –1
]
2
2` , x , ` A bounded interval with no endpoints is called an open interval. Figure A-9(a)
shows the open interval between 23 and 2. A bounded interval with one endpoint
is called a half-open interval. Figure A-9(b) shows the half-open interval between 22
and 3, including 22.
Intervals that include two endpoints are called closed intervals. Figure A-10
shows the graph of a closed interval from 22 to 4.
]
4
[−2, 4]
−2 ≤ x ≤ 4
FIGURE A-10
Open Intervals, Half-Open Intervals, and Closed Intervals
Interval
Inequality
Graph
Open
1a, b2 a , x , b
122, 32 22 , x , 3
3 a, b2 a # x , b
(
a
)
(
b
–4 –3 –2 –1
0
1
2
0
1
2
0
1
2
0
1
2
)
3
4
5
4
5
4
5
4
5
Half-Open
3 22, 32 22 # x , 3
1a, b 4 a , x # b
[
a
)
[
b
–4 –3 –2 –1
)
3
Half-Open
122, 3 4 22 , x # 3
3 a, b 4 a # x # b
(
a
]
(
b
–4 –3 –2 –1
]
3
Closed
3 22, 3 4 22 # x # 3
[
a
]
[
b
–4 –3 –2 –1
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]
3
A10
Appendix
Not For Sale
A Review of Basic Algebra
EXAMPLE 7 Writing an Inequality in Interval Notation and Graphing the Inequality
Write the inequality 3 # x in interval notation and graph it.
SOLUTION The inequality 3 # x can be written in the form x $ 3. This is the interval 3 3, ` 2 . Its
graph includes all real numbers greater than or equal to 3, as shown in Figure A-11.
[
3
FIGURE A-11
Now Try Exercise 65.
EXAMPLE 8 Writing an Inequality in Interval Notation and Graphing the Inequality
Write the inequality 5 $ x $ 21 in interval notation and graph it.
SOLUTION The inequality 5 $ x $ 21 can be written in the form
21 # x # 5
This is the interval 3 21, 5 4 . Its graph includes all real numbers from 21 to 5. The
graph is shown in Figure A-12.
[
]
−1
5
FIGURE A-12
Self Check 8 Write the inequality 0 # x # 3 in interval notation and graph it. Now Try Exercise 69.
The expression
x , 22 or x $ 3 Read as “x is less than 22 or x is greater than or equal to 3.”
represents the union of two intervals. In interval notation, it is written as
12`, 222 c 3 3, ` 2 Read the symbol c as “union.”
Its graph is shown in Figure A-13.
)
[
−2
3
FIGURE A-13
5. Define Absolute Value
The absolute value of a real number x (denoted as 0 x 0 ) is the distance on a number line
between 0 and the point with a coordinate of x. For example, points with coordinates
of 4 and 24 both lie four units from 0, as shown in Figure A-14. Therefore, it follows
that
0 24 0 5 0 4 0 5 4
4 units
–5 –4 –3 –2 –1
4 units
0
1
FIGURE A-14
2
3
4
5
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Self Check 7 Write the inequality 5 . x in interval notation and graph it. Section
.1 Sets of Real Numbers A11
In general, for any real number x,
0 2x 0 5 0 x 0
We can define absolute value algebraically as follows.
Absolute Value
If x is a real number, then
0 x 0 5 x when x $ 0
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
0 x 0 5 2x when x , 0
Caution
Remember that x is not always
positive and 2x is not always
negative.
This definition indicates that when x is positive or 0, then x is its own absolute
value. However, when x is negative, then 2x (which is positive) is its absolute value.
Thus, 0 x 0 is always nonnegative.
0 x 0 $ 0 for all real numbers x
EXAMPLE 9 Using the Definition of Absolute Value
Write each number without using absolute value symbols:
a. 0 3 0 b. 0 24 0 c. 0 0 0 d. 2 0 28 0
SOLUTION In each case, we will use the definition of absolute value.
a. 0 3 0 5 3 b. 0 24 0 5 4 c. 0 0 0 5 0 d. 2 0 28 0 5 2 182 5 28
Self Check 9 Write each number without using absolute value symbols:
a. 0 210 0 b. 0 12 0 c. 2 0 6 0 Now Try Exercise 85.
In Example 10, we must determine whether the number inside the absolute
value is positive or negative.
EXAMPLE 10Simplifying an Expression with Absolute Value Symbols
Write each number without using absolute value symbols:
a. 0 p 2 1 0 b. 0 2 2 p 0 c. 0 2 2 x 0 if x $ 5
SOLUTION a. Since p < 3.1416, p 2 1 is positive, and p 2 1 is its own absolute value.
0p 2 10 5 p 2 1
b. Since 2 2 p is negative, its absolute value is 2 12 2 p2 .
0 2 2 p 0 5 2 12 2 p2 5 22 2 1 2 p2 5 22 1 p 5 p 2 2
c. Since x $ 5, the expression 2 2 x is negative, and its absolute value is 2 12 2 x2 .
0 2 2 x 0 5 2 12 2 x2 5 22 1 x 5 x 2 2 provided x $ 5
Self Check 10 Write each number without using absolute value symbols. 1Hint: "5 < 2.236.2
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a. P 2 2 "5 P Now Try Exercise 89.
b. 0 2 2 x 0 if x # 1 A12
Appendix
Not For Sale
A Review of Basic Algebra
6. Find Distances on the Number Line
On the number line shown in Figure A-15, the distance between the points with
coordinates of 1 and 4 is 4 2 1, or 3 units. However, if the subtraction were done
in the other order, the result would be 1 2 4, or 23 units. To guarantee that the
distance between two points is always positive, we can use absolute value symbols.
Thus, the distance d between two points with coordinates of 1 and 4 is
d 5 04 2 10 5 01 2 40 5 3
0
1
2
3
4
5
FIGURE A-15
In general, we have the following definition for the distance between two points
on the number line.
Distance between Two Points
If a and b are the coordinates of two points on the number line, the distance
between the points is given by the formula
d 5 0b 2 a0
EXAMPLE 11 Finding the Distance between Two Points on a Number Line
Find the distance on a number line between points with coordinates of a. 3 and 5
b. 22 and 3 c. 25 and 21
SOLUTION We will use the formula for finding the distance between two points.
a. d 5 0 5 2 3 0 5 0 2 0 5 2
b. d 5 0 3 2 1222 0 5 0 3 1 2 0 5 0 5 0 5 5
c. d 5 0 21 2 1252 0 5 0 21 1 5 0 5 0 4 0 5 4
Self Check 11 Find the distance on a number line between points with coordinates of a. 4 and 10 b. 22 and 27 Now Try Exercise 99.
Self Check Answers1. a. no b. {a, c, d, e, i, o, u} c. {a, e, i} 2. a. repeats b. terminates 3. a. 23, 1, 5 b. 4, 6 c. "5 4. a. Commutative
Property of Multiplication b. Associative Property of Multiplication
3
–2
3/4
c. Commutative Property of Addition 5. –3
6. 122, 5 4 8. 3 0, 3 4 (
]
−2
[
0
5
]
7. 12`, 52 –2
–1
)
5
9. a. 10 b. 12 c. 26
3
10. a. "5 2 2 b. 2 2 x 11. a. 6 b. 5
0
1
2
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
–1
d = |4 − 1| = 3
Section
.1 Sets of Real Numbers A13
Exercises A.1
Getting Ready
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
You should be able to complete these vocabulary and
concept statements before you proceed to the practice
exercises.
Fill in the blanks.
1. A
is a collection of objects.
2. If every member of one set B is also a member of a
second set A, then B is called a
of A.
3. If A and B are two sets, the set that contains all
members that are in sets A and B or both is called
the
of A and B.
4. If A and B are two sets, the set that contains
all members that are in both sets is called the
of A and B.
5. A real number is any number that can be expressed
as a
.
6. A
is a letter that is used to represent a
number.
7. The smallest prime number is
.
8. All integers that are exactly divisible by 2 are called
integers.
9. Natural numbers greater than 1 that are not prime
are called
numbers.
2 8
10. Fractions such as 3 , 2 , and 279 are called
numbers.
11. Irrational numbers are
that don’t terminate and don’t repeat.
12. The symbol
is read as “is less than or equal to.”
13. On a number line, the
numbers are to the
left of 0.
14. The only integer that is neither positive nor negative
is
.
15. The Associative Property of Addition states that
1x 1 y2 1 z 5
.
16. The Commutative Property of Multiplication states
that xy 5
.
17. Use the Distributive Property to complete the state.
ment: 5 1m 1 22 5
18. The statement 1m 1 n2 p 5 p 1m 1 n2 illustrates the
Property of
.
19. The graph of an
is a portion of a number
line.
20. The graph of an open interval has
endpoints.
21. The graph of a closed interval has
endpoints.
22. The graph of a
interval has one endpoint.
23. Except for 0, the absolute value of every number is
.
24. The
between two distinct points on a
number line is always positive.
Let
N 5 the set of natural numbers
W 5 the set of whole numbers
Z 5 the set of integers
Q 5 the set of rational numbers
R 5 the set of real numbers
Determine whether each statement is true or false. Read
the symbol ( as “is a subset of.”
25. N ( W 26. Q ( R 27. Q ( N 28. Z ( Q 29. W ( Z 30. R ( Z Practice
Let A 5 5 a,b,c,d,e 6 , B 5 1d,e,f,g 6 , and
C 5 5 a,c,e,f 6 . Find each set.
31. A c B 33. A d C 32. A d B 34. B c C Determine whether the decimal form of each fraction
terminates or repeats.
9
3
35. 36. 16
8
3
5
37. 38. 11
12
Consider the following set:
5 25,24,223,0,1,"2,2,2.75,6,7 6 .
39. Which numbers are natural numbers? 40. Which numbers are whole numbers? 41. Which numbers are integers? 42. Which numbers are rational numbers? 4 3. Which numbers are irrational numbers? 44. Which numbers are prime numbers? 45. Which numbers are composite numbers? 46. Which numbers are even integers? 47. Which numbers are odd integers? 48. Which numbers are negative numbers? Graph each subset of the real numbers on a number line.
49. The natural numbers between 1 and 5
Not For Sale
Not For Sale
Appendix
A Review of Basic Algebra
50. The composite numbers less than 10
51. The prime numbers between 10 and 20
52. The integers from –2 to 4
53. The integers between –5 and 0
54. The even integers between –9 and –1
55. The odd integers between –6 and 4
56. 20.7, 1.75, and
378
Write each inequality in interval notation and graph the
interval.
57. x . 2 58. x , 4 59. 0 , x , 5 61. x . 24 67. 25 , x # 0 60. 22 , x , 3 62. x , 3 63. 22 # x , 2 65. x # 5 64. 24 , x # 1 66. x $ 21 68. 23 # x , 4 69. 22 # x # 3 70. 24 # x # 4 71. 6 $ x $ 2 72. 3 $ x $ 22 Write each pair of inequalities as the intersection of two
intervals and graph the result.
73. x . 25 and x , 4 Write each inequality as the union of two intervals and
graph the result.
77. x , 22 or x . 2 78. x # 25 or x . 0 79. x # 21 or x $ 3 80. x , 23 or x $ 2 Write each expression without using absolute value
symbols.
82. 0 217 0 81. 0 13 0 83. 0 0 0 84. 2 0 63 0 86. 0 225 0 85. 2 0 28 0 87. 2 0 32 0 88. 89. 0 p 2 5 0 90. 91. 0 p 2 p 0 92. 93. 0 x 1 1 0 and x $ 2 94. 2 0 26 0 08 2 p0 0 2p 0 0 x 1 1 0 and x # 2 2
95. 0 x 2 4 0 and x , 0 96. 0 x 2 7 0 and x . 10
Find the distance between each pair of points on the number
line.
97. 3 and 8 98. 25 and 12 99. 28 and 23 100. 6 and 220 Applications
101. What subset of the real numbers would you use to
describe the populations of several cities?
102. What subset of the real numbers would you use to
describe the subdivisions of an inch on a ruler?
103. What subset of the real numbers would you use to
report temperatures in several cities?
104. What subset of the real numbers would you use to
describe the financial condition of a business?
74. x $ 23 and x , 6
75. x $ 28 and x # 23 76. x . 1 and x # 7 Discovery and Writing
105. Explain why 2x could be positive.
106. Explain why every integer is a rational number.
107. Is the statement 0 ab 0 5 0 a 0 ? 0 b 0 always true?
Explain.
0a0
a
1b 2 02 always true?
108. Is the statement ` ` 5
0b0
b
Explain.
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A14
Section
109. Is the statement 0 a 1 b 0 5 0 a 0 1 0 b 0 always true?
Explain.
111. Explain why it is incorrect to write a , b . c if
a , b and b . c.
110. Under what conditions will the statement given in
Exercise 109 be true?
112. Explain why 0 b 2 a 0 5 0 a 2 b 0 .
A.2 Integer Exponents and Scientific Notation
In this section, we will learn to
1.Define natural-number exponents.
2.Apply the rules of exponents.
3.Apply the rules for order of operations to evaluate expressions.
4.Express numbers in scientific notation.
5.Use scientific notation to simplify computations.
bioraven/Shutterstock.com
The number of cells in the human body is approximated to be one hundred trillion or 100,000,000,000,000. One hundred trillion is 1102 1102 1102 # # # 1102 , where ten
occurs fourteen times. Fourteen factors of ten can be written as 1014.
In this section, we will use integer exponents to represent repeated multiplication of numbers.
1. ​Define Natural-Number Exponents
When two or more quantities are multiplied together, each quantity is called a
factor of the product. The exponential expression x4 indicates that x is to be used
as a factor four times.
4 factors of x







x4 5 x ? x ? x ? x
In general, the following is true.
Natural-Number Exponents
For any natural number n,
n factors of x
xn 5 x ? x ? x ? c ? x











© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
.2 Integer Exponents and Scientific Notation A15
In the exponential expression xn, x is called the base, and n is called the exponent
or the power to which the base is raised. The expression xn is called a power of x.
From the definition, we see that a natural-number exponent indicates how many
times the base of an exponential expression is to be used as a factor in a product. If
an exponent is 1, the 1 is usually not written:
x1 5 x
EXAMPLE 1 Using the Definition of Natural-Number Exponents
Not For Sale
Write each expression without using exponents:
a. 42 b. 1242 2 c. 253 d. 1252 3 e. 3x4 f. 13x2 4
A16
Appendix
Not For Sale
A Review of Basic Algebra
SOLUTION In each case, we apply the definition of natural-number exponents.
a. 42 5 4 ? 4 5 16 Read 42 as “four squared.”
b. 1242 2 5 1242 1242 5 16 c. 253 5 25 152 152 5 2125 Read 1242 2 as “negative four squared.”
Read 253 as “the negative of five cubed.”
d. 1252 3 5 1252 1252 1252 5 2125 e. 3x4 5 3 ? x ? x ? x ? x Read 1252 3 as “negative five cubed.”
Read 3x4 as “3 times x to the fourth power.”
Self Check 1 Write each expression without using exponents:
a. 73 b. 1232 2 c. 5a3 d. 15a2 4 Now Try Exercise 19.
Comment









n factors of x
axn 5 a ? x ? x ? x ? c ? x
n factors of ax
1ax2 n 5 1ax2 1ax2 1ax2 ? c ? 1ax2













Note the distinction between axn and 1ax2 n:
Also note the distinction between 2xn and 12x2 n:
Accent on Technology















n factors of 2x
12x2 n 5 12x2 12x2 12x2 ? c ? 12x2









n factors of x
2xn 5 2 1x ? x ? x ? c ? x2 Using a Calculator to Find Powers
We can use a graphing calculator to find powers of numbers. For example,
consider 2.353.
• Input 2.35 and press the
• Input 3 and press ENTER .
key.
Comment
To find powers on a scientific calculator
use the yx key.
Figure A-16
We see that 2.353 5 12.977875 as the figure shows above.
2. ​Apply the Rules of Exponents
We begin the review of the rules of exponents by considering the product xmxn.
Since xm indicates that x is to be used as a factor m times, and since xn indicates that
x is to be used as a factor n times, there are m 1 n factors of x in the product xmxn.
m 1 n factors of x























m factors of x
n factors of x






















x x 5 x ? x ? x ? c ? x ? x ? x ? x ? c ? x 5 xm1n
m n
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
f. 13x2 4 5 13x2 13x2 13x2 13x2 5 81 ? x ? x ? x ? x Read 13x2 4 as “3x to the fourth power.”
Section
.2 Integer Exponents and Scientific Notation A17
This suggests that to multiply exponential expressions with the same base, we
keep the base and add the exponents.
Product Rule for Exponents
If m and n are natural numbers, then
xmxn 5 xm1n
mn factors of x















The Product Rule applies to exponential
expressions with the same base. A product of two powers with different
bases, such as x4y3, cannot be simplified.
To find another property of exponents, we consider the exponential expression
1xm2 n. In this expression, the exponent n indicates that xm is to be used as a factor n
times. This implies that x is to be used as a factor mn times.
n factors of xm















1x 2 5 1x 2 1xm2 1xm2 ? c ? 1xm2 5 xmn
m n
m
This suggests that to raise an exponential expression to a power, we keep the
base and multiply the exponents.
To raise a product to a power, we raise each factor to that power.
n factors of xy
n factors of x
n factors of y






















1xy2 n 5 1xy2 1xy2 1xy2 ? c ? 1xy2 5 1x ? x ? x ? c ? x2 1y ? y ? y ? c ? y2 5 xnyn













To raise a fraction to a power, we raise both the numerator and the denominator to that power. If y 2 0, then
n factors of
x
y













x n
x x x
x
a b 5 a ba ba b ? c ? a b
y
y y y
y
n factors of x
xxx ? c ? x
yyy ? c ? y







5







© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Comment
n factors of y
5
xn
yn
The previous three results are called the Power Rules of Exponents.
Power Rules of Exponents
If m and n are natural numbers, then
x n xn
1xm2 n 5 xmn 1xy2 n 5 xnyn a b 5 n 1y 2 02
y
y
EXAMPLE 2 Using Exponent Rules to Simplify Expressions with Natural-Number
Exponents
Simplify: a. x5x7 b. x2y3x5y c. 1x42 9 d. 1x2x52 3
Not For Sale
e. a
5x2y 2
x 5
b
2 b f. a
z3
y
A18
Appendix
Not For Sale
A Review of Basic Algebra
SOLUTION In each case, we will apply the appropriate rule of exponents.
a. x5x7 5 x517 5 x12 b. x2y3x5y 5 x215y311 5 x7y4
c. 1x42 9 5 x4?9 5 x36 d. 1x2x52 3 5 1x7 2 3 5 x21
f. a
x5
x5
x 5
1y 2 02
2b 5
2 5 5 10 1y 2
y
y
5x2y 2 52 1x22 2y2
25x4y2
5
5
1z 2 02
b
1z32 2
z3
z6
Self Check 2 Simplify:
a. 1y32 2 b. 1a2a42 3 Now Try Exercise 49.
c. 1x22 3 1x32 2 d. a
3a3b2 3
b 1c 2 02 c3
If we assume that the rules for natural-number exponents hold for exponents
of 0, we can write
x0xn 5 x01n 5 xn 5 1xn
Since x0xn 5 1xn, it follows that if x 2 0, then x0 5 1.
Zero Exponent
x0 5 1 1x 2 02
If we assume that the rules for natural-number exponents hold for exponents
that are negative integers, we can write
x2nxn 5 x2n1n 5 x0 5 1 1x 2 02
However, we know that
1
? xn 5 1 1x 2 02 xn
xn
1
n
n ? x 5 n , and any nonzero number divided by itself is 1.
x
x
Since x2nxn 5 x1n ? xn , it follows that x2n 5 x1n 1x 2 02 .
Negative Exponents
If n is an integer and x 2 0, then
x2n 5
1
1
5 xn
n and x
x2n
Because of the previous definitions, all of the rules for natural-number exponents will hold for integer exponents.
EXAMPLE 3Simplifying Expressions with Integer Exponents
Simplify and write all answers without using negative exponents:
1
a. 13x2 0 b. 3 1x02 c. x24 d. 26 e. x23x f. 1x24x82 25
x
SOLUTION We will use the definitions of zero exponent and negative exponents to simplify
each expression.
1
1
a. 13x2 0 5 1 b. 3 1x02 5 3 112 5 3 c. x24 5 4 d. 26 5 x6 x
x
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
e. a
Section
.2 Integer Exponents and Scientific Notation A19
e. x23x 5 x2311 f. 1x24x82 25 5 1x42 25
5 x22
5
5 x220
1
x2
5
1
x20
Self Check 3 Simplify and write all answers without using negative exponents: a. 7a0 b. 3a22 c. a24a2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Now Try Exercise 59.
d. 1a3a272 3 To develop the Quotient Rule for Exponents, we proceed as follows:
xm
m 1
m 2n
5 xm1 1 2n2 5 xm2n 1x 2 02
n 5 x a nb 5 x x
x
x
This suggests that to divide two exponential expressions with the same nonzero
base, we keep the base and subtract the exponent in the denominator from the exponent in the numerator.
Quotient Rule for Exponents
If m and n are integers, then
xm
5 xm2n 1x 2 02
xn
EXAMPLE 4Simplifying Expressions with Integer Exponents
Simplify and write all answers without using negative exponents:
a.
x8
x2x4
b. x5
x25
SOLUTION We will apply the Product and Quotient Rules of Exponents.
a.
x8
x2x4
x6
825
5
x
b. 5
x5
x25
x25
5 x62 1252
5 x3
5 x11
Self Check 4 Simplify and write all answers without using negative exponents: a.
x26
x2
b. x4x23
x2
Now Try Exercise 69.
EXAMPLE 5Simplifying Expressions with Integer Exponents
Simplify and write all answers without using negative exponents:
Not For Sale
a. a
x3y22 22
x 2n
b.
b
a
b
x22y3
y
A20
Appendix
Not For Sale
A Review of Basic Algebra
SOLUTION We will apply the appropriate rules of exponents.
x3y22 22
1 2
b 5 1x32 22 y22232 22
x22y3
5 1x5y252 22
5 x210y10
5
y10
x10
x 2n x2n
b. a b 5 2n
y
y
5
x2nxnyn
xnyn
2n n n Multiply numerator and denominator by 1 in the form xnyn .
y xy
5
x0yn
y0xn
5
yn
xn
y n
5a b
x
x2nxn 5 x0 and y2nyn 5 y0.
x0 5 1 and y0 5 1.
Self Check 5 Simplify and write all answers without using negative exponents: a. a
x4y23 2
b x23y2
b. a
Now Try Exercise 75.
2a 23
b 3b
Part b of Example 5 establishes the following rule.
A Fraction to a Negative Power
If n is a natural number, then
x 2n
y n
a b 5 a b (x 2 0 and y 2 0)
y
x
3. ​Apply the Rules for Order of Operations to Evaluate
Expressions
When several operations occur in an expression, we must perform the operations in
the following order to get the correct result.
Strategy for Evaluating Expressions
Using Order of Operations
If an expression does not contain grouping symbols such as parentheses or
brackets, follow these steps:
1. Find the values of any exponential expressions.
2. Perform all multiplications and/or divisions, working from left to right.
3. Perform all additions and/or subtractions, working from left to right.
• If an expression contains grouping symbols such as parentheses, brackets,
or braces, use the rules above to perform the calculations within each pair of
grouping symbols, working from the innermost pair to the outermost pair.
• In a fraction, simplify the numerator and the denominator of the fraction
separately. Then simplify the fraction, if possible.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
a. a
Section
Comment
.2 Integer Exponents and Scientific Notation A21
Many students remember the Order of Operations Rule with the acronym PEMDAS:
• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
For example, to simplify
3 3 4 2 16 1 102 4
, we proceed as follows:
22 2 16 1 72
3 3 4 2 16 1 102 4
3 14 2 162
5 2
22 2 16 1 72
2 2 16 1 72
3 12122
5 2
2 2 13
3 12122
5
4 2 13
236
5
29
54
Simplify within the inner parentheses: 6 1 10 5 16.
Simplify within the parentheses: 4 2 16 5 212, and 6 1 7 5 13.
Evaluate the power: 22 5 4.
3 12122 5 236. 4 2 13 5 29.
EXAMPLE 6Evaluating Algebraic Expressions
If x 5 22, y 5 3, and z 5 24, evaluate
a. 2x2 1 y2z b. 2z3 2 3y2
5x2
SOLUTION In each part, we will substitute the numbers for the variables, apply the rules of
order of operations, and simplify.
a. 2x2 1 y2z 5 2 1222 2 1 32 1242
Evaluate the powers.
5 2 142 1 9 1242
Do the multiplication.
5 240
Do the addition.
b.
5 24 1 12362
2z3 2 3y2
2 1242 3 2 3 132 2
5
2
5x
5 1222 2
2 12642 2 3 192
5
5 142
2128 2 27
5
20
2155
5
20
31
52
4
Self Check 6 If x 5 3 and y 5 22, evaluate
Evaluate the powers.
Do the multiplications.
Do the subtraction.
Simplify the fraction.
2x2 2 3y2
. x2y
Now Try Exercise 91.
4. ​Express Numbers in Scientific Notation
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Scientists often work with numbers that are very large or very small. These numbers
can be written compactly by expressing them in scientific notation.
A22
Appendix
Not For Sale
A Review of Basic Algebra
Scientific Notation
A number is written in scientific notation when it is written in the form
N 3 10n
Light travels 29,980,000,000 centimeters per second.
To express this number in scientific notation, we must write it as the product of
a number between 1 and 10 and some integer power of 10. The number 2.998 lies
between 1 and 10. To get 29,980,000,000, the decimal point in 2.998 must be moved
ten places to the right. This is accomplished by multiplying 2.998 by 1010.
29,980,000,000 5 2.998 3 1010 Scientific notation
▲
▲
Standard notation ▲
Standard notation 0.0006214 5 6214 3 1024 Scientific notation
▲
ollirg/Shutterstock.com
One meter is approximately 0.0006214 mile. To express this number in scientific
notation, we must write it as the product of a number between 1 and 10 and some
integer power of 10. The number 6.214 lies between 1 and 10. To get 0.0006214, the
decimal point in 6.214 must be moved four places to the left. This is accomplished
by multiplying 6.214 by 101 4 or by multiplying 6.214 by 1024.
To write each of the following numbers in scientific notation, we start to the
right of the first nonzero digit and count to the decimal point. The exponent gives
the number of places the decimal point moves, and the sign of the exponent indicates the direction in which it moves.
a. 3 7 2 0 0 0 5 3.72 3 105
5 places to the right.
b. 0 . 0 0 0 5 3 7 5 5.37 3 1024 4 places to the left.
c. 7.36 5 7.36 3 100
No movement of the decimal point.
EXAMPLE 7 Writing Numbers in Scientific Notation
Write each number in scientific notation: a. 62,000 b. 20.0027
SOLUTION a. We must express 62,000 as a product of a number between 1 and 10 and some
integer power of 10. This is accomplished by multiplying 6.2 by 104.
62,000 5 6.2 3 104
b. We must express 20.0027 as a product of a number whose absolute value is
between 1 and 10 and some integer power of 10. This is accomplished by multiplying 22.7 by 1023.
20.0027 5 22.7 3 1023
Self Check 7 Write each number in scientific notation:
b. 0.0000087 a. 293,000,000 Now Try Exercise 103.
EXAMPLE 8 Writing Numbers in Standard Notation
Write each number in standard notation:
a. 7.35 3 102 b. 3.27 3 1025
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
where 1 # 0 N 0 , 10 and n is an integer.
Section
.2 Integer Exponents and Scientific Notation A23
SOLUTION a. The factor of 102 indicates that 7.35 must be multiplied by 2 factors of 10.
Because each multiplication by 10 moves the decimal point one place to the
right, we have
7.35 3 102 5 735
b. The factor of 1025 indicates that 3.27 must be divided by 5 factors of 10. Because
each division by 10 moves the decimal point one place to the left, we have
3.27 3 1025 5 0.0000327
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Self Check 8 Write each number in standard notation: a. 6.3 3 103 b. 9.1 3 1024 Now Try Exercise 111.
5. ​Use Scientific Notation to Simplify Computations
Another advantage of scientific notation becomes evident when we multiply and
divide very large and very small numbers.
EXAMPLE 9 Using Scientific Notation to Simplify Computations
Use scientific notation to calculate
13,400,0002 10.000022
.
170,000,000
SOLUTION After changing each number to scientific notation, we can do the arithmetic on the
numbers and the exponential expressions separately.
13,400,0002 10.000022
13.4 3 1062 12.0 3 10252
5
170,000,000
1.7 3 108
6.8
5
3 1061 1 252 28
1.7
5 4.0 3 1027
5 0.0000004
Self Check 9 Use scientific notation to simplify
Now Try Exercise 119.
Accent on Technology
1192,0002 10.00152
. 10.00322 14,5002
Scientific Notation
Graphing calculators will often give an answer in scientific notation. For example,
consider 218.
Figure A-17
Not For Sale
We see in the figure that the answer given is 3.782285936E10, which means
3.782285936 3 1010.
Appendix A
A24
Not For Sale
A Review of Basic Algebra
Comment
218
on a scientific calculator, it is necessary to convert
0.000000000061
the denominator to scientific notation because the number has too many digits to fit on the screen.
To calculate an expression like
• For scientific notation, we enter these numbers and press these keys: 6.1 EXP 11 1/2 .
• To evaluate the expression above, we enter these numbers and press these keys:
21 yx 8 5 4 6.1 EXP 11 1/2 5
Self Check Answers 1. a. 7 ? 7 ? 7 5 343 b. 1232 1232 5 9 c. 5 ? a ? a ? a d. 15a2 15a2 15a2 15a2 5 625 ? a ? a ? a ? a 2. a. y6 b. a18 c. x12 27a9b6
3
1
1
1
1
3. a. 7 b. 2 c. 2 d. 12 4. a. 8 b. 9
c
a
a
a
x
x
x14
27b3
6
5. a. 10 b. 6. 7. a. 29.3 3 107 b. 8.7 3 1026 y
8a3
5
8. a. 6,300 b. 0.00091 9. 20
d. Exercises A.2
Getting Ready
15. 252 You should be able to complete these vocabulary and
concept statements before you proceed to the practice
exercises.
Fill in the blanks.
1. Each quantity in a product is called a
of
the product.
2. A
number exponent tells how many times
a base is used as a factor.
3. In the expression 12x2 3 ,
is the exponent and
is the base.
4. The expression xn is called an
expression.
5. A number is in
notation when it is written in the form N 3 10n , where 1 # 0 N 0 , 10 and n
is an
.
6. Unless
indicate otherwise,
are performed before additions.
Complete each exponent rule. Assume x 2 0.
7. xmxn 5
8. 1xm2 n 5
9. 1xy2 n 5
11. x0 5
10. xm
5
xn
12. x2n 5
Practice
Write each number or expression without using exponents.
13. 132 14. 103
16. 1252 2 18. 14x2 3 1 7. 4x3 19. 125x2 4 21. 28x4 20. 26x2 22. 128x2 4 Write each expression using exponents.
23. 7xxx 25. 12x2 12x2 27. 13t2 13t2 123t2 29. xxxyy 24. 28yyyy 26. 12a2 12a2 12a2 28. 2 12b2 12b2 12b2 12b2
30. aaabbbb Use a calculator to simplify each expression.
31. 2.23 33. 20.54 32. 7.14 34. 120.22 4 Simplify each expression. Write all answers without using
negative exponents. Assume that all variables are restricted
to those numbers for which the expression is defined.
35. x2x3 37. 1z22 3 39. 1y y 2 2 3 4 5
41. 1z 2 1z 2 2 3 4 2
43. 1a 2 1a 2 5 2 3
45. 13x2 3 47. 1x2y2 3 36. y3y4 38. 1t62 7 40. 1a3a62 a4 42. 1t32 4 1t52 2 44. 1a22 4 1a32 3 46. 122y2 4 48. 1x3z42 6 © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
The display will read 6.20046874820 . In standard notation, the answer is approximately
620,046,874,800,000,000,000.
Section
49. a
a2 3
b b
50. a
51. 12x2 0 53. 14x2 0 55. z24
57. y y 54. 22x0 1
56. 22 t
58. 2m22m3
59. 1x x 2 3 24 22
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
61.
63.
65.
67.
69.
71.
75. a
77. a
79. a
82.
r5
62. 2 r
t13
64. 4 t
s9s3
66. 2 2 1s 2
t4 3
68. a 3 b t
r9r23
70. 22 3 1r 2
x
x3
a21
a17
1x22 2
x2x
3 3
m
a 2b n
1a32 22
aa2
a23 24
a 21 b b
73. a
81.
60. 1y y 2 72. a
2
rr
b r3r23
x5y22 4
b x23y2
74. 22
78. a
3x5y23 22
b 6x25y3
80. a
1822z23y2 21
15y2z222 3 15yz222 21
t24 22
b t23
Simplify each expression.
5 3 62 1 19 2 52 4
83. 2
4 12 2 32 2
95.
23x23z22
6x2z23
102. 0.00052
103. 20.000000693
104. 20.000000089
105. one trillion
106. one millionth
x27y5 3
b x7y24
2 25
109. 2.21 3 1025
110. 2.774 3 1022
111. 0.00032 3 104
112. 9,300.0 3 1024
113. 23.2 3 1023
114. 27.25 3 103
116.
3x y
b 2x22y26
12x24y3z25 3
b 4x4y23z5
117.
118.
119.
120.
165,0002 145,0002
250,000
10.0000000452 10.000000122
45,000,000
10.000000352 1170,0002
0.00000085
10.00000001442 112,0002
600,000
145,000,000,0002 1212,0002
0.00018
10.000000002752 14,7502
500,000,000,000
Applications
84. 6 3 3 2 14 2 72 2 4
25 12 2 422
3
108. 4.26 3 109
23
88. 2x3 107. 9.37 3 105
115.
Let x 5 22, y 5 0, and z 5 3 and evaluate each
expression.
85. x2 86. 2x2 93. 5x2 2 3y3z 101. 0.007
Use the method of Example 9 to do each calculation.
Write all answers in scientific notation.
23 2 2
1m22n3p42 22 1mn22p32 4
1mn22p32 24 1mn2p2 21
87. x3 3
89. 12xz2 2 1x2z32
91. 2
z 2 y2
100. 223,470,000,000
Express each number in standard notation.
1x x 2
1x2x252 23
76. a
5x y
b 3x2y23
23 22
99. 2177,000,000
22 3 24
7
4 26
Express each number in scientific notation.
97. 372,000 98. 89,500
52. 4x0 22 23
x 4
b y3
.2 Integer Exponents and Scientific Notation A25
90. 2xz z2 1x2 2 y22
92. x3z
94. 3 1x 2 z2 2 1 2 1y 2 z2 3 Use scientific notation to compute each answer. Write all
answers in scientific notation.
121. Speed of sound The speed of sound in air is
3.31 3 104 centimeters per second. Compute the
speed of sound in meters per minute.
122. Volume of a box Calculate the volume of a box
that has dimensions of 6,000 by 9,700 by 4,700
millimeters. 123. Mass of a proton The mass of one proton is
0.00000000000000000000000167248 gram. Find
the mass of one billion protons. Not For Sale
96. 125x2z232 2
5xz22
Not For Sale
A Review of Basic Algebra
124. Speed of light The speed of light in a vacuum
is approximately 30,000,000,000 centimeters per
second. Find the speed of light in miles per hour.
1160,934.4 cm 5 1 mile.2 125. Astronomy The distance d, in miles, of the nth
planet from the sun is given by the formula
mass within 1 millimeter (0.04 inch) a year by using
a combination of four space-based techniques.
d 5 9,275,200 3 3 12n222 1 4 4
To the nearest million miles, find the distance of
Earth and the distance of Mars from the sun. Give
each answer in scientific notation. Jupiter
Mars
Earth
Venus
The distance from the Earth’s center to the North
Pole (the polar radius) measures approximately
6,356.750 km, and the distance from the center
to the equator (the equatorial radius ) measures
approximately 6,378.135 km. Express each distance
using scientific notation. 128. Refer to Exercise 127. Given that 1 km is
approximately equal to 0.62 miles, use scientific
notation to express each distance in miles. Mercury
126. License Plates The number of different
license plates of the form three digits followed
by three letters, as in the illustration, is
10 ? 10 ? 10 ? 26 ? 26 ? 26. Write this expression
using exponents. Then evaluate it and express the
result in scientific notation. WB
COUNTY
UTAH
12
123ABC
Discovery and Writing
127. New way to the center of the earth The spectacular
“blue marble’ image is the most detailed true-color
image of the entire Earth to date. A new NASAdeveloped technique estimates Earth’s center of
Write each expression with a single base.
xm
129. xnx2 130. 3 x
m 2
x x
x3m15
131. 3 132.
x
x2
133. xm11x3 134. an23a3 4
135. Explain why 2x and 12x2 4 represent different
numbers.
136. Explain why 32 3 102 is not in scientific notation.
Review
137. Graph the interval 122,42 .
138. Graph the interval 12`,23 4 c 3 3,` 2 .
139. Evaluate 0 p 2 5 0 . 140. Find the distance between 27 and 25 on the
number line. © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Appendix
NASA Goddard Space Flight Center
A26
Section
.3 Rational Exponents and Radicals A27
A.3 Rational Exponents and Radicals
In this section, we will learn to
“Dead Man’s Curve” is a 1964 hit song by the rock and roll duo Jan Berry and Dean
Torrence. The song details a teenage drag race that ends in an accident. Today, dead
man’s curve is a commonly used expression given to dangerous curves on our roads.
Every curve has a “critical speed.” If we exceed this speed, regardless of how skilled
a driver we are, we will lose control of the vehicle.
The radical expression 3.9"r gives the critical speed in miles per hour when
we travel a curved road with a radius of r feet. A knowledge of square roots and
radicals is important and used in the construction of safe highways and roads. We
will study the topic of radicals in this section.
Medioimages/Photodisc/Jupiter Images
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
1.Define rational exponents whose numerators are 1.
2.Define rational exponents whose numerators are not 1.
3.Define radical expressions.
4.Simplify and combine radicals.
5.Rationalize denominators and numerators.
1. Define Rational Exponents Whose Numerators Are 1
If we apply the rule 1xm2 n 5 xmn to 1251/22 2 , we obtain
1251/22 2 5 2511/222 5 251
Caution
Keep the base and multiply the exponents.
1
2
?251
5 25
In the expression a , there is no
real-number nth root of a when n
is even and a , 0. For example,
12642 1/2 is not a real number,
because the square of no real
number is 264.
1 /n
Rational Exponents
Thus, 251/2 is a real number whose square is 25. Although both 52 5 25 and
1252 2 5 25, we define 251/2 to be the positive real number whose square is 25:
251/2 5 5 Read 251/2 as “the square root of 25.”
In general, we have the following definition.
If a $ 0 and n is a natural number, then a1/n (read as “the nth root of a”) is the
nonnegative real number b such that
bn 5 a
Since b 5 a1/n, we have bn 5 1a1/n2 n 5 a.
EXAMPLE 1Simplifying Expressions with Rational Exponents
In each case, we will apply the definition of rational exponents.
a. 161/2 5 4 Because 42 5 16. Read 161/2 as “the square root of 16.”
b. 271/3 5 3 Because 33 5 27. Read 271/3 as “the cube root of 27.”
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c. a
1 1/4 1
b 5 3
81
1
1 1 /4
1
1 4
.”
Because a b 5 . Read a b as “the fourth root of
81
3
81
81
A28
Appendix
Not For Sale
A Review of Basic Algebra
d. 2321/5 5 2 1321/52 5 2 122 Read 321/5 as “the fifth root of 32.”
Because 25 5 32.
5 22
Self Check 1 Simplify: a. 1001/2 b. 2431/5 Now Try Exercise 15.
149x22 1/2 5 7 0 x 0 Because
17 0 x 0 2 2 5 49x2 . Since x could be negative, absolute value sym-
bols are necessary to guarantee that the square root is nonnegative.
116x42 1/4 5 2 0 x 0 Because
12 0 x 0 2 4 5 16x4 . Since x could be negative, absolute value sym-
bols are necessary to guarantee that the fourth root is nonnegative.
1729x122 1/6 5 3x2 Because
13x22 6 5 729x12 . Since x2 is always nonnegative, no absolute
value symbols are necessary. Read 1729x122 1/6 as “the sixth root of
729x12 .”
If n is an odd number in the expression a1/n , the base a can be negative.
EXAMPLE 2Simplifying Expressions with Rational Exponents
Simplify by using the definition of rational exponent.
a. 1282 1/3 5 22 b. 123,1252 1/5 5 25 c. a2
1/3
1
1
b 5 2 1,000
10
Self Check 2 Simplify: a. 121252 1/3 Now Try Exercise 19.
Caution
Because 1222 3 5 28.
Because 1252 5 5 23,125.
Because a2
1 3
1
.
b 52
10
1,000
b. 12100,0002 1/5 If n is odd in the expression a1/n , we don’t need to use absolute value symbols, because odd
roots can be negative.
1227x32 1/3 5 23x Because 123x2 3 5 227x3.
12128a72 1/7 5 22a Because 122a2 7 5 2128a7.
We summarize the definitions concerning a1/n as follows.
Summary of Definitions of a1/n
If n is a natural number and a is a real number in the expression a1/n, then
If a $ 0, then a1/n is the nonnegative real number b such that bn 5 a.
If a , 0 e
and n is odd, then a1/n is the real number b such that bn 5 a.
and n is even, then a1/n is not a real number.
The following chart also shows the possibilities that can occur when simplifying a1/n .
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
If n is even in the expression a1/n and the base contains variables, we often use
absolute value symbols to guarantee that an even root is nonnegative.
Section
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Strategy for Simplifying
Expressions of the Form a1/n
.3 Rational Exponents and Radicals A29
n
a1/n
a50
n is a
natural
number.
01/n is the real
number 0
because 0n 5 0.
01/2 5 0 because 02 5 0.
01/5 5 0 because 05 5 0.
a.0
n is a
natural
number.
161/2 5 4 because 42 5 16.
271/3 5 3 because 33 5 27.
a,0
n is an
odd
natural
number.
a1/n is the nonnegative real
number such
that 1a1/n2 n 5 a.
a,0
n is an
even
natural
number.
a1/n is not a real
number.
1292 1/2 is not a real number.
12812 1/4 is not a real number.
a
Examples
a1/n is the real
number such
that 1a1/n2 n 5 a.
12322 1/5 5 22 because 1222 5 5 232.
121252 1/3 5 25 because 1252 3 5 2125.
2. Define Rational Exponents Whose Rational Exponents Are
Not 1
The definition of a1/n can be extended to include rational exponents whose numerators are not 1. For example, 43/2 can be written as either
141/22 3 or 1432 1/2 Because of the Power Rule, 1xm2 n 5 xmn.
This suggests the following rule.
Rule for Rational Exponents
If m and n are positive integers, the fraction mn is in lowest terms, and a1/n is a
real number, then
am/n 5 1a1/n2 m 5 1am2 1/n
In the previous rule, we can view the expression am/n in two ways:
1. 1a1/n2 m: the mth power of the nth root of a
2. 1am2 1/n: the nth root of the mth power of a
For example, 12272 2/3 can be simplified in two ways:
12272 2/3 5 3 12272 1/3 4 2
5 1232 2
59
12272 2/3 5 3 12272 2 4 1/3
59
or
5 17292 1/3
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As this example suggests, it is usually easier to take the root of the base first to
avoid large numbers.
Appendix A
Not For Sale
A Review of Basic Algebra
Comment
Negative Rational Exponents
It is helpful to think of the phrase power over root when we see a rational exponent. The numerator
of the fraction represents the power and the denominator represents the root. Begin with the root
when simplifying to avoid large numbers.
If m and n are positive integers, the fraction mn is in lowest terms and a1/n is a
real number, then
1
1
1a 2 02
a2m/n 5 m/n and 2m/n 5 am/n a
a
EXAMPLE 3 Simplifying Expressions with Rational Exponents
We will apply the rules for rational exponents.
a. 253/2 5 1251/22 3 b. a2
5 53 5 125
x6 2/3
x6 1/3 2
b 5 c a2
b d
1,000
1,000
5 a2
5
x2 2
b
10
x4
100
1
1
d. 23/4 5 813/4
2/5
32
81
1
5
5 1811/42 3
1321/52 2
c. 3222/5 5
1
22
1
5 4
5 33
5
Self Check 3 Simplify: a. 493/2 b. 1623/4 Now Try Exercise 43.
c. 5 27
1
127x32 22/3
Because of the definition, rational exponents follow the same rules as integer
exponents.
EXAMPLE 4 Using Exponent Rules to Simplify Expressions with Rational Exponents
Simplify each expression. Assume that all variables represent positive numbers, and
write answers without using negative exponents.
a.
136x2 1/2 5 361/2x1/2 b. 5 6x1/2 1a1/3b2/32 6
a6/3b12/3
5
3 2
1y 2
y6
5
a2b4
y6
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
A30
Section A.3
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
ax/2ax/4
c. x/6 5 ax/21x/42x/6 a
Rational Exponents and Radicals
d. c
A31
2c22/5 5/3
d 5 12c22/524/52 5/3
c4/5
5 a6x/1213x/1222x/12 5 a7x/12 5 3 1212 1c26/52 4 5/3
5 21c230/15
5 2c22
1
5 2 2
c
5 1212 5/3 1c26/52 5/3 Self Check 4 Use the directions for Example 4:
19r2s2 1/2
y2 1/2
b3/7b2/7
a. a b b. c. 49
b4/7
rs23/2
Now Try Exercise 59.
3. Define Radical Expressions
Radical signs can also be used to express roots of numbers.
n
Definition of "a
If n is a natural number greater than 1 and if a1/n is a real number, then
n
"a 5 a1/n
n
In the radical expression "a, the symbol " is the radical sign, a is the radicand,
and n is the index (or the order) of the radical expression. If the order is 2, the expression is a square root, and we do not write the index.
2
"a 5 "
a
If the index of a radical is 3, we call the radical a cube root.
nth Root of a Nonnegative
Number
n
If n is a natural number greater than 1 and a $ 0, then "a is the nonnegative
number whose nth power is a.
n
n
Q"aR 5 a
Caution
n
In the expression "a, there is no real-number nth root of a when n is even and a , 0. For
example, "264 is not a real number, because the square of no real number is 264.
n
n
If 2 is substituted for n in the equation Q"aR 5 a, we have
2
2
2
Q"
aR 5 Q"aR 5 "a"a 5 a for a $ 0
This shows that if a number a can be factored into two equal factors, either of those factors
is a square root of a. Furthermore, if a can be factored into n equal factors, any one of those
factors is an nth root of a.
Not For Sale
n
If n is an odd number greater than 1 in the expression "a, the radicand can be negative.
A32
Appendix
Not For Sale
A Review of Basic Algebra
EXAMPLE 5 Finding nth Roots of Real Numbers
We apply the definitions of cube root and fifth root.
3
b. "
28 5 22 c.
27
3
5 2 Å 1,000
10
3
2
Because 1232 3 5 227.
Because 1222 3 5 28.
Because a2
5
5
2243 5 2Q"
2243R
d. 2"
1
5 2 232
53
3
Self Check 5 Find each root: a. "
216 b. Now Try Exercise 69.
27
3 3
.
b 52
10
1,000
1
Å 32
5
2
n
We summarize the definitions concerning "a as follows.
n
Summary of Definitions of "a
If n is a natural number greater than 1 and a is a real number, then
n
n
n
If a $ 0, then "a is the nonnegative real number such that Q"aR 5 a.
n
If a , 0 e
n
n
and n is odd, then "a is the real number such that Q"aR 5 a.
n
and n is even, then "a is not a real number.
The following chart also shows the possibilities that can occur when simplifyn
ing "a.
Strategy for Simplifying
n
Expressions of the Form "a
a
n
n
Examples
"a
a50
n is a natural "0 is the real
number
number 0
greater than because 0n 5 0.
1.
"0 5 0 because 03 5 0.
5
"0 5 0 because 05 5 0.
a.0
n is a natural
number
greater than
1.
"16 5 4 because 42 5 16.
3
"
27 5 3 because 33 5 27.
a,0
a,0
n
n
"a is the nonnegative real
number such
n
n
that Q"aR 5 a.
3
n
5
n is an odd
"a is the real
"
232 5 22 because 1222 5 5 232.
3
natural num- number such
"2125 5 25 because 1252 3 5 2125.
n
n
ber greater
that Q"aR 5 a.
than 1.
n is an even
natural
number.
n
"a is not a real "29 is not a real number.
4
number.
"
281 is not a real number.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
3
a. "227 5 23 Section
.3 Rational Exponents and Radicals A33
We have seen that if a1/n is real, then am/n 5 1a1/n2 m 5 1am2 1/n .
This same fact can be stated in radical notation.
n
n
am/n 5 1"a2 m 5 "am
Thus, the mth power of the nth root of a is the same as the nth root of the mth power
3
of a. For example, to find "
272, we can proceed in either of two ways:
2
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
3
3
3
3
272 5 "
729 5 9
"
272 5 Q"
27R 5 32 5 9 or "
By definition, "a2 represents a nonnegative number. If a could be negative, we
must use absolute value symbols to guarantee that "a2 will be nonnegative. Thus,
if a is unrestricted,
"a2 5 0 a 0
A similar argument holds when the index is any even natural number. The sym4
bol "a4, for example, means the positive fourth root of a4. Thus, if a is unrestricted,
4 4
"
a 5 0a0
EXAMPLE 6Simplifying Radical Expressions
6
3 3
If x is unrestricted, simplify a. "
64x6 b. "
x c. "9x8
SOLUTION We apply the definitions of sixth roots, cube roots, and square roots.
6
a. "64x6 5 2 0 x 0 3 3
b. "
x 5 x Use absolute value symbols to guarantee that the result will be nonnegative.
Because the index is odd, no absolute value symbols are needed.
c. "9x8 5 3x4 Because 3x4 is always nonnegative, no absolute value symbols are needed.
Self Check 6 Use the directions for Example 6:
4
a. "
16x4 3
b. "
27y3 Now Try Exercise 73.
4 8
c. "
x 4. Simplify and Combine Radicals
Many properties of exponents have counterparts in radical notation. For example,
1 /n
since a1/nb1/n 5 1ab2 1/n and ab1/n 5 1ab2 1/n and 1b 2 02 , we have the following.
Multiplication and Division
Properties of Radicals
If all expressions represent real numbers,
n
n
n
"a"b 5 "ab n
"a
n
"b
5
a
1b 2 02
Åb
n
In words, we say
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The product of two nth roots is equal to the nth root of their product.
The quotient of two nth roots is equal to the nth root of their quotient.
Appendix
Not For Sale
A Review of Basic Algebra
Caution
These properties involve the nth root of the product of two numbers or the nth root of the quotient of two numbers. There is no such property for sums or differences. For example,
"9 1 4 2 "9 1 "4, because
"9 1 4 5 "13 but "9 1 "4 5 3 1 2 5 5
and "13 2 5. In general,
"a 1 b 5 "a 1 "b and "a 2 b 5 "a 2 "b
Numbers that are squares of positive integers, such as
1, 4, 9, 16, 25, and 36
are called perfect squares. Expressions such as 4x2 and 19x6 are also perfect squares,
because each one is the square of another expression with integer exponents and
rational coefficients.
2
1
1
4x2 5 12x2 2 and x6 5 a x3 b
9
3
Numbers that are cubes of positive integers, such as
1, 8, 27, 64, 125, and 216
1 9
x are also perfect cubes,
are called perfect cubes. Expressions such as 64x3 and 27
because each one is the cube of another expression with integer exponents and rational coefficients.
64x3 5 14x2 3 and
3
1 9
1
x 5 a x3 b
3
27
There are also perfect fourth powers, perfect fifth powers, and so on.
We can use perfect powers and the Multiplication Property of Radicals to simplify many radical expressions. For example, to simplify "12x5, we factor 12x5 so
that one factor is the largest perfect square that divides 12x5. In this case, it is 4x4.
We then rewrite 12x5 as 4x4 ? 3x and simplify.
"12x5 5 "4x4 ? 3x Factor 12x5 as 4x4 ? 3x.
5 "4x4"3x Use the Multiplication Property of Radicals: "ab 5 "a"b.
5 2x2"3x
"4x4 5 2x2
3
To simplify "
432x9y, we find the largest perfect-cube factor of 432x9y (which is
216x9) and proceed as follows:
3
3
"432x9y 5 "216x9 ? 2y Factor 432x9y as 216x9 ? 2y.
3
3
3
3
3
5"
216x9 "
2y Use the Multiplication Property of Radicals: "
ab 5 "
a"
b.
3
5 6x3"
2y
3
"
216x9 5 6x3
Radical expressions with the same index and the same radicand are called like
or similar radicals. We can combine the like radicals in 3"2 1 2"2 by using the
Distributive Property.
3"2 1 2"2 5 13 1 22 "2
5 5"2
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A34
Section
.3 Rational Exponents and Radicals A35
This example suggests that to combine like radicals, we add their numerical
coefficients and keep the same radical.
When radicals have the same index but different radicands, we can often change
them to equivalent forms having the same radicand. We can then combine them. For
example, to simplify "27 2 "12, we simplify both radicals and combine like radicals.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
"27 2 "12 5 "9 ? 3 2 "4 ? 3 Factor 27 and 12.
5 "9"3 2 "4"3 5 3"3 2 2"3
"ab 5 "a"b
5 "3
Combine like radicals.
"9 5 3 and "4 5 2.
EXAMPLE 7Adding and Subtracting Radical Expressions
5
5
Simplify: a. "50 1 "200 b. 3z"
64z 2 2"
2z6
SOLUTION We will simplify each radical expression and then combine like radicals.
a. "50 1 "200 5 "25 ? 2 1 "100 ? 2
5 "25"2 1 "100"2
5 15"2
5 5"2 1 10"2
5
5
5
5 5
b. 3z"
64z 2 2"
2z6 5 3z"
32 ? 2z 2 2"
z ? 2z
5
5
5 5 5
5 3z"
32 "
2z 2 2"
z "2z
5
5
5 3z 122 "
2z 2 2z"
2z
5
5
2z
5 6z"
2z 2 2z"
5
5 4z"
2z
Self Check 7 Simplify: a. "18 2 "8 3
3
b. 2"
81a4 1 a"
24a Now Try Exercise 85.
5. Rationalize Denominators and Numerators
By rationalizing the denominator, we can write a fraction such as
"5
"3
as a fraction with a rational number in the denominator. All that we must do is
multiply both the numerator and the denominator by "3. (Note that "3"3 is the
rational number 3.)
"5
"3
5
"5"3
"3"3
5
"15
3
To rationalize the numerator, we multiply both the numerator and the denominator by "5. (Note that "5"5 is the rational number 5.)
Not For Sale
"5
"3
5
"5"5
"3"5
5
5
"15
A36
Appendix
Not For Sale
A Review of Basic Algebra
EXAMPLE 8Rationalizing the Denominator of a Radical Expression
Rationalize each denominator and simplify. Assume that all variables represent
positive numbers.
a.
1
"7
b. 3
3
3a3
c. d. Å4
Åx
Å 5x5
3
SOLUTION We will multiply both the numerator and the denominator by a radical that will
make the denominator a rational number.
1
"7
5
5
1"7
"7"7
"7
7
b. 3
"
3
3
5 3
Å4
"4
3
5
5
5
c.
3
"3
5
Åx
"x
d. 3
3
"
3"
2
3
3
"4"2
Multiply numerator and
3
denominator by "2, because
3
6
"
3
3
"
8
3
"
6
2
3a3
"3a3
5 5
Å 5x
"5x5
"3a3"5x
"3x
x
5
5
"a2"15ax
5x3
5
a"15ax
5x3
5
5
"x"x
3
"4"2 5 "8 5 2.
"3"x
3
5
"5x5"5x
"15a3x
Multiply numerator and
denominator by "5x, because
"5x5 "5x 5 "25x6 5 5x3.
"25x6
Self Check 8 Use the directions for Example 8:
a. 6
"6
b. 3 2
Å 5x
Now Try Exercise 101.
EXAMPLE 9Rationalizing the Numerator of a Radical Expression
Rationalize each numerator and simplify. Assume that all variables represent posi3
"x
2"
9x
tive numbers: a. b. 7
3
SOLUTION We will multiply both the numerator and the denominator by a radical that will
make the numerator a rational number.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
a.
Section
a.
"x
"x ? "x
5
7
7"x
x
b. .3 Rational Exponents and Radicals A37
3
3
3
2"
9x
2"
9x ? "
3x2
5
3
3
3"
3x2
5
5
5
5
7"x
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Self Check 9 Use the directions for Example 9:
a.
"2x
5
b. 3
2"
27x3
3
3"
3x2
2 13x2
3
3"
3x2
2x
3
"3x2
Divide out the 3’s.
3
3"
2y2
6
Now Try Exercise 111.
After rationalizing denominators, we often can simplify an expression.
EXAMPLE 10Rationalizing Denominators and Simplifying
Simplify: 1
1
1
.
Å2 Å8
SOLUTION We will rationalize the denominators of each radical and then combine like
radicals.
1
1
1
1
1
5
1
Å2 Å8
"2
"8
5
5
5
5
Self Check 10 Simplify: 1"2
"2"2
1
1"2
1
"1
1
1
"1
1
;
5
5
5
5
Å2
Å
8
"2
"2
"8
"8
"8"2
"2
"2
1
2
"16
"2
"2
1
2
4
3"2
4
x
3 x
2 3 . Å2
Å 16
Now Try Exercise 115.
Another property of radicals can be derived from the properties of exponents.
If all of the expressions represent real numbers,
n m
n
mn
#"x 5 "x1/m 5 1x1/m2 1/n 5 x1/ 1mn2 5 "x
m
m
mn
n
#"x
5 "x1/n 5 1x1/n2 1/m 5 x1/1nm2 5 "x
Not For Sale
These results are summarized in the following theorem (a fact that can be proved).
A38
Appendix
Not For Sale
A Review of Basic Algebra
Theorem
If all of the expressions involved represent real numbers, then
m
n
mn
n m
#"x 5 #"x 5 "x
We can use the previous theorem to simplify many radicals. For example,
3
3
#"8 5 #"8 5 "2
EXAMPLE 11Simplifying Radicals Using the Previously Stated Theorem
Simplify. Assume that x and y are positive numbers.
12
9
6
a. "4 b. "x3 c. "8y3
SOLUTION In each case, we will write the radical as an exponential expression, simplify the
resulting expression, and write the final result as a radical.
6
3
a. "4 5 41/6 5 1222 1/6 5 22/6 5 21/3 5 "2
12
4
b. "x3 5 x3/12 5 x1/4 5 "
x
9
3
c. "
8y3 5 123y32 1/9 5 12y2 3/9 5 12y2 1/3 5 "
2y
4
Self Check 11 Simplify: a. "
4 9
b. "
27x3 Now Try Exercise 117.
1
Self Check Answers 1. a. 10 b. 3 2. a. 25 b. 210 3. a. 343 b. 8
y
1
c. 9x2 4. a. b. b1/7 c. 3s2 5. a. 6 b. 2 6. a. 2 0 x 0 7
2
3
"
50x2
3
b. 3y c. x2 7. a. "2 b. 8a"3a 8. a. "6 b. 5x
3
y
"
4x
2x
3
b. 3 10. 11. a. "2 b. "3x
9. a. 4
"4y
5"2x
Exercises A.3
Getting Ready
3. If a , 0 and n is an even number, then a1/n is
a real number.
You should be able to complete these vocabulary and
concept statements before you proceed to the practice
exercises.
Fill in the blanks.
1. If a 5 0 and n is a natural number, then a1/n 5
2. If a . 0 and n is a natural number, then a1/n is a
number.
4. 62/3 can be written as
n
.
6. "a 5
n
8.
7. "a"b 5
9. "x 1 y
m
n
.
2
5. "a 5
n
or
n
"x 1 "y
m
a
5
Åb
n
10. #"x or #"x can be written as
.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Rational exponents can be used to simplify many radical expressions, as shown
in the following example.
Section
Practice
Simplify each expression.
11. 91/2 1 1/2
13. a b 25
12. 81/3 16 1/4
14. a
b 625
15. 2811/4 16. 2a
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
17. 110,0002 1/4 19. a2
8 1/3
b 27
18. 1,0241/5 27 1/3
b 8
20. 2641/3 21. 12642 1/2 22. 121252 1/3 Simplify each expression. Use absolute value symbols
when necessary.
23. 116a22 1/2 24. 125a42 1/2 25. 116a42 1/4 27. 1232a52 1/5 29. 12216b 2 6 1/3
31. a
16a
b 25b2
33. a2
30. 1256t 2 8 1/4
1/2
4
26. 1264a32 1/3 28. 164a62 1/6 1,000x6 1/3
b 27y3
32. a2
34. a
a5 1/5
b 32b10
49t2 1/2
b 100z4
Simplify each expression. Write all answers without using
negative exponents.
35. 43/2 3/2
37. 216 39. 21,0002/3 41. 6421/2 40. 1003/2 42. 2521/2 43. 6423/2 44. 4923/2 45. 2923/2 4 5/2
47. a b 9
49. a2
36. 82/3 38. 1282 2/3 46. 12272 22/3 48. a
27 22/3
b
64
50. a
25 3/2
b 81
125 24/3
b
8
Simplify each expression. Assume that all variables represent positive numbers. Write all answers without using
negative exponents.
51. 1100s42 1/2 52. 164u6v32 1/3 53. 132y10z52 21/5 55. 1x10y52 3/5 57. 1r8s162 23/4 59. a2
54. 1625a4b82 21/4 56. 164a6b122 5/6 58. 128x9y122 22/3 61. a
63.
.3 Rational Exponents and Radicals A39
27r6 22/3
b
1,000s12
a2/5a4/5
a1/5
62. a2
64. 60. a
16x4 3/4
b 625y8
x6/7x3/7
x2/7x5/7
Simplify each radical expression.
65. "49 66. "81 3
3
67. "
125 68. "
264 3
5
69. "
2125 70. "
2243 32
256
71. 5 2
72. 4
Å 100,000
Å 625
Simplify each expression, using absolute value symbols when
necessary. Write answers without using negative exponents.
73. "36x2 75. "9y4 3
77. "
8y3 79.
x4y8
Å z12
4
74. 2"25y2 76. "a4b8 3
78. "
227z9 80. a10b5
Å c15
5
Simplify each expression. Assume that all variables represent
positive numbers so that no absolute value symbols are needed.
81. "8 2 "2 82. "75 2 2"27 2
2
83. "200x 1 "98x 84. "128a3 2 a"162a 85. 2"48y5 2 3y"12y3 86. y"112y 1 4"175y3 3
3
87. 2"
81 1 3"
24 4
4
5
89. "768z 1 "48z5 4
4
88. 3"
32 2 2"
162 5
5
2
90. 22"64y 1 3"
486y2 91. "8x2y 2 x"2y 1 "50x2y 92. 3x"18x 1 2"2x3 2 "72x3 3
3
3
93. "
16xy4 1 y"
2xy 2 "
54xy4 4
4
4
94. "
512x5 2 "
32x5 1 "
1,250x5 Rationalize each denominator and simplify. Assume that all
variables represent positive numbers.
95.
97.
3
"3
2
96. 98. "x
2
99. 3 100. "2
5a
101. 3
102. "25a
103.
2b
Not For Sale
8a6 2/3
b 125b9
32m10 22/5
b
243n15
105.
4
2
"3a
2u4
Å 9v
3
104. 106. 6
"5
8
"y
4d
3
"
9
7
3
"
36c
x
Å 2y
3s5
Å 4r2
3
2
Not For Sale
A Review of Basic Algebra
Rationalize each numerator and simplify. Assume that all
variables are positive numbers.
"5
"y
107.
108. 10
3
3
109.
111.
"9
3
3
110.
5
"
16b3
64a
112.
2
"16b
16
3x
Å 57
Rationalize each denominator and simplify.
113.
115.
1
1
2
Å 3 Å 27
114. x
x
x
2
1
Å2
Å 32
Å8
116. 4
1
1
1 3 Å 2 Å 16
3
y
y
y
1 3
2 3
Å 4 Å 32 Å 500
3
6
118. "
27 10
119. "16x6 6
120. "
27x9 Discovery and Writing
We often can multiply and divide radicals with different indi3
ces. For example, to multiply "3 by "
5, we first write each
radical as a sixth root
6
6 3
3 5"
27
"3 5 31/2 5 33/6 5 "
3
6 2
6
"
5 5 51/3 5 52/6 5 "
5 5"
25
and then multiply the sixth roots.
3
6
6
6
3
121. "2"
2 123.
4
"
3
6
3
122. "3"
5 124.
3
"
2
"5
"2
4 4
125. For what values of x does "
x 5 x? Explain.
126. If all of the radicals involved represent real
numbers and y 2 0, explain why
n
x
"x
5 n
Åy
"y
127. If all of the radicals involved represent real
numbers and there is no division by 0, explain why
n
Simplify each radical expression.
117. "9 Use this idea to write each of the following expressions as
a single radical.
m
x 2m/n
n y
5
a b
y
Å xm
n
128. The definition of xm/n requires that "x be a real
number. Explain why this is important. (Hint:
Consider what happens when n is even, m is odd,
and x is negative.)
Review
1 29. Write 22 , x # 5 using interval notation. 130. Write the expression 0 3 2 x 0 without using
absolute value symbols. Assume that x . 4. Evaluate each expression when x 5 22 and y 5 3.
xy 1 4y
131. x2 2 y2 132.
x
133. Write 617,000,000 in scientific notation.
134. Write 0.00235 3 104 in standard notation. "3"5 5 "27"25 5 " 1272 1252 5 "675
Division is similar.
A.4 Polynomials
In this section, we will learn to
1.Define polynomials.
2.Add and subtract polynomials.
3.Multiply polynomials.
4.Rationalize denominators.
5.Divide polynomials.
© PCN Photography/Alamy
Football is one of the most popular sports in the United States. Brett Favre, one
of the most talented quarterbacks ever to play the game, holds the record for the
most career NFL touchdown passes, the most NFL pass completions, and the most
passing yards. He led the Green Bay Packers to the Super Bowl and most recently
played for the Minnesota Vikings. His nickname is Gunslinger.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Appendix
A40
Section
.4 Polynomials A41
An algebraic expression can be used to model the trajectory or path of a football when passed by Brett Favre. Suppose the height in feet, t seconds after the
football leaves Brett’s hand, is given by the algebraic expression
20.1t2 1 t 1 5.5
At a time of t 5 3 seconds, we see that the height of the football is
20.1 132 2 1 3 1 5.5 5 7.6 ft.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Algebraic expressions like 20.1t2 1 t 1 5.5 are called polynomials, and we will
study them in this section.
1. Define Polynomials
A monomial is a real number or the product of a real number and one or more
variables with whole-number exponents. The number is called the coefficient of the
variables. Some examples of monomials are
3x, 7ab2, 25ab2c4, x3, and 212
with coefficients of 3, 7, 25, 1, and 212, respectively.
The degree of a monomial is the sum of the exponents of its variables. All nonzero constants (except 0) have a degree of 0.
The degree of 3x is 1.
The degree of 7ab2 is 3.
The degree of 25ab2c4 is 7.
The degree of x3 is 3.
The degree of 212 is 0 (since 212 5 212x0).
0 has no defined degree.
A monomial or a sum of monomials is called a polynomial. Each monomial in
that sum is called a term of the polynomial. A polynomial with two terms is called
a binomial, and a polynomial with three terms is called a trinomial.
Monomials
2
Binomials
Trinomials
3x 2a 1 3b
x2 1 7x 2 4
225xy
4x3 2 3x2 4y4 2 2y 1 12
a2b3c
22x3 2 4y2 12x3y2 2 8xy 2 24
The degree of a polynomial is the degree of the term in the polynomial with
highest degree. The only polynomial with no defined degree is 0, which is called the
zero polynomial. Here are some examples.
• 3x2y3 1 5xy2 1 7 is a trinomial of 5th degree, because its term with highest
degree (the first term) is 5.
• 3ab 1 5a2b is a binomial of degree 3.
4
3z4 2 "7 is a polynomial, because its variables have whole• 5x 1 3y2 1 "
number exponents. It is of degree 4.
5
• 27y1/2 1 3y2 1 "
3z is not a polynomial, because one of its variables (y in the
first term) does not have a whole-number exponent.
If two terms of a polynomial have the same variables with the same exponents,
they are like or similar terms. To combine the like terms in the sum 3x2y 1 5x2y or
the difference 7xy2 2 2xy2, we use the Distributive Property:
3x2y 1 5x2y 5 13 1 52 x2y
5 8x2y
7xy2 2 2xy2 5 17 2 22 xy2
Not For Sale
5 5xy2
This illustrates that to combine like terms, we add (or subtract) their coefficients
and keep the same variables and the same exponents.
A42
Appendix
Not For Sale
A Review of Basic Algebra
2. ​Add and Subtract Polynomials
Recall that we can use the Distributive Property to remove parentheses enclosing
the terms of a polynomial. When the sign preceding the parentheses is +, we simply
drop the parentheses:
1 1a 1 b 2 c2 5 11 1a 1 b 2 c2
5 1a 1 1b 2 1c
When the sign preceding the parentheses is 2, we drop the parentheses and the
2 sign and change the sign of each term within the parentheses.
2 1a 1 b 2 c2 5 21 1a 1 b 2 c2
5 21a 1 1212 b 2 1212 c
5 2a 2 b 1 c
We can use these facts to add and subtract polynomials. To add (or subtract)
polynomials, we remove parentheses (if necessary) and combine like terms.
EXAMPLE 1Adding Polynomials
Add: 13x3y 1 5x2 2 2y2 1 12x3y 2 5x2 1 3x2 .
Solution To add the polynomials, we remove parentheses and combine like terms.
13x3y 1 5x2 2 2y2 1 12x3y 2 5x2 1 3x2
5 3x3y 1 5x2 2 2y 1 2x3y 2 5x2 1 3x
5 3x3y 1 2x3y 1 5x2 2 5x2 2 2y 1 3x Use the Commutative Property to
rearrange terms.
5 5x3y 2 2y 1 3x
Combine like terms.
We can add the polynomials in a vertical format by writing like terms in a column and adding the like terms, column by column.
3x3y 1 5x2 2 2y
2x3y 2 5x2
1 3x
3
5x y
2 2y 1 3x
Self Check 1 Add: 14x2 1 3x 2 52 1 13x2 2 5x 1 72 . Now Try Exercise 21.
EXAMPLE 2Subtracting Polynomials
Subtract: 12x2 1 3y22 2 1x2 2 2y2 1 72 .
Solution To subtract the polynomials, we remove parentheses and combine like terms.
12x2 1 3y22 2 1x2 2 2y2 1 72
5 2x2 1 3y2 2 x2 1 2y2 2 7
5 2x2 2 x2 1 3y2 1 2y2 2 7 Use the Commutative Property to
rearrange terms.
5 x2 1 5y2 2 7
Combine like terms.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
5a1b2c
Section
.4 Polynomials A43
We can subtract the polynomials in a vertical format by writing like terms in a
column and subtracting the like terms, column by column.
2x2 1 3y2
2 1x2 2 2y2 1 72 x2 1 5y2 2 7
2x2 2 x2 5 12 2 12 x2
3y2 2 1222 y2 5 3y2 1 2y2 5 13 1 22 y2
0 2 7 5 27
Self Check 2 Subtract: 14x2 1 3x 2 52 2 13x2 2 5x 1 72 . © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Now Try Exercise 23.
We can also use the Distributive Property to remove parentheses enclosing several terms that are multiplied by a constant. For example,
4 13x2 2 2x 1 62 5 4 13x22 2 4 12x2 1 4 162
5 12x2 2 8x 1 24
This example suggests that to add multiples of one polynomial to another, or to
subtract multiples of one polynomial from another, we remove parentheses and combine like terms.
EXAMPLE 3 Using the Distributive Property and Combining Like Terms
Simplify: 7x 12y2 1 13x22 2 5 1xy2 2 13x32 .
Solution 7x 12y2 1 13x22 2 5 1xy2 2 13x32
5 14xy2 1 91x3 2 5xy2 1 65x3 Use the Distributive Property to remove parentheses.
5 14xy 2 5xy 1 91x 1 65x Use the Commutative Property to rearrange terms.
5 9xy2 1 156x3
Combine like terms.
2
2
3
3
Self Check 3 Simplify: 3 12b2 2 3a2b2 1 2b 1b 1 a22 . Now Try Exercise 30.
3. ​Multiply Polynomials
To find the product of 3x2y3z and 5xyz2, we proceed as follows:
13x2y3z2 15xyz22 5 3 ? x2 ? y3 ? z ? 5 ? x ? y ? z2
5 3 ? 5 ? x2 ? x ? y3 ? y ? z ? z2 Use the Commutative Property to
rearrange terms.
5 15x3y4z3
This illustrates that to multiply two monomials, we multiply the coefficients and
then multiply the variables.
To find the product of a monomial and a polynomial, we use the Distributive
Property.
3xy2 12xy 1 x2 2 7yz2 5 3xy2 12xy2 1 13xy22 1x22 2 13xy22 17yz2
5 6x2y3 1 3x3y2 2 21xy3z
Not For Sale
This illustrates that to multiply a polynomial by a monomial, we multiply each
term of the polynomial by the monomial.
A44
Appendix
Not For Sale
A Review of Basic Algebra
To multiply one binomial by another, we use the Distributive Property twice.
EXAMPLE 4 Multiplying Binomials
Multiply: a. 1x 1 y2 1x 1 y2 b. 1x 2 y2 1x 2 y2 c. 1x 1 y2 1x 2 y2
5 x2 1 xy 1 xy 1 y2
5 x2 1 2xy 1 y2
b. 1x 2 y2 1x 2 y2 5 1x 2 y2 x 2 1x 2 y2 y
5 x2 2 xy 2 xy 1 y2
5 x2 2 2xy 1 y2
c. 1x 1 y2 1x 2 y2 5 1x 1 y2 x 2 1x 1 y2 y
5 x2 1 xy 2 xy 2 y2
5 x2 2 y2
Self Check 4 Multiply: a. 1x 1 22 1x 1 22 c. 1x 1 42 1x 2 42 b. 1x 2 32 1x 2 32 Now Try Exercise 45.
The products in Example 4 are called special products. Because they occur so
often, it is worthwhile to learn their forms.
Special Product Formulas
1x 1 y2 2 5 1x 1 y2 1x 1 y2 5 x2 1 2xy 1 y2
1x 2 y2 2 5 1x 2 y2 1x 2 y2 5 x2 2 2xy 1 y2
1x 1 y2 1x 2 y2 5 x2 2 y2
Caution
Remember that 1x 1 y2 2 and 1x 2 y2 2 have trinomials for their products and that
For example,
1x 1 y2 2 2 x2 1 y2 and 1x 2 y2 2 2 x2 2 y2
13 1 52 2 2 32 1 52 and 13 2 52 2 2 32 2 52
82 2 9 1 25 64 2 34
1222 2 2 9 2 25
4 2 216
We can use the FOIL method to multiply one binomial by another. FOIL is
an acronym for First terms, Outer terms, Inner terms, and Last terms. To use this
method to multiply 3x 2 4 by 2x 1 5, we write
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Solution a. 1x 1 y2 1x 1 y2 5 1x 1 y2 x 1 1x 1 y2 y
Section
.4 Polynomials A45
First terms Last terms
13x 2 42 12x 1 52 5 3x 12x2 1 3x 152 2 4 12x2 2 4 152
Inner terms
5 6x2 1 15x 2 8x 2 20
Outer terms
5 6x2 1 7x 2 20
In this example,
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
•
•
•
•
the product of
the product of
the product of
the product of
the first terms is 6x2,
the outer terms is 15x,
the inner terms is 28x, and
the last terms is 220.
The resulting like terms of the product are then combined.
EXAMPLE 5 Using the FOIL Method to Multiply Polynomials
Use the FOIL method to multiply: Q"3 1 xRQ2 2 "3xR.
Solution
1"3 1 x2 12 2 "3x2 5 2"3 2 "3"3x 1 2x 2 x"3x
5 2"3 2 3x 1 2x 2 "3x2
5 2"3 2 x 2 "3x2
Self Check 5 Multiply: Q2x 1 "3RQx 2 "3R. Now Try Exercise 63.
To multiply a polynomial with more than two terms by another polynomial, we
multiply each term of one polynomial by each term of the other polynomial and
combine like terms whenever possible.
EXAMPLE 6 Multiplying Polynomials
Multiply: a. 1x 1 y2 1x2 2 xy 1 y22 b. 1x 1 32 3
Solution a. 1x 1 y2 1x2 2 xy 1 y22 5 x3 2 x2y 1 xy2 1 yx2 2 xy2 1 y3
5 x3 1 y3
b. 1x 1 32 3 5 1x 1 32 1x 1 32 2
5 1x 1 32 1x2 1 6x 1 92
5 x3 1 6x2 1 9x 1 3x2 1 18x 1 27
5 x3 1 9x2 1 27x 1 27
Self Check 6 Multiply: 1x 1 22 12x2 1 3x 2 12 . Now Try Exercise 67.
We can use a vertical format to multiply two polynomials, such as the polynomials given in Self Check 6. We first write the polynomials as follows and draw a
line beneath them. We then multiply each term of the upper polynomial by each
term of the lower polynomial and write the results so that like terms appear in each
column. Finally, we combine like terms column by column.
Not For Sale
Appendix
Not For Sale
A Review of Basic Algebra
2x2 1 3x 2 1
x12
4x2 1 6x 2 2 2x 1 3x2 2 x
Multiply 2x2 1 3x 2 1 by 2.
2x3 1 7x2 1 5x 2 2 In each column, combine like terms.
3
Multiply 2x2 1 3x 2 1 by x.
If n is a whole number, the expressions an 1 1 and 2an 2 3 are polynomials and
we can multiply them as follows:
1an 1 12 12an 2 32 5 2a2n 2 3an 1 2an 2 3
5 2a2n 2 an 2 3
Combine like terms.
We can also use the methods previously discussed to multiply expressions that
are not polynomials, such as x22 1 y and x2 2 y21.
1x22 1 y2 1x2 2 y212 5 x2212 2 x22y21 1 x2y 2 y121
1
1 x2y 2 y0
x2y
1
5 1 2 2 1 x2y 2 1
xy
5 x0 2
5 x2y 2
x0 5 1 and y0 5 1.
1
x2y
4. ​Rationalize Denominators
If the denominator of a fraction is a binomial containing square roots, we can use
the product formula 1x 1 y2 1x 2 y2 to rationalize the denominator. For example,
to rationalize the denominator of
6
"7 1 2
To rationalize a denominator means to change
the denominator into a rational number.
we multiply the numerator and denominator by "7 2 2 and simplify.
6
"7 1 2
5
6 1"7 2 22
1"7 1 22 1"7 2 22
6 1"7 2 22
724
6 1"7 2 22
5
3
5
"7 2 2
"7 2 2
51
Here the denominator is a rational number.
5 2 1"7 2 22
In this example, we multiplied both the numerator and the denominator of the
given fraction by "7 2 2. This binomial is the same as the denominator of the
given fraction "7 1 2, except for the sign between the terms. Such binomials are
called conjugate binomials or radical conjugates.
Conjugate Binomials
Conjugate binomials are binomials that are the same except for the sign between
their terms. The conjugate of a 1 b is a 2 b, and the conjugate of a 2 b is a 1 b.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
A46
Section
.4 Polynomials A47
EXAMPLE 7Rationalizing the Denominator of a Radical Expression
Rationalize the denominator: "3x 2 "2
"3x 1 "2
1x . 02 .
Solution We multiply the numerator and the denominator by "3x 2 "2 (the conjugate of
"3x 1 "22 and simplify.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
"3x 2 "2
"3x 1 "2
5
Q"3x 2 "2RQ"3x 2 "2R
5
"3x"3x 2 "3x"2 2 "2"3x 1 "2"2
5
3x 2 "6x 2 "6x 1 2
3x 2 2
5
3x 2 2"6x 1 2
3x 2 2
Q"3x 1 "2RQ"3x 2 "2R
2
"3x 2 "2
"3x 2 "2
51
2
Q"3xR 2 Q"2R
Self Check 7 Rationalize the denominator: Now Try Exercise 89.
"x 1 2
"x 2 2
. In calculus, we often rationalize a numerator.
EXAMPLE 8Rationalizing the Numerator of a Radical Expression
Rationalize the numerator: "x 1 h 2 "x
.
h
Solution To rid the numerator of radicals, we multiply the numerator and the denominator
by the conjugate of the numerator and simplify.
Q"x 1 h 2 "xRQ"x 1 h 1 "xR
"x 1 h 2 "x
5
h
hQ"x 1 h 1 "xR
5
5
5
x1h2x
"x 1 h 1 "x
Here the numerator has no radicals.
Divide out the common factor of h.
hQ"x 1 h 1 "xR
h
hQ"x 1 h 1 "xR
1
"x 1 h 1 "x
Self Check 8 Rationalize the numerator: "4 1 h 2 2
. h
Not For Sale
Now Try Exercise 99.
"x 1 h 1 "x
51
Appendix
Not For Sale
A Review of Basic Algebra
5. ​Divide Polynomials
To divide monomials, we write the quotient as a fraction and simplify by using the
rules of exponents. For example,
6x2y3
5 23x223y321
22x3y
5 23x21y2
3y2
x
To divide a polynomial by a monomial, we write the quotient as a fraction,
write the fraction as a sum of separate fractions, and simplify each one. For
example, to divide 8x5y4 1 12x2y5 2 16x2y3 by 4x3y4, we proceed as follows:
52
8x5y4
12x2y5
216x2y3
8x5y4 1 12x2y5 2 16x2y3
5 3 41
3 4
3 4 1
4x y
4x y
4x y
4x3y4
5 2x2 1
3y
4
2
x
xy
To divide two polynomials, we can use long division. To illustrate, we consider
the division
2x2 1 11x 2 30
x17
which can be written in long division form as
x 1 7q2x2 1 11x 2 30
The binomial x 1 7 is called the divisor, and the trinomial 2x2 1 11x 2 30 is called
the dividend. The final answer, called the quotient, will appear above the long division symbol.
We begin the division by asking “What monomial, when multiplied by x, gives
2x2?” Because x ? 2x 5 2x2, the answer is 2x. We place 2x in the quotient, multiply
each term of the divisor by 2x, subtract, and bring down the –30.
2x
x 1 7q2x 1 11x 2 30
2x2 1 14x
2 3x 2 30
2
We continue the division by asking “What monomial, when multiplied by x,
gives 23x?” We place the answer, 23, in the quotient, multiply each term of the
divisor by 23, and subtract. This time, there is no number to bring down.
2x 2 3
x 1 7q2x 1 11x 2 30
2x2 1 14x
2 3x 2 30
2 3x 2 21
29
2
Because the degree of the remainder, 29, is less than the degree of the divisor,
the division process stops, and we can express the result in the form
quotient 1
remainder
divisor
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
A48
Section
.4 Polynomials A49
Thus,
2x2 1 11x 2 30
29
5 2x 2 3 1
x17
x17
EXAMPLE 9 Using Long Division to Divide Polynomials
Divide 6x3 2 11 by 2x 1 2.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Solution We set up the division, leaving spaces for the missing powers of x in the dividend.
Comment
2x 1 2q6x3 2 11
In Example 9, we could write the missing
powers of x using coefficients of 0.
2x 1 2q6x3 1 0x2 1 0x 2 11
The division process continues as usual, with the following results:
3x2 2 3x 1 3
2x 1 2q6x
2 11
3
2
6x 1 6x
2 6x2
2 6x2 2 6x
1 6x 2 11
1 6x 1 6
2 17
3
6x 2 11
217
Thus,
.
5 3x2 2 3x 1 3 1
2x 1 2
2x 1 2
3
Self Check 9 Divide: 3x 1 1q9x2 2 1. Now Try Exercise 113.
EXAMPLE 10 Using Long Division to Divide Polynomials
Divide 23x3 2 3 1 x5 1 4x2 2 x4 by x2 2 3.
Solution The division process works best when the terms in the divisor and dividend are written with their exponents in descending order.
x3 2 x2 1 1
x2 2 3qx5 2 x4 2 3x3 1 4x2 2 3
x5
2 3x3
2 x4
1 4x2
4
2x
1 3x2
x2 2 3
x2 2 3
0
Thus,
23x3 2 3 1 x5 1 4x2 2 x4
5 x3 2 x2 1 1.
x2 2 3
Self Check 10 Divide: x2 1 1q3x2 2 x 1 1 2 2x3 1 3x4. Not For Sale
Now Try Exercise 115.
Appendix
A50
Not For Sale
A Review of Basic Algebra
Self Check Answers1. 7x2 2 2x 1 2 2. x2 1 8x 2 12 3. 8b2 2 7a2b 4. a. x2 1 4x 1 4 b. x2 2 6x 1 9 c. x2 2 16 5. 2x2 2 x"3 2 3 x 1 4"x 1 4
1
8. x24
"4 1 h 1 2
x11
2
9. 3x 2 1 10. 3x 2 2x 1 2
x 11
6. 2x3 1 7x2 1 5x 2 2 7. Getting Ready
You should be able to complete these vocabulary and
concept statements before you proceed to the practice
exercises.
Fill in the blanks.
1. A
is a real number or the product of a
real number and one or more
.
2. The
of a monomial is the sum of the exponents of its
.
3. A
is a polynomial with three terms.
4. A
is a polynomial with two terms.
5. A monomial is a polynomial with
term.
6. The constant 0 is called the
polynomial.
7. Terms with the same variables with the same exponents are called
terms.
8. The
of a polynomial is the same as the
degree of its term of highest degree.
9. To combine like terms, we add their
and keep the same
and the same exponents.
10. The conjugate of 3"x 1 2 is
.
Determine whether the given expression is a polynomial.
If so, tell whether it is a monomial, a binomial, or a trinomial, and give its degree.
11. x2 1 3x 1 4 12. 5xy 2 x3 1/2
3
13. x 1 y 14. x23 2 5y22 2
3
15. 4x 2 "5x 2 3
16. x y 17. "15 x
5
18. 1 1 5 x
5
19. 0 20. 3y3 2 4y2 1 2y 1 2 Practice
Perform the operations and simplify.
21. 1x3 2 3x22 1 15x3 2 8x2 22. 12x4 2 5x32 1 17x3 2 x4 1 2x2 23. 1y5 1 2y3 1 72 2 1y5 2 2y3 2 72 24. 13t7 2 7t3 1 32 2 17t7 2 3t3 1 72 25. 2 1x 1 3x 2 12 2 3 1x 1 2x 2 42 1 4 26. 5 1x3 2 8x 1 32 1 2 13x2 1 5x2 2 7 2
2
27. 8 1t2 2 2t 1 52 1 4 1t2 2 3t 1 22 2 6 12t2 2 82 28. 23 1x3 2 x2 1 2 1x2 1 x2 1 3 1x3 2 2x2 29. y 1y2 2 12 2 y2 1y 1 22 2 y 12y 2 22 30. 24a2 1a 1 12 1 3a 1a2 2 42 2 a2 1a 1 22 31. xy 1x 2 4y2 2 y 1x2 1 3xy2 1 xy 12x 1 3y2 32. 3mn 1m 1 2n2 2 6m 13mn 1 12 2 2n 14mn 2 12 33. 2x2y3 14xy42 34. 215a3b 122a2b32 mn
b 12
3r2s3 2r2s 15rs2
36. 2
a
ba
b 5
3
2
35. 23m2n 12mn22 a2
37. 24rs 1r2 1 s22 38. 6u2v 12uv2 2 y2 39. 6ab2c 12ac 1 3bc2 2 4ab2c2 mn2
14mn 2 6m2 2 82 2
41. 1a 1 22 1a 1 22 42. 1y 2 52 1y 2 52
45. 1x 1 42 1x 2 42 46. 1z 1 72 1z 2 72
40. 2
43. 1a 2 62 2
44. 1t 1 92 2
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Exercises A.4
Section
47. 1x 2 32 1x 1 52 49. 1u 1 22 13u 2 22 51. 15x 2 12 12x 1 32 91.
"2 2 "3
52. 14x 2 12 12x 2 72
93.
"x 2 "y
50. 14x 1 12 12x 2 32
53. 13a 2 2b2 54. 14a 1 5b2 14a 2 5b2
57. 12y 2 4x2 13y 2 2x2 58. 122x 1 3y2 13x 1 y2
2
55. 13m 1 4n2 13m 2 4n2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
48. 1z 1 42 1z 2 62
59. 19x 2 y2 1x2 2 3y2 61. 15z 1 2t2 1z2 2 t2 63. Q"5 1 3xRQ2 2 "5xR
65. 13x 2 12 3
67. 13x 1 12 12x2 1 4x 2 32 68. 12x 2 52 1x2 2 3x 1 22 56. 14r 1 3s2 2
60. 18a2 1 b2 1a 1 2b2
1 2 "3
"x 1 "y
95.
"2 1 1
2
97.
y 2 "3
66. 12x 2 32 3
99.
"x 1 3 2 "x
3
69. 13x 1 2y2 12x2 2 3xy 1 4y22 70. 14r 2 3s2 12r2 1 4rs 2 2s22 Multiply the expressions as you would multiply
polynomials.
71. 2yn 13yn 1 y2n2 92. "3 2 "2
94. "2x 1 y
96. "x 2 3
3
1 1 "2
"2x 2 y
Rationalize each numerator.
62. 1y 2 2x22 1x2 1 3y2
64. Q"2 1 xRQ3 1 "2xR
.4 Polynomials A51
y 1 "3
98. "a 2 "b
100. "2 1 h 2 "2
h
"a 1 "b
Perform each division and write all answers without using
negative exponents.
36a2b3
245r2s5t3
101.
102. 6 18ab
27r6s2t8
103.
16x6y4z9
224x9y6z0
104. 32m6n4p2
26m6n7p2
72. 3a2n 12an 1 3an212 105.
5x3y2 1 15x3y4
10x2y3
106. 9m4n9 2 6m3n4
12m3n3
75. 1xn 1 32 1xn 2 42 76. 1an 2 52 1an 2 32 107.
73. 25x2nyn 12x2ny2n 1 3x22nyn2 74. 22a3nb2n 15a23nb 2 ab22n2 24x5y7 2 36x2y5 1 12xy
60x5y4
3 4
2 4
2 3
9a b 1 27a b 2 18a b
108.
18a2b7
77. 12rn 2 72 13rn 2 22 78. 14zn 1 32 13zn 1 12 Perform each division. If there is a nonzero remainder,
write the answer in quotient 1 remainder
divisor form.
2
109. x 1 3q3x 1 11x 1 6 110. 3x 1 2q3x2 1 11x 1 6 79. x1/2 1x1/2y 1 xy1/22 80. ab1/2 1a1/2b1/2 1 b1/22 81. 1a1/2 1 b1/22 1a1/2 2 b1/22 82. 1x3/2 1 y1/22 2 Rationalize each denominator.
2
1
84. 83.
"3 2 1
"5 1 2
3x
14y
85.
86. "7 1 2
"2 2 3
x
y
87.
88. x 2 "3
2y 1 "7
89.
y 1 "2
y 2 "2
111. 2x 2 5q2x2 2 19x 1 37 112. x 2 7q2x2 2 19x 1 35 2x3 1 1
113.
x21
2x3 2 9x2 1 13x 2 20
114.
2x 2 7
115. x2 1 x 2 1qx3 2 2x2 2 4x 1 3 Not For Sale
90. x 2 "3
x 1 "3
116. x2 2 3qx3 2 2x2 2 4x 1 5 117.
x5 2 2x3 2 3x2 1 9
x3 2 2
Not For Sale
A Review of Basic Algebra
x5 2 2x3 2 3x2 1 9
x3 2 3
x5 2 32
119.
x22
x4 2 1
120.
x11
126. Travel Complete the following table, which shows
the rate (mph), time traveled (hr), and distance
traveled (mi) by a family on vacation.
121. 11x 2 10 1 6x2 q36x4 2 121x2 1 120 1 72x3 2 142x Discovery and Writing
118.
122. x 1 6x2 2 12q2121x2 1 72x3 2 142x 1 120 1 36x4 Applications
123. Geometry Find an expression that represents the area
of the brick wall. r
?
3x 1 4
5
t
d
3x2 1 19x 1 20
127. Show that a trinomial can be squared by using the
formula
1a 1 b 1 c2 2 5 a2 1 b2 1 c2 1 2ab 1 2bc 1 2ac.
128. Show that 1a 1 b 1 c 1 d2 2 5 a2 1 b2 1 c2 1
d 2 1 2ab 1 2ac 1 2ad 1 2bc 1 2bd 1 2cd .
1 29. Explain the FOIL method.
130. Explain how to rationalize the numerator of
131. Explain why 1a 1 b2 2 2 a2 1 b2.
"x 1 2
.
x
132. Explain why "a2 1 b2 2 "a2 1 "b2.
(x – 2) ft
Review
(x + 5) ft
124. Geometry The area of the triangle shown in the
illustration is represented as 1x2 1 3x 2 402 square
feet. Find an expression that represents its height. Height
(x + 8) ft
125. Gift Boxes The corners of a 12 in.-by-12 in. piece of
cardboard are folded inward and glued to make a box.
Write a polynomial that represents the volume of the
resulting box. Crease here and
fold inward
x
x
x
x
12 in.
x
x
x
x
12 in.
Simplify each expression. Assume that all variables represent positive numbers.
8 22/3
133. 93/2 134. a
b
125
625x4 3/4
135. a
136. "80x4 b 16y8
3
3
137. "16ab4 2 b"54ab
4
4
138. x"
1,280x 1 "
80x5 © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Appendix
A52
Section
.5 Factoring Polynomials A53
A.5 Factoring Polynomials
1.Factor
2.Factor
3.Factor
4.Factor
5.Factor
6.Factor
7.Factor
© Peter Coombs/Alamy
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
In this section, we will learn to
out a common monomial.
by grouping.
the difference of two squares.
trinomials.
trinomials by grouping.
the sum and difference of two cubes.
miscellaneous polynomials.
The television network series CSI: Crime Scene Investigation and its spinoffs, CSI:
NY and CSI: Miami, are favorite television shows of many people. Their fans enjoy
watching a team of forensic scientists uncover the circumstances that led to an
unusual death or crime. These mysteries are captivating because many viewers are
interested in the field of forensic science.
In this section, we will investigate a mathematics mystery. We will be given a
polynomial and asked to unveil or uncover the two or more polynomials that were
multiplied together to obtain the given polynomial. The process that we will use to
solve this mystery is called factoring.
When two or more polynomials are multiplied together, each one is called a
factor of the resulting product. For example, the factors of 7 1x 1 22 1x 1 32 are
7, x 1 2, and x 1 3
The process of writing a polynomial as the product of several factors is called
factoring.
In this section, we will discuss factoring where the coefficients of the polynomial and the polynomial factors are integers. If a polynomial cannot be factored by
using integers only, we call it a prime polynomial.
1. ​Factor Out a Common Monomial
The simplest type of factoring occurs when we see a common monomial factor in
each term of the polynomial. In this case, our strategy is to use the Distributive
Property and factor out the greatest common factor.
EXAMPLE 1 Factoring by Removing a Common Monomial
Factor: 3xy2 1 6x.
SOLUTION We note that each term contains a greatest common factor of 3x:
3xy2 1 6x 5 3x 1y22 1 3x 122
We can then use the Distributive Property to factor out the common factor of 3x:
3xy2 1 6x 5 3x 1y2 1 22 We can check by multiplying: 3x 1y2 1 22 5 3xy2 1 6x.
Self Check 1 Factor: 4a2 2 8ab. Now Try Exercise 11.
Not For Sale
A54
Appendix
Not For Sale
A Review of Basic Algebra
EXAMPLE 2 Factoring by Removing a Common Monomial
Factor: x2y2z2 2 xyz.
SOLUTION We factor out the greatest common factor of xyz:
x2y2z2 2 xyz 5 xyz 1xyz2 2 xyz 112
5 xyz 1xyz 2 12
We can check by multiplying.
2 2 2
Self Check 2 Factor: a2b2c2 1 a3b3c3. Now Try Exercise 13.
2. ​Factor by Grouping
A strategy that is often helpful when factoring a polynomial with four or more
terms is called grouping. Terms with common factors are grouped together and
then their greatest common factors are factored out using the Distributive Property.
EXAMPLE 3 Factoring by Grouping
Factor: ax 1 bx 1 a 1 b.
SOLUTION Although there is no factor common to all four terms, we can factor x out of the
first two terms and write the expression as
ax 1 bx 1 a 1 b 5 x 1a 1 b2 1 1a 1 b2
We can now factor out the common factor of a 1 b.
ax 1 bx 1 a 1 b 5 x 1a 1 b2 1 1a 1 b2
5 x 1a 1 b2 1 1 1a 1 b2
5 1a 1 b2 1x 1 12
We can check by multiplying.
Self Check 3 Factor: x2 1 xy 1 2x 1 2y. Now Try Exercise 17.
3. ​Factor the Difference of Two Squares
A binomial that is the difference of the squares of two quantities factors easily.
The strategy used is to write the polynomial as the product of two factors. The
first factor is the sum of the quantities and the other factor is the difference of the
quantities.
EXAMPLE 4 Factoring the Difference of Two Squares
Factor: 49x2 2 4.
SOLUTION We observe that each term is a perfect square:
49x2 2 4 5 17x2 2 2 22
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
The last term in the expression x y z 2 xyz has an understood coefficient of 1.
When the xyz is factored out, the 1 must be written.
Section
.5 Factoring Polynomials A55
The difference of the squares of two quantities is the product of two factors. One
is the sum of the quantities, and the other is the difference of the quantities. Thus,
49x2 2 4 factors as
49x2 2 4 5 17x2 2 2 22
5 17x 1 22 17x 2 22 We can check by multiplying.
Self Check 4 Factor: 9a2 2 16b2. © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Now Try Exercise 21.
Example 4 suggests a formula for factoring the difference of two squares.
Factoring the Difference
of Two Squares
x2 2 y2 5 1x 1 y2 1x 2 y2
EXAMPLE 5 Factoring the Difference of Two Squares Twice in One Problem
Factor: 16m4 2 n4.
SOLUTION The binomial 16m4 2 n4 can be factored as the difference of two squares:
Caution
If you are limited to integer
coefficients, the sum of two
squares cannot be factored.
For example, x2 1 y2 is a prime
polynomial.
16m4 2 n4 5 14m22 2 2 1n22 2
5 14m2 1 n22 14m2 2 n22
The first factor is the sum of two squares and is prime. The second factor is a difference of two squares and can be factored:
16m4 2 n4 5 14m2 1 n22 3 12m2 2 2 n2 4
5 14m2 1 n22 12m 1 n2 12m 2 n2 We can check by multiplying.
Self Check 5 Factor: a4 2 81b4. Now Try Exercise 23.
EXAMPLE 6Removing a Common Factor and Factoring the Difference of Two Squares
Factor: 18t2 2 32.
SOLUTION We begin by factoring out the common monomial factor of 2.
18t2 2 32 5 2 19t2 2 162 Since 9t2 2 16 is the difference of two squares, it can be factored.
18t2 2 32 5 2 19t2 2 162
5 2 13t 1 42 13t 2 42 Self Check 6 Factor: 23x2 1 12. We can check by multiplying.
Not For Sale
Now Try Exercise 61.
A56
Appendix
Not For Sale
A Review of Basic Algebra
4. ​Factor Trinomials
Trinomials that are squares of binomials can be factored by using the following
formulas.
Factoring Trinomial Squares
(1) x2 1 2xy 1 y2 5 1x 1 y2 1x 1 y2 5 1x 1 y2 2
For example, to factor a2 2 6a 1 9, we note that it can be written in the form
a2 2 2 13a2 1 32 x 5 a and y 5 3
which matches the left side of Equation 2 above. Thus,
a2 2 6a 1 9 5 a2 2 2 13a2 1 32
5 1a 2 32 1a 2 32
5 1a 2 32 2
We can check by multiplying.
Factoring trinomials that are not squares of binomials often requires some
guesswork. If a trinomial with no common factors is factorable, it will factor into
the product of two binomials.
EXAMPLE 7 Factoring a Trinomial with Leading Coefficient of 1
Factor: x2 1 3x 2 10.
SOLUTION To factor x2 1 3x 2 10, we must find two binomials x 1 a and x 1 b such that
x2 1 3x 2 10 5 1x 1 a2 1x 1 b2
where the product of a and b is –10 and the sum of a and b is 3.
ab 5 210 and a 1 b 5 3
To find such numbers, we list the possible factorizations of –10:
10 1212 5 1222 210 112 25 122
Only in the factorization 5 1222 do the factors have a sum of 3. Thus, a 5 5 and
b 5 22, and
(3) x2 1 3x 2 10 5 1x 1 a2 1x 1 b2
x2 1 3x 2 10 5 1x 1 52 1x 2 22 We can check by multiplying.
Because of the Commutative Property of Multiplication, the order of the factors in Equation 3 is not important. Equation 3 can also be written as
x2 1 3x 2 10 5 1x 2 22 1x 1 52
Self Check 7 Factor: p2 2 5p 2 6. Now Try Exercise 29.
EXAMPLE 8 Factoring a Trinomial with Leading Coefficient not 1
Factor: 2x2 2 x 2 6.
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(2) x2 2 2xy 1 y2 5 1x 2 y2 1x 2 y2 5 1x 2 y2 2
Section
.5 Factoring Polynomials A57
SOLUTION Since the first term is 2x2, the first terms of the binomial factors must be 2x and x:
2x2 2 x 2 6 5 12x
2 1x
2
The product of the last terms must be 26, and the sum of the products of the outer
terms and the inner terms must be 2x. Since the only factorization of 26 that will
cause this to happen is 3 1222 , we have
2x2 2 x 2 6 5 12x 1 32 1x 2 22 We can check by multiplying.
Self Check 8 Factor: 6x2 2 x 2 2. © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Now Try Exercise 33.
It is not easy to give specific rules for factoring trinomials, because some guesswork is often necessary. However, the following hints are helpful.
Strategy for Factoring
a General Trinomial
with Integer Coefficients
1. Write the trinomial in descending powers of one variable.
2. Factor out any greatest common factor, including 21 if that is necessary to
make the coefficient of the first term positive.
3. When the sign of the first term of a trinomial is 1 and the sign of the third
term is 1, the sign between the terms of each binomial factor is the same
as the sign of the middle term of the trinomial.
When the sign of the first term is 1 and the sign of the third term is 2,
one of the signs between the terms of the binomial factors is 1 and the
other is 2.
4. Try various combinations of first terms and last terms until you find one
that works. If no possibilities work, the trinomial is prime.
5. Check the factorization by multiplication.
EXAMPLE 9 Factoring a Trinomial Completely
Factor: 10xy 1 24y2 2 6x2.
SOLUTION We write the trinomial in descending powers of x and then factor out the common
factor of 22.
10xy 1 24y2 2 6x2 5 26x2 1 10xy 1 24y2
5 22 13x2 2 5xy 2 12y22
Since the sign of the third term of 3x2 2 5xy 2 12y2 is 2, the signs between
the binomial factors will be opposite. Since the first term is 3x2, the first terms of
the binomial factors must be 3x and x:
22 13x2 2 5xy 2 12y22 5 22 13x
2 1x
2
The product of the last terms must be 212y2, and the sum of the outer terms
and the inner terms must be 25xy. Of the many factorizations of 212y2, only
4y 123y2 leads to a middle term of 25xy. So we have
10xy 1 24y2 2 6x2 5 26x2 1 10xy 1 24y2
5 22 13x2 2 5xy 2 12y22
Not For Sale
5 22 13x 1 4y2 1x 2 3y2 We can check by multiplying.
A58
Appendix
Not For Sale
A Review of Basic Algebra
Self Check 9 Factor: 26x2 2 15xy 2 6y2. Now Try Exercise 69.
5. ​Factor Trinomials by Grouping
1. Find the product ac: 6 1262 5 236. This number is called the key number.
2. Find two factors of the key number 12362 whose sum is b 5 5. Two such numbers are 9 and 24.
9 1242 5 236 and 9 1 1242 5 5
3. Use the factors 9 and 24 as coefficients of two terms to be placed between 6x2
and 26.
6x2 1 5x 2 6 5 6x2 1 9x 2 4x 2 6
4. Factor by grouping:
6x2 1 9x 2 4x 2 6 5 3x 12x 1 32 2 2 12x 1 32
5 12x 1 32 13x 2 22
Factor out 2x 1 3.
EXAMPLE 10 Factoring a Trinomial by Grouping
Factor: 15x2 1 x 2 2.
SOLUTION Since a 5 15 and c 5 22 in the trinomial, ac 5 230. We now find factors of 230
whose sum is b 5 1. Such factors are 6 and 25. We use these factors as coefficients
of two terms to be placed between 15x2 and –2.
15x2 1 6x 2 5x 2 2
Finally, we factor by grouping.
3x 15x 1 22 2 15x 1 22 5 15x 1 22 13x 2 12
Self Check 10 Factor: 15a2 1 17a 2 4. Now Try Exercise 39.
We can often factor polynomials with variable exponents. For example, if n is
a natural number,
a2n 2 5an 2 6 5 1an 1 12 1an 2 62
because
1an 1 12 1an 2 62 5 a2n 2 6an 1 an 2 6
5 a2n 2 5an 2 6
Combine like terms.
6. Factor the Sum and Difference of Two Cubes
Two other types of factoring involve binomials that are the sum or the difference of
two cubes. Like the difference of two squares, they can be factored by using a formula.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Another way of factoring trinomials involves factoring by grouping. This method
can be used to factor trinomials of the form ax2 1 bx 1 c. For example, to factor
6x2 1 5x 2 6, we note that a 5 6, b 5 5, and c 5 26, and proceed as follows:
Section
Factoring the Sum and Difference
of Two Cubes
.5 Factoring Polynomials A59
x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22
x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22
EXAMPLE 11 Factoring the Sum of Two Cubes
Factor: 27x6 1 64y3.
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SOLUTION We can write this expression as the sum of two cubes and factor it as follows:
27x6 1 64y3 5 13x22 3 1 14y2 3
5 13x2 1 4y2 3 13x22 2 2 13x22 14y2 1 14y2 2 4
5 13x2 1 4y2 19x4 2 12x2y 1 16y22
We can check by multiplying.
Self Check 11 Factor: 8a3 1 1,000b6. Now Try Exercise 41.
EXAMPLE 12 Factoring the Difference of Two Cubes
Factor: x3 2 8.
SOLUTION This binomial can be written as x3 2 23, which is the difference of two cubes. Substituting into the formula for the difference of two cubes gives
x3 2 23 5 1x 2 22 1x2 1 2x 1 222
5 1x 2 22 1x2 1 2x 1 42 We can check by multiplying.
Self Check 12 Factor: p3 2 64. Now Try Exercise 43.
7. ​Factor Miscellaneous Polynomials
EXAMPLE 13 Factoring a Miscellaneous Polynomial
Factor: x2 2 y2 1 6x 1 9.
SOLUTION Here we will factor a trinomial and a difference of two squares.
x2 2 y2 1 6x 1 9 5 x2 1 6x 1 9 2 y2
5 1x 1 32 2 2 y2
Use the Commutative Property to rearrange the terms.
Factor x2 1 6x 1 9.
5 1x 1 3 1 y2 1x 1 3 2 y2 Factor the difference of two squares.
We could try to factor this expression in another way.
x2 2 y2 1 6x 1 9 5 1x 1 y2 1x 2 y2 1 3 12x 1 32 Factor x2 2 y2 and 6x 1 9.
However, we are unable to finish the factorization. If grouping in one way doesn’t
work, try various other ways.
Not For Sale
Self Check 13 Factor: a2 1 8a 2 b2 1 16. Now Try Exercise 97.
A60
Appendix
Not For Sale
A Review of Basic Algebra
EXAMPLE 14 Factoring a Miscellaneous Trinomial
Factor: z4 2 3z2 1 1.
SOLUTION This trinomial cannot be factored as the product of two binomials, because no
combination will give a middle term of 23z2. However, if the middle term were
22z2, the trinomial would be a perfect square, and the factorization would be easy:
z4 2 2z2 1 1 5 1z2 2 12 1z2 2 12
We can change the middle term in z4 2 3z2 1 1 to 22z2 by adding z2 to it.
However, to make sure that adding z2 does not change the value of the trinomial,
we must also subtract z2. We can then proceed as follows.
z4 2 3z2 1 1 5 z4 2 3z2 1 z2 1 1 2 z2 Add and subtract z2.
5 z4 2 2z2 1 1 2 z2
Combine 23z2 and z2.
5 1z2 2 12 2 2 z2
Factor z4 2 2z2 1 1.
5 1z2 2 1 1 z2 1z2 2 1 2 z2 Factor the difference of two squares.
In this type of problem, we will always try to add and subtract a perfect square
in hopes of making a perfect-square trinomial that will lead to factoring a difference of two squares.
Self Check 14 Factor: x4 1 3x2 1 4. Now Try Exercise 103.
It is helpful to identify the problem type when we must factor polynomials that
are given in random order.
Factoring Strategy
1. Factor out all common monomial factors.
2. If an expression has two terms, check whether the problem type is
a. The difference of two squares:
x2 2 y2 5 1x 1 y2 1x 2 y2
b. The sum of two cubes:
x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22
c. The difference of two cubes:
x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22
3. If an expression has three terms, try to factor it as a trinomial.
4. If an expression has four or more terms, try factoring by grouping.
5. Continue until each individual factor is prime.
6. Check the results by multiplying.
Self Check Answers
1. 4a 1a 2 2b2 2. a2b2c2 11 1 abc2 3. 1x 1 y2 1x 1 22 4. 13a 1 4b2 13a 2 4b2 5. 1a2 1 9b22 1a 1 3b2 1a 2 3b2 6. 23 1x 1 22 1x 2 22 7. 1p 2 62 1p 1 12 8. 13x 2 22 12x 1 12
9. 23 1x 1 2y2 12x 1 y2 10. 13a 1 42 15a 2 12 11. 8 1a 1 5b22 1a2 2 5ab2 1 25b42 12. 1p 2 42 1p2 1 4p 1 162
13. 1a 1 4 1 b2 1a 1 4 2 b2 14. 1x2 1 2 1 x2 1x2 1 2 2 x2
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
5 1z2 2 12 2
Section
.5 Factoring Polynomials A61
Exercises A.5
Getting Ready
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You should be able to complete these vocabulary and
concept statements before you proceed to the practice
exercises.
Fill in the blanks.
1. When polynomials are multiplied together, each
polynomial is a
of the product.
2. If a polynomial cannot be factored using
coefficients, it is called a
polynomial.
Complete each factoring formula.
3. ax 1 bx 5
2
2
4. x 2 y 5
3 3. 12x2 2 xy 2 6y2 34. 8x2 2 10xy 2 3y2 In each expression, factor the trinomial by grouping.
35. x2 1 10x 1 21 36. x2 1 7x 1 10 37. x2 2 4x 2 12 38. x2 2 2x 2 63 39. 6p2 1 7p 2 3 40. 4q2 2 19q 1 12 In each expression, factor the sum of two cubes.
41. t3 1 343 42. r3 1 8s3 In each expression, factor the difference of two cubes.
5. x2 1 2xy 1 y2 5
4 3. 8z3 2 27 6. x2 2 2xy 1 y2 5
7. x3 1 y3 5
44. 125a3 2 64 8. x3 2 y3 5
Factor each expression completely. If an expression is
prime, so indicate.
Practice
45. 3a2bc 1 6ab2c 1 9abc2 In each expression, factor out the greatest common
monomial.
9. 3x 2 6 2
3
1 1. 8x 1 4x 13. 7x2y2 1 14x3y2 10. 5y 2 15 12. 9y3 1 6y2 14. 25y2z 2 15yz2 In each expression, factor by grouping.
15. a 1x 1 y2 1 b 1x 1 y2 16. b 1x 2 y2 1 a 1x 2 y2 17. 4a 1 b 2 12a2 2 3ab 18. x2 1 4x 1 xy 1 4y In each expression, factor the difference of two squares.
20. 36z2 2 49 19. 4x2 2 9 21. 4 2 9r2 22. 16 2 49x2 23. 81x4 2 1 24. 81 2 x4 25. 1x 1 z2 2 2 25 26. 1x 2 y2 2 2 9 In each expression, factor the trinomial.
27. x2 1 8x 1 16 28. a2 2 12a 1 36 29. b2 2 10b 1 25 30. y2 1 14y 1 49 31. m2 1 4mn 1 4n2 32. r2 2 8rs 1 16s2 46. 5x3y3z3 1 25x2y2z2 2 125xyz 47. 3x3 1 3x2 2 x 2 1 48. 4x 1 6xy 2 9y 2 6 49. 2txy 1 2ctx 2 3ty 2 3ct 50. 2ax 1 4ay 2 bx 2 2by 51. ax 1 bx 1 ay 1 by 1 az 1 bz 5 2. 6x2y3 1 18xy 1 3x2y2 1 9x 53. x2 2 1y 2 z2 2 54. z2 2 1y 1 32 2 55. 1x 2 y2 2 2 1x 1 y2 2 56. 12a 1 32 2 2 12a 2 32 2 57. x4 2 y4 58. z4 2 81 59. 3x2 2 12 60. 3x3y 2 3xy 61. 18xy2 2 8x 62. 27x2 2 12 63. x2 2 2x 1 15 64. x2 1 x 1 2 65. 215 1 2a 1 24a2 66. 232 2 68x 1 9x2 67. 6x2 1 29xy 1 35y2
68. 10x2 2 17xy 1 6y2
69. 12p2 2 58pq 2 70q2
70. 3x2 2 6xy 2 9y2
71. 26m2 1 47mn 2 35n2
73. 26x3 1 23x2 1 35x
72. 214r2 2 11rs 1 15s2
Not For Sale
74. 2y3 2 y2 1 90y
Not For Sale
A Review of Basic Algebra
75. 6x4 2 11x3 2 35x2
76. 12x 1 17x2 2 7x3
77. x4 1 2x2 2 15
78. x4 2 x2 2 6
79. a2n 2 2an 2 3
80. a2n 1 6an 1 8
81. 6x2n 2 7xn 1 2
82. 9x2n 1 9xn 1 2
83. 4x2n 2 9y2n
84. 8x2n 2 2xn 2 3
2n
4n
n
110. Movie Stunts The formula that gives the distance
a stuntwoman is above the ground t seconds after
she falls over the side of a 144-foot tall building is
f 5 144 2 16t2. Factor the right side.
144 ft
2n
85. 10y 2 11y 2 6
86. 16y 2 25y 87. 2x3 1 2,000
88. 3y3 1 648
89. 1x 1 y2 3 2 64 90. 1x 2 y2 3 1 27 Discovery and Writing
111. Explain how to factor the difference of two squares.
112. Explain how to factor the difference of two cubes.
91. 64a6 2 y6 92. a6 1 b6 113. Explain how to factor a2 2 b2 1 a 1 b.
93. a3 2 b3 1 a 2 b 94. 1a2 2 y22 2 5 1a 1 y2 114. Explain how to factor x2 1 2x 1 1.
95. 64x6 1 y6 Factor the indicated monomial from the given expression.
96. z2 1 6z 1 9 2 225y2 115. 3x 1 2; 2 97. x2 2 6x 1 9 2 144y2 98. x2 1 2x 2 9y2 1 1 99. 1a 1 b2 2 2 3 1a 1 b2 2 10 100. 2 1a 1 b2 2 2 5 1a 1 b2 2 3 102. x6 2 13x4 1 36x2 103. x4 1 x2 1 1 104. x4 1 3x2 1 4 4
118. 3x2 2 2x 2 5; 3 119. a 1 b; a 120. a 2 b; b 1/2
101. x6 1 7x3 2 8 116. 5x 2 3; 5 117. x2 1 2x 1 4; 2 1/2
121. x 1 x ; x 122. x3/2 2 x1/2 ; x1/2 123. 2x 1 "2y; "2 124. "3a 2 3b; "3 125. ab3/2 2 a3/2b; ab 126. ab2 1 b; b21 2
105. x 1 7x 1 16 106. y4 1 2y2 1 9 107. 4a4 1 1 1 3a2 Factor each expression by grouping three terms and two
terms.
108. x4 1 25 1 6x2 127. x2 1 x 2 6 1 xy 2 2y 109. Candy To find the amount of chocolate used in
the outer coating of one of the malted-milk balls
shown, we can find the volume V of the chocolate
shell using the formula V 5 43pr31 2 43pr32. Factor
the expression on the right side of the formula. 128. 2x2 1 5x 1 2 2 xy 2 2y 129. a4 1 2a3 1 a2 1 a 1 1 130. a4 1 a3 2 2a2 1 a 2 1 Review
Outer radius r1
© Istockphoto.com/maureenpr
Inner radius r2
131. Which natural number is neither prime nor
composite? 132. Graph the interval 3 22,32 . 133. Simplify: 1x3x22 4. 1a32 3 1a22 4
134. Simplify: . 1a2a32 3
136. Simplify: "20x5. 135. a
137. Simplify: "20x 2 "125x. 138. Rationalize the denominator:
3x4x3 0
b 6x22x4
3
. !3
3
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Appendix
A62
Section
.6 RationalExpressions A63
A.6 Rational Expressions
In this section, we will learn to
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
1.Define rational expressions.
2.Simplify rational expressions.
3.Multiple and divide rational expressions.
4.Add and subtract rational expressions.
5.Simplify complex fractions.
Photo by Tim Boyle/Getty Images
Abercrombie and Fitch (A&F) is a very successful American clothing company
founded in 1892 by David Abercrombie and Ezra Fitch. Today the stores are popular shopping destinations for university students wanting to keep up with the latest
styles and trends.
Suppose that a clothing manufacturer finds that the cost in dollars of producing x fleece vintage shirts is given by the algebraic expression 13x 1 1,000. The
average cost of producing each shirt could be obtained by dividing the production
cost, 13x 1 1,000, by the number of shirts produced, x. The algebraic fraction
13x 1 1,000
x
represents the average cost per shirt. We see that the average cost of producing 200
shirts would be $18.
13 12002 1 1,000
5 18
200
An understanding of algebraic fractions is important in solving many real-life
problems.
1. ​Define Rational Expressions
Caution
If x and y are real numbers, the quotient xy 1y 2 02 is called a fraction. The number
x is called the numerator, and the number y is called the denominator.
Algebraic fractions are quotients of algebraic expressions. If the expressions
are polynomials, the fraction is called a rational expression. The first two of the
following algebraic fractions are rational expressions. The third is not, because the
numerator and denominator are not polynomials.
Not For Sale
Remember that the denominator of
a fraction cannot be zero.
5y2 1 2y
8ab2 2 16c3
x1/2 1 4x
y2 2 3y 2 7
2x 1 3
x3/2 2 x1/2
A64
Appendix
Not For Sale
A Review of Basic Algebra
We summarize some of the properties of fractions as follows:
Properties of Fractions
If a, b, c, and d are real numbers and no denominators are 0, then
Equality of Fractions
a
c
5 if and only if ad 5 bc
b
d
Fundamental Property of Fractions
Multiplication and Division of Fractions
a c
ac
a
c
a d
ad
? 5 and 4 5 ? 5
b d
bd
b
d
b c
bc
Addition and Subtraction of Fractions
a
c
a1c
a
c
a2c
and 1 5
2 5
b
b
b
b
b
b
The first two examples illustrate each of the previous properties of fractions.
EXAMPLE 1Illustrating the Properties of Fractions
Assume that no denominators are 0.
2a
4a
5 Because 2a 162 5 3 14a2
a.
3
6
b.
6xy
3 12xy2
5
Factor the numerator and denominator and divide out the common factors.
10xy
5 12xy2
5
Self Check 1 a. Is
3
5
3y
15z
5
? 5
25
b. Simplify: 15a2b
.
25ab2
Now Try Exercise 9.
EXAMPLE 2Illustrating the Properties of Fractions
Assume that no denominators are 0.
a.
c.
2r 3r
2r ? 3r
?
5
7s 5s
7s ? 5s
5
6r2
35s2
b. 2ab
ab
2ab 1 ab
1
5
5xy
5xy
5xy
5
3ab
5xy
d. 3mn
2pq
3mn 7mn
4
5
?
4pq
7mn
4pq 2pq
5
21m2n2
8p2q2
6uv2
3uv2
6uv2 2 3uv2
2 2
2 5
7w
7w
7w2
5
3uv2
7w2
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
ax
a
5
bx
b
Section
Self Check 2 Perform each operation:
3a 2a
2ab
2rs
a.
? b.
4
5b 7b
3rs
4ab
c. .6 RationalExpressions A65
5pq
3pq
1
3t
3t
d. 5mn2
mn2
2
3w
3w
Now Try Exercise 19.
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To add or subtract rational expressions with unlike denominators, we write
each expression as an equivalent expression with a common denominator. We can
then add or subtract the expressions. For example,
3x
2x
3x 172
2x 152
1
5
1
5
7
5 172
7 152
4a2
3a2
4a2 122
3a2 132
2
5
2
15
10
15 122
10 132
5
8a2
9a2
2
30
30
5
8a2 2 9a2
30
5
5
21x
10x
1
35
35
5
21x 1 10x
35
5
31x
35
2a2
30
a2
52
30
A rational expression is in lowest terms if all factors common to the numerator
and the denominator have been removed. To simplify a rational expression means
to write it in lowest terms.
2. ​Simplify Rational Expressions
To simplify rational expressions, we use the Fundamental Property of Fractions.
This enables us to divide out all factors that are common to the numerator and the
denominator.
EXAMPLE 3Simplifying a Rational Expression
Simplify: x2 2 9
1x 2 0, 32 .
x2 2 3x
SOLUTION We factor the difference of two squares in the numerator, factor out x in the denominator, and divide out the common factor of x 2 3.
1x 1 32 1x 2 32
x2 2 9
5
2
x 2 3x
x 1x 2 32
5
Self Check 3 Simplify: x23
51
x23
x13
x
a2 2 4a
1a 2 4,232 . a2 2 a 2 12
Not For Sale
Now Try Exercise 23.
We will encounter the following properties of fractions in the next examples.
A66
Appendix
Not For Sale
A Review of Basic Algebra
Properties of Fractions
If a and b represent real numbers and there are no divisions by 0, then
• a
5 a
1
• • a
2a
a
2a
5
52
52 b
2b
2b
b
a
a
2a
2a
• 2 5
5
52
b
2b
b
2b
a
51
a
Simplify: x2 2 2xy 1 y2
1x 2 y2 .
y2x
SOLUTION We factor the trinomial in the numerator, factor 21 from the denominator, and
divide out the common factor of x 2 y.
1x 2 y2 1x 2 y2
x2 2 2xy 1 y2
5
y2x
21 1x 2 y2
x2y
5
21
x2y
52
1
5 2 1x 2 y2
Self Check 4 Simplify: x2y
x2y
51
a2 2 ab 2 2b2
12b 2 a 2 02 .
2b 2 a
Now Try Exercise 25.
EXAMPLE 5Simplifying a Rational Expression
Simplify: x2 2 3x 1 2
1x 2 2,212 .
x2 2 x 2 2
SOLUTION We factor the numerator and denominator and divide out the common factor of
x 2 2.
1x 2 12 1x 2 22
x2 2 3x 1 2
5
2
1x 1 12 1x 2 22
x 2x22
5
Self Check 5 Simplify: x22
51
x22
x21
x11
a2 1 3a 2 4
1a 2 1,232 . a2 1 2a 2 3
Now Try Exercise 27.
3. Multiply and Divide Rational Expressions
EXAMPLE 6 Multiplying Rational Expressions
Multiply: x2 2 x 2 2 x2 1 2x 2 3
?
1x 2 1, 21, 22 .
x2 2 1
x22
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EXAMPLE 4Simplifying a Rational Expression
Section
.6 RationalExpressions A67
SOLUTION To multiply the rational expressions, we multiply the numerators, multiply the
denominators, and divide out the common factors.
1x2 2 x 2 22 1x2 1 2x 2 32
x2 2 x 2 2 x2 1 2x 2 3
?
5
2
1x2 2 12 1x 2 22
x 21
x22
5
1x 2 22 1x 1 12 1x 2 12 1x 1 32
1x 1 12 1x 2 12 1x 2 22
x22
x11
x21
5 1,
5 1,
51
x22
x11
x21
5x13
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Self Check 6 Simplify: x2 2 9 x 2 1
?
1x 2 0, 1, 32 . x2 2 x x2 2 3x
Now Try Exercise 33.
EXAMPLE 7Dividing Rational Expressions
Divide: x2 2 2x 2 3
x2 1 2x 2 15
4 2
1x 2 2, 22, 3, 252 .
2
x 24
x 1 3x 2 10
SOLUTION To divide the rational expressions, we multiply by the reciprocal of the second rational expression. We then simplify by factoring the numerator and denominator and
dividing out the common factors.
x2 2 2x 2 3
x2 1 2x 2 15
x2 2 2x 2 3 x2 1 3x 2 10
4
5
? 2
x2 2 4
x2 1 3x 2 10
x2 2 4
x 1 2x 2 15
5
5
5
Self Check 7 Simplify: 1x2 2 2x 2 32 1x2 1 3x 2 102
1x2 2 42 1x2 1 2x 2 152
1x 2 32 1x 1 12 1x 2 22 1x 1 52
1x 1 22 1x 2 22 1x 1 52 1x 2 32
x23
x22
x15
5 1,
5 1,
51
x23
x22
x15
x11
x12
a2 2 a
a2 2 2a
1a 2 0, 2, 222 . 4 2
a12
a 24
Now Try Exercise 39.
EXAMPLE 8 Using Multiplication and Division to Simplify a Rational Expression
Simplify: 2x2 2 5x 2 3 3x2 1 2x 2 1
2x2 1 x
1
1
? 2
4
ax 2 , 21, 3, 0, 2 b.
3x 2 1
x 2 2x 2 3
3x
3
2
SOLUTION We can change the division to a multiplication, factor, and simplify.
2x2 2 5x 2 3 3x2 1 2x 2 1
2x2 1 x
? 2
4
3x 2 1
x 2 2x 2 3
3x
2x2 2 5x 2 3 3x2 1 2x 2 1
3x
? 2
? 2
3x 2 1
x 2 2x 2 3 2x 1 x
2
12x 2 5x 2 32 13x2 1 2x 2 12 13x2
5
13x 2 12 1x2 2 2x 2 32 12x2 1 x2
53
5
5
1x 2 32 12x 1 12 13x 2 12 1x 1 12 3x
13x 2 12 1x 1 12 1x 2 32 x 12x 1 12
x23
2x 1 1
3x 2 1
x11
x
5 1,
5 1,
5 1,
5 1, 5 1
x23
2x 1 1
3x 2 1
x11
x
Not For Sale
A68
Appendix
Not For Sale
A Review of Basic Algebra
Self Check 8 Simplify: x2 2 25
x2 2 5x x2 1 2x
4 2
?
1x 2 0, 2, 5, 252 . x22
x 2 2x x2 1 5x
Now Try Exercise 45.
4. ​Add and Subtract Rational Expressions
EXAMPLE 9Adding Rational Expression with Like Deonominators
Add: SOLUTION
2x 1 5
3x 1 20
1
1x 2 252 .
x15
x15
2x 1 5
3x 1 20
5x 1 25
1
5
x15
x15
x15
5 1x 1 52
Factor out 5 and divide out the common factor of x 1 5.
1 1x 1 52
5
55
Self Check 9 Add: Add the numerators and keep the common denominator.
3x 2 2
x26
1
1x 2 22 . x22
x22
Now Try Exercise 49.
To add (or subtract) rational expressions with unlike denominators, we must find
a common denominator, called the least (or lowest) common denominator (LCD).
Suppose the unlike denominators of three rational expressions are 12, 20, and 35.
To find the LCD, we first find the prime factorization of each number.
12 5 4 ? 3 20 5 4 ? 5 35 5 5 ? 7
5 22 ? 3 5 22 ? 5
Because the LCD is the smallest number that can be divided by 12, 20, and 35,
it must contain factors of 22, 3, 5, and 7. Thus, the
LCD 5 22 ? 3 ? 5 ? 7 5 420
Comment
Remember to find the least common
denominator, use each factor the greatest
number of times that it occurs.
That is, 420 is the smallest number that can be divided without remainder by 12,
20, and 35.
When finding an LCD, we always factor each denominator and then create the
LCD by using each factor the greatest number of times that it appears in any one
denominator. The product of these factors is the LCD.
This rule also applies if the unlike denominators of the rational expressions
contain variables. Suppose the unlike denominators are x2 1x 2 52 and x 1x 2 52 3.
To find the LCD, we use x2 and 1x 2 52 3. Thus, the LCD is the product x2 1x 2 52 3 .
EXAMPLE 10Adding Rational Expressions with Unlike Denominators
Add: 1
2
1 2
1x 2 2, 222 .
x2 2 4
x 2 4x 1 4
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
To add (or subtract) rational expressions with like denominators, we add (or subtract) the numerators and keep the common denominator.
Section
.6 RationalExpressions A69
SOLUTION We factor each denominator and find the LCD.
x2 2 4 5 1x 1 22 1x 2 22
x2 2 4x 1 4 5 1x 2 22 1x 2 22 5 1x 2 22 2
The LCD is 1x 1 22 1x 2 22 2. We then write each rational expression with its denominator in factored form, convert each rational expression into an equivalent expression with a denominator of 1x 1 22 1x 2 22 2, add the expressions, and simplify.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
1
2
1
2
1 2
5
1
1x 1 22 1x 2 22
1x 2 22 1x 2 22
x2 2 4
x 2 4x 1 4
5
5
Comment
5
Always attempt to simplify the final result.
In this case, the final fraction is already
in lowest terms.
5
Self Check 10 Add: 1 1x 2 22
2 1x 1 22
1
1x 1 22 1x 2 22 1x 2 22
1x 2 22 1x 2 22 1x 1 22
x22
x12
5 1,
51
x22
x12
1 1x 2 22 1 2 1x 1 22
1x 1 22 1x 2 22 1x 2 22
x 2 2 1 2x 1 4
1x 1 22 1x 2 22 1x 2 22
3x 1 2
1x 1 22 1x 2 22 2
3
1
1 2
1x 2 3,232 . x 2 6x 1 9
x 29
2
Now Try Exercise 59.
EXAMPLE 11 Combining and Simplifying Rational Expressions with Unlike Denominators
Simplify: x22
x13
3
1x 2 1, 21, 222 .
2 2
1 2
2
x 21
x 1 3x 1 2
x 1x22
SOLUTION We factor the denominators to find the LCD.
x2 2 1 5 1x 1 12 1x 2 12
x2 1 3x 1 2 5 1x 1 22 1x 1 12
x2 1 x 2 2 5 1x 1 22 1x 2 12
The LCD is 1x 1 12 1x 1 22 1x 2 12 . We now write each rational expression as an
equivalent expression with this LCD, and proceed as follows:
x22
x13
3
2 2
1 2
2
x 21
x 1 3x 1 2
x 1x22
x22
x13
3
5
2
1
1x 1 12 1x 2 12
1x 1 12 1x 1 22
1x 2 12 1x 1 22
1x 2 22 1x 1 22
1x 1 32 1x 2 12
3 1x 1 12
5
2
1
1x 1 12 1x 2 12 1x 1 22
1x 1 12 1x 1 22 1x 2 12
1x 2 12 1x 1 22 1x 1 12
1x2 2 42 2 1x2 1 2x 2 32 1 13x 1 32
5
1x 1 12 1x 1 22 1x 2 12
5
5
x12
x21
x11
5 1,
5 1,
51
x12
x21
x11
x2 2 4 2 x2 2 2x 1 3 1 3x 1 3
1x 1 12 1x 1 22 1x 2 12
x12
1x 1 12 1x 1 22 1x 2 12
1
5
1x 1 12 1x 2 12
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Divide out the common factor of x 1 2.
x12
51
x12
A70
Appendix
Not For Sale
A Review of Basic Algebra
Self Check 11 Simplify: 4y
2
2
1 2 1y 2 1, 212 . y2 2 1
y11
Now Try Exercise 61.
5. ​Simplify Complex Fractions
Strategies for Simplifying
Complex Fractions
Method 1: Multiply the Complex Fraction by 1
• Determine the LCD of all fractions in the complex fraction.
• Multiply both numerator and denominator of the complex fraction by the
LCD. Note that when we multiply by LCD
LCD we are multiplying by 1.
Method 2: Simplify the Numerator and Denominator and then Divide
• Simplify the numerator and denominator so that both are single fractions.
• Perform the division by multiplying the numerator by the reciprocal of the
denominator.
EXAMPLE 12Simplifying Complex Fractions
1
1
1
x
y
Simplify: 1x, y 2 02 .
x
y
Method 1: We note that the LCD of the three fractions in the complex fraction is
xy. So we multiply the numerator and denominator of the complex fraction by xy
and simplify:
1
1
1
xy
xy
1
1
xya 1 b
1
x
y
x
y
x
y
y1x
5
5
5
x
x
xyx
x2
xya b
y
y
y
Method 2: We combine the fractions in the numerator of the complex fraction to
obtain a single fraction over a single fraction.
1
1 1y2
1 1x2
y1x
1
1
1
x
y
x 1y2
y 1x2
xy
5
5
x
x
x
y
y
y
Then we use the fact that any fraction indicates a division:
y1x
1y 1 x2 y
xy
y1x
x
y1x y
y1x
5
4 5
? 5
5
x
xy
y
xy
x
xyx
x2
y
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
A complex fraction is a fraction that has a fraction in its numerator or a fraction in
its denominator. There are two methods generally used to simplify complex fractions. These are stated here for you.
Section
.6 RationalExpressions A71
1
1
2
x
y
Self Check 12 Simplify: 1x, y 2 02 . 1
1
1
x
y
Now Try Exercise 81.
3a
6a2
4a2b2
8pq
4mn2
2. a. d. 2 b. 2 2 c. 5b
35b
3r s
3t
3w
a
a14
x13
3. 4. 2 1a 1 b2 5. 6. 7. a 2 1 a13
a13
x2
4x 1 6
2y
y2x
8. x 1 2 9. 4 10. 11. 12. 1x 1 32 1x 2 32 2
y21
y1x
Self Check Answers 1. a. no b. © Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Exercises A.6
Getting Ready
Perform the operations and simplify, whenever possible.
Assume that no denominators are 0.
4x 2
25y 4
15.
? 16. ? 7 5a
2z y2
8m
3m
15p
25p
17.
4
18. 4
5n
10n
8q
16q2
3z
2z
7a
3a
19.
1 20. 2 5c
5c
4b
4b
You should be able to complete these vocabulary and
concept statements before you proceed to the practice
exercises.
Fill in the blanks.
1. In the fraction ab, a is called the
.
2. In the fraction ab , b is called the
.
a
c
3. b 5 d if and only if
.
4. The denominator of a fraction can never be
Complete each formula.
a c
5. ? 5 b d
7.
a
c
1 5 b
b
a
c
6. 4 5 b
d
8. a
c
2 5 b
b
.
21.
9.
8x 16x
,
3y 6y
10. 1 1.
25xyz 50a2bc
,
12ab2c 24xyz
12. Practice
2
3x 12y
,
4y2 16x2
15rs2 37.5a3
,
4rs2 10a3
Simplify each rational expression. Assume that no denominators are 0.
7a2b
35p3q2
13.
14. 2 21ab
49p4q
22. 8rst2
7rst2
4 2 1
15m t
15m4t2
Simplify each fraction. Assume that no denominators are 0.
2x 2 4
x2 2 16
23. 2
24. 2
x 24
x 2 8x 1 16
25.
4 2 x2
x2 2 5x 1 6
26. 25 2 x2
x2 1 10x 1 25
27.
6x3 1 x2 2 12x
4x3 1 4x2 2 3x
28. 6x4 2 5x3 2 6x2
2x3 2 7x2 2 15x
29.
x3 2 8
x 1 ax 2 2x 2 2a
30. xy 1 2x 1 3y 1 6
x3 1 27
Determine whether the fractions are equal. Assume that
no denominators are 0.
2
15x2y
x2y
2 3 2
7a b
7a2b3
2
Perform the operations and simplify, whenever possible.
Assume that no denominators are 0.
x2 2 1
x2
31.
? 2
x
x 1 2x 1 1
Not For Sale
Not For Sale
A Review of Basic Algebra
32.
y2 2 2y 1 1
y12
? 2
y
y 1y22
54.
3 3.
3x2 1 7x 1 2 x2 2 x
? 2
x2 1 2x
3x 1 x
55.
34.
x2 1 x
2x2 1 x 2 3
?
2
2x 1 3x
x2 2 1
35.
x2 1 x x2 2 1
?
x21 x12
x2 1 5x 1 6 x 1 2
36. 2
?
x 1 6x 1 9 x2 2 4
2x2 1 32
x2 1 16
37.
4
8
2
38.
x2 1 x 2 6
x2 2 4
4 2
2
x 2 6x 1 9
x 29
39.
z2 1 z 2 20
z2 2 25
4
2
z 24
z25
40.
ax 1 bx 1 a 1 b
x2 2 1
4
a2 1 2ab 1 b2
x2 2 2x 1 1
41.
3x2 1 5x 2 2
6x2 1 13x 2 5
4
x3 1 2x2
2x3 1 5x2
42.
x2 1 13x 1 12
2x2 2 x 2 3
4
8x2 2 6x 2 5
8x2 2 14x 1 5
43.
x2 1 7x 1 12 x2 2 3x 2 10 x3 2 4x2 1 3x
?
? 2
x3 2 x2 2 6x x2 1 2x 2 3
x 2 x 2 20
44.
45.
x 1x 2 22 2 3
x 1x 1 12 2 2
?
1
2
1
2
1
x x 1 7 2 3 x 2 1 x x 2 72 1 3 1x 1 12
x2 2 2x 2 3
3x 2 8
x2 1 6x 1 5
?
4 2
2
21x 2 50x 2 16 x 2 3
7x 2 33x 2 10
46.
47.
48.
49.
50.
51.
52.
a13
a
1 2
a 1 7a 1 12
a 2 16
a
2
56. 2
1 2
a 1a22
a 2 5a 1 4
x
1
5 7. 2
2
x 24
x12
58.
3
x12
1
x13
x13
3
x11
4x
x21
6x
x22
2
52x
3
x26
1
2
2
1
2
x12
x11
4
x21
3
x22
1
x25
2
62x
3
2
53.
1
x11
x21
2
b2
4
2 2
b 24
b 1 2b
2
3x 2 2
x
2 2
x 1 2x 1 1
x 21
2t
t11
60. 2
2 2
t 2 25
t 1 5t
2
1
61. 2
131
y 21
y11
4
1
62. 2 1 2
2
t 24
t22
1
3
3x 2 2
63.
1
2 2
x22
x12
x 24
x
5
3 13x 2 12
64.
2
1
x23
x13
x2 2 9
1
1
x23
65. a
1
b?
x22
x23
2x
1
1
1
2
66. a
b4
x11
x22
x22
3x
x
3x 1 1
67.
2
2
x24
x14
16 2 x2
7x
3x
3x 2 1
68.
1
1 2
x25
52x
x 2 25
59.
69.
x3 1 27
x2 1 4x 1 3
x2 1 x 2 6
4
4
a
b x2 2 4
x2 1 2x
x2 2 3x 1 9
3
x
1
x14
x24
70.
71.
2
1
2
1
2 2
1 2
x 1 3x 1 2
x 1 4x 1 3
x 1 5x 1 6
2
22
2
2z 2 2y
1
2
1y 2 x2 1z 2 x2
x2y
x2z
3x 2 2
4x2 1 2
3x2 2 25
2
1
x2 1 x 2 20
x2 2 25
x2 2 16
72.
3x 1 2
x14
1
1 2
2
8x2 2 10x 2 3
6x 2 11x 1 3
4x 1 1
Simplify each complex fraction. Assume that no denominators are 0.
3a
b
73.
6ac
b2
3t2
9x
74. t
18x
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Appendix
A72
Section
2
3a b
ab
27
x2y
ab
77.
y2x
ab
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
75.
x2 2 5x 1 6
2x2y
78. x2 2 9
2x2y
1
1
1
x
y
79.
xy
1
1
1
x
y
81.
1
1
2
x
y
80. xy
11
11
1
x
y
1
1
2
x
y
82. 1
1
1
x
y
2
3a
4a
2
b
x
83.
1
1
1
b
ax
12
6
x112
x
85.
6
x151
x
87.
89.
1
x1
x
86. x
y
x2
21
y2
2z
3
z
88. k5
1
1
1
1
k1
k2
Section 1
Section 2
98. Electronics The combined resistance R of three
resistors with resistances of R1, R2, and R3 is given
by the following formula. Simplify the complex
fraction on the right side of the formula.
R5
1
1
1
1
1
1
R1
R2
R3
Discovery and Writing
1
x
1
2 2x13
x
2x2 1 4
90. 4x
21
5
2x
1
1
x23
x22
92. 3
x
2
x23
x22
Simplify each complex fraction. Assume that no denominators are 0.
x
ab
99.
100. 1
3
1 1 21
2 1 21
3x
2a
1
y
101.
102. 1
2
11
21
1
2
11
21
x
y
a
c
ad 1 bc
1 5
is valid.
b
d
bd
a
c
a d
104. Explain why the formula 4 5 ? is valid.
b
d
b c
105. Explain the Commutative Property of Addition
and explain why it is useful.
106. Explain the Distributive Property and explain why
it is useful.
103. Explain why the formula
Review
Write each expression without using negative exponents,
and simplify the resulting complex fraction. Assume that
no denominators are 0.
1
y21
93.
94. 1 1 x21
x21 1 y21
3 1x 1 22 21 1 2 1x 2 12 21
95.
1x 1 22 21
96.
97. Engineering The stiffness k of the shaft shown in
the illustration is given by the following formula
where k1 and k2 are the individual stiffnesses of
each section. Simplify the complex fraction on the
right side of the formula.
x231
x
2
2
x12
x21
91.
3
x
1
x12
x21
84. 12
3xy
1
12
xy
3x
Applications
3u2v
4t
76. 3uv
.6 RationalExpressions A73
Write each expression without using absolute value symbols.
107. 0 26 0 108. 0 5 2 x 0 , given that x , 0 Simplify each expression.
109. a
x3y22 23
b x21y
Not For Sale
2x 1x 2 32 21 2 3 1x 1 22 21
1x 2 32 21 1x 1 22 21
111. "20 2 "45 112. 2 1x2 1 42 2 3 12x2 1 52 110. 127x62 2/3 A74
Appendix
Not For Sale
A Review of Basic Algebra
APPENDIXpReview
Sets of Real Numbers
Definitions and Concepts
Examples
Natural numbers: The numbers that we count with.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, %
Whole numbers: The natural numbers and 0.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, %
Integers: The whole numbers and the negatives of the
natural numbers.
% , 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, %
Rational numbers: {x 0 x can be written in the form ab
1b 2 02 , where a and b are integers.}
5 2 7
1
2
2 5 , 25 5 2 , , 2 , 0.25 5
1
1 3 3
4
All decimals that either terminate or repeat.
13
2
7
3
5 0.75,
5 2.6, 5 0.666 % 5 0.6,
5 0.63
4
5
3
11
Irrational numbers: Nonrational real numbers.
All decimals that neither terminate nor repeat.
"2, 2"26, p, 0.232232223 %
Real numbers: Any number that can be expressed as a
decimal.
26, 2
9
, 0, p, "31, 10.73
13
Prime numbers: A natural number greater than 1 that is
divisible only by itself and 1.
2, 3, 5, 7, 11, 13, 17, %
Composite numbers: A natural number greater than 1
that is not prime.
4, 6, 8, 9, 10, 12, 14, 15, %
Even integers: The integers that are exactly divisible by 2.
% , 26, 24, 22, 0, 2, 4, 6, %
Odd integers: The integers that are not exactly divisible
by 2.
% , 25, 23, 21, 1, 3, 5, %
Associative Properties:
of addition 1a 1 b2 1 c 5 a 1 1b 1 c2
5 1 14 1 72 5 15 1 42 1 7
Commutative Properties:
of addition a 1 b 5 b 1 a
4 1 7 5 7 1 4 1.7 1 2.5 5 2.5 1 1.7
of multiplication 1ab2 c 5 a 1bc2
of multiplication ab 5 ba
15 ? 42 ? 7 5 5 14 ? 72
4 ? 7 5 7 ? 4 1.7 12.52 5 2.5 11.72 Distributive Property:
a 1b 1 c2 5 ab 1 ac
3 1x 1 62 5 3x 1 3 ? 6 0.2 1y 2 102 5 0.2y 2 0.2 1102
Open intervals have no endpoints.
123, 22 Double Negative Rule:
2 12a2 5 a
Closed intervals have two endpoints.
Half-open intervals have one endpoint.
2 1252 5 5 2 12a2 5 a
3 23, 2 4 123, 2 4 (
)
[
]
(
]
–3
–3
–3
2
2
2
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
section A.1
Appendix Review A75
Definitions and Concepts
Examples
Absolute value:
If x $ 0, then 0 x 0 5 x.
7
7
0 7 0 5 7 0 3.5 0 5 3.5 ` ` 5 0 0 0 5 0
2
2
Distance:
The distance between points a and b on a number line is
d 5 0b 2 a0.
The distance on the number line between points with
coordinates of 23 and 2 is d 5 0 2 2 1232 0 5 0 5 0 5 5.
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
If x , 0, then 0 x 0 5 2x.
Exercises
7
7
0 27 0 5 7 0 23.5 0 5 3.5 ` 2 ` 5 2
2
Consider the set 5 26,23,0,12,3,p,"5,6,8 6 . List the
numbers in this set that are
1. natural numbers. 16. 12x 1 y2 1 z 5 1y 1 2x2 1 z
2. whole numbers. 3. integers.
6. real numbers. Graph each subset of the real numbers:
18. the prime numbers between 10 and 20
19. the even integers from 6 to 14
Consider the set 5 26,23,0,12,3,p,"5,6,8 6 . List the
numbers in this set that are
7. prime numbers. Graph each interval on the number line.
20. 23 , x # 5
8. composite numbers. 9. even integers.
10. odd integers. 2 1. x $ 0 or x , 21
4. rational numbers. 5. irrational numbers. Determine which property of real numbers justifies each
statement.
11. 1a 1 b2 1 2 5 a 1 1b 1 22
12. a 1 7 5 7 1 a
13. 4 12x2 5 14 ? 22 x
14. 3 1a 1 b2 5 3a 1 3b
15. 15a2 7 5 7 15a2
17. 2 1262 5 6
2 2. 122, 4 4
2 3. 12`, 22 d 125,` 2
24. 12`,242 c 3 6,` 2
Write each expression without absolute value symbols.
26. 0 225 0 25. 0 6 0 2 7. 0 1 2 "2 0 28. 0 "3 2 1 0 29. On a number line, find the distance between points
with coordinates of 25 and 7. Not For Sale
Appendix
A76
Not For Sale
A Review of Basic Algebra
Integer Exponents and Scientific Notation
section A.2
Definitions and Concepts
Examples
Natural-number exponents:
n factors of x











x 5 x ? x ? x ? c? x
x5 5 x ? x ? x ? x ? x
Rules of exponents:
If there are no divisions by 0,
• 1xm2 n 5 xmn
• xmxn 5 xm1n
• 1xy2 n 5 xnyn
• x0 5 1 1x 2 02 m
• x
5 xm2n
xn
n
1
5 n
x
x
• a b
y
2n
y
5a b
x
n
Scientific notation:
A number is written in scientific notation when it is written in the form N 3 10n, where 1 # 0 N 0 , 10.
Exercises
Write each expression without using exponents.
31. 125a2 2 30. 25a3 Write each expression using exponents.
32. 3ttt 33. 122b2 13b2 Simplify each expression.
35. 1p32 2 34. n2n4 36. 1x3y22 4 38. 1m n 2 23 0 2
a5
40. 8 a
4
1
1
2 5
6
36
60 5 1
622 5
x6
5 x624 5 x2
x4
x 23
6 3
63
216
a b 5a b 5 35 3
6
x
x
x
Write each number in scientific notation.
386,000 5 3.86 3 105 0.0025 5 2.5 3 1023
Write each number in standard form.
7.3 3 103 5 7,300 5.25 3 1024 5 0.000525
3x2y22 22
b x2y2
23x3y 22
44. a
b xy3
42. a
a23b2 22
b ab23
2m22n0 23
45. a2 2 21 b 4m n
43. a
46. If x 5 23 and y 5 3, evaluate 2x2 2 xy2 Write each number in scientific notation.
47. 6,750 48. 0.00023 3
a
b b2
p22q2 3
39. a
b 2
a2 22
41. a 3 b b
37. a
x 5 x5
a b 5 5
y
y
1xy2 5 5 x5y5
n
x
x
• a b 5 n y
y
• x2n
1x22 7 5 x2?7 5 x14
x2x5 5 x215 5 x7
Write each number in standard notation.
49. 4.8 3 102 50. 0.25 3 1023 51. Use scientific notation to simplify
145,0002 1350,0002
. 0.000105
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
n
Appendix Review A77
Rational Exponents and Radicals
section A.3
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Definitions and Concepts
Examples
Summary of a1/n definitions:
• If a $ 0, then a1/n is the nonnegative number b
such that bn 5 a.
• If a , 0 and n is odd, then a1/n is the real number
b such that bn 5 a.
• If a , 0 and n is even, then a1/n is not a real
number.
Rule for rational exponents:
If m and n are positive integers,
a1/n is a real number, then
m
n
n
Properties of radicals:
If all radicals are real numbers and there are no divisions by 0, then
n
n
n
• "ab 5 "a"b
n
a
"a
5 n
Åb
"b
m
n
mn
n m
Exercises
54. 132x52 1/5 56. 121,000x62 1/3 58. 1x12y22 1/2 60. a
53. a
2c2/3c5/3 1/3
b c22/3
Simplify each expression.
2/3
62. 64 64. a
66. a
3/4
16
b 81
8 22/3
b
27
68. 12216x32 2/3 1/3
27
b 125
55. 181a42 1/4 57. 1225x22 1/2 59. a
61. a
x12 21/2
b
y4
a21/4a3/4 21/2
b
a9/2
23/5
63. 32
65. a
5
5
5 10
"
32x10 5 "
32"
x 5 2x2
4 12
x12
"
x
x12/4
x3
5 4
5
5
Å 625
5
5
"625
4
70. "36 72.
9
Å 25
73. 74. "x2y4 76.
m8n4
Å p16
4
77. Simplify and combine terms.
78. "50 1 "8 3
3
4
3 27
Å 125
3 3
75. "
x 4
a15b10
Å c5
5
79. "12 1 "3 2 "27 Rationalize each denominator.
2/5
32
b 243
81.
83.
"7
"5
1
3
"
2
Not For Sale
6
"64 5 "
64 5 2
71. 2"49 80. "24x 2 "3x 16 23/4
b
625
pa/2pa/3
69. pa/6
67. a
2#3
Simplify each expression.
Simplify each expression, if possible.
3
"
125 5 1251/3 5 5
3
3
3
#
"64 5 "
8 5 2 #"
64 5 "4 5 2
• #"a 5 #"a 5"a
52. 1211/2 12162 1/2 is not a real number because no real number
squared is 216.
82/3 5 181/32 2 5 22 5 4 or 1822 1/3 5 641/3 5 4
Definition of "a:
n
"a 5 a1/n
n
12272 1/3 5 23 because 1232 3 5 227
is in lowest terms, and
am/n 5 1a1/n2 m 5 1am2 1/n
• 161/2 5 4 because 42 5 16
82. 84. 8
"8
2
3
"
25
Appendix
A78
Not For Sale
A Review of Basic Algebra
Rationalize each numerator.
"2
5
section A.4
86. "5
5
87.
"2x
3
88. 3
3"
7x
2
Polynomials
Definitions and Concepts
Examples
Monomial: A polynomial with one term.
2, 3x, 4x2y, 2x3y2z
Binomial: A polynomial with two terms.
2t 1 3, 3r2 2 6r, 4m 1 5n
Trinomial: A polynomial with three terms.
3p2 2 7p 1 8, 3m2 1 2n 2 p
The degree of a monomial is the sum of the exponents on
its variables.
The degree of 4x2y3 is 2 1 3 5 5
The degree of a polynomial is the degree of the term in
the polynomial with highest degree.
The degree of the first term of 3p2q3 2 6p3q4 1 9pq2 is
2 1 3 5 5, the degree of the second term is 3 1 4 5 7,
and the degree of the third term is 1 1 2 5 3. The
degree of the polynomial is the largest of these. It is 7.
Multiplying a monomial times a polynomial:
a 1b 1 c 1 d 1 c 2 5 ab 1 ac 1 ad 1 c
Addition and subtraction of polynomials:
To add or subtract polynomials, remove
parentheses and combine like terms.
3 1x 1 22 5 3x 1 6
2x 13x2 2 2y 1 32 5 6x3 2 4xy 1 6x
Add: 13x2 1 5x2 1 12x2 2 2x2 .
13x2 1 5x2 1 12x2 2 2x2 5 3x2 1 5x 1 2x2 2 2x
5 3x2 1 2x2 1 5x 2 2x
5 5x2 1 3x
Subtract: 14a2 2 5b2 2 13a2 2 7b2 .
14a2 2 5b2 2 13a2 2 7b2 5 4a2 2 5b 2 3a2 1 7b
5 4a2 2 3a2 2 5b 1 7b
5 a2 1 2b
Special products:
1x 1 y2 2 5 x2 1 2xy 1 y2
1x 2 y2 2 5 x2 2 2xy 1 y2
1x 1 y2 1x 2 y2 5 x2 2 y2
Multiplying a binomial times a binomial:
Use the FOIL method.
12m 1 32 2 5 4m2 1 12m 1 9
14t 2 3s2 2 5 16t2 2 24ts 1 9s2
12m 1 n2 12m 2 n2 5 4m2 2 2mn 1 2mn 2 n2 5 4m2 2 n2
12a 1 b2 1a 2 b2 5 2a 1a2 1 2a 12b2 1 ba 1 b 12b2
5 2a2 2 2ab 1 ab 2 b2
5 2a2 2 ab 2 b2
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
85.
Appendix Review A79
Definitions and Concepts
Examples
The conjugate of a 1 b is a 2 b.
To rationalize the denominator of a radical expression,
multiply both the numerator and denominator of the
rational expression by the conjugate of the denominator.
Rationalize the denominator:
x
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
"x 1 2
5
x 1"x 2 22
x
"x 1 2
1"x 1 22 1"x 2 22
x"x 2 2x
5
x24
Division of polynomials:
To divide polynomials, use long division.
Give the degree of each polynomial and tell whether the
polynomial is a monomial, a binomial, or a trinomial.
89. x3 2 8 90. 8x 2 8x2 2 8 91. "3x2 92. 4x4 2 12x2 1 1 Perform the operations and simplify.
93. 2 1x 1 32 1 3 1x 2 42 94. 3x2 1x 2 12 2 2x 1x 1 32 2 x2 1x 1 22 95. 13x 1 22 13x 1 22 96. 13x 1 y2 12x 2 3y2 97. 14a 1 2b2 12a 2 3b2 98. 1z 1 32 13z2 1 z 2 12 99. 1an 1 22 1an 2 12 2
100. Q"2 1 xR 101. Q"2 1 1RQ"3 1 1R 3
3
3
3 2 2RQ"
9 1 2"
3 1 4R
102. Q"
Multiply the numerator and denominator
by the conjugate of
"x 1 2.
Simplify.
Divide: 2x 1 3q6x3 1 7x2 2 x 1 3.
3x2
2x 1 3q 6x3 1 7x2
6x3 1 9x2
2 2x2
2 2x2
Exercises
.
2 x11
2 x13
2 x
2 3x
1 2x 1 3
1 2x 1 3
0
Rationalize each denominator.
2
22
104. 103.
"3 2 1
"3 2 "2
105.
2x
"x 2 "y
106. Rationalize each numerator.
"x 1 2
107.
5
108. 1 2 "a
a
110. 4a2b3 1 6ab4
2b2
"x 2 2
Perform each division.
3x2y2
109. 3 6x y
"x 1 "y
111. 2x 1 3q2x3 1 7x2 1 8x 1 3 112. x2 2 1qx5 1 x3 2 2x 2 3x2 2 3 Not For Sale
Appendix
A80
Not For Sale
A Review of Basic Algebra
Factoring Polynomials
section A.5
Definitions and Concepts
Examples
ab 1 ac 5 a 1b 1 c2
3p3 2 6p2q 1 9p 5 3p 1p2 2 2pq 1 32
x2 2 y2 5 1x 1 y2 1x 2 y2
4x2 2 9 5 12x 1 32 12x 2 32
Factoring the difference of two squares:
Factoring trinomials:
• Trinomial squares
x2 1 2xy 1 y2 5 1x 1 y2 2
9a2 1 12ab 1 4b2 5 13a 1 2b2 13a 1 2b2 5 13a 1 2b2 2
x2 2 2xy 1 y2 5 1x 2 y2 2
• To factor general trinomials use trial and error or
grouping.
Factoring the sum and difference of two cubes:
x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22
27a3 2 8b3 5 13a 2 2b2 19a2 1 6ab 1 4b22
Exercises
Factor each expression completely, if possible.
2
6x2 2 5x 2 6 5 12x 2 32 13x 1 22
r3 1 8 5 1r 1 22 1r2 2 2r 1 42
x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22
113. 3t3 2 3t r2 2 4rs 1 4s2 5 1r 2 2s2 1r 2 2s2 5 1r 2 2s2 2
114. 5r3 2 5 119. x2 1 6x 1 9 2 t2
120. 3x2 2 1 1 5x
121. 8z3 1 343
122. 1 1 14b 1 49b2
123. 121z2 1 4 2 44z
124. 64y3 2 1,000
2
115. 6x 1 7x 2 24
116. 3a 1 ax 2 3a 2 x
117. 8x3 2 125
118. 6x2 2 20x 2 16
125. 2xy 2 4zx 2 wy 1 2zw 126. x8 1 x4 1 1
D
section A.6
efinitions and Concepts
Properties of fractions:
If there are no divisions by 0, then
a
c
• 5 if and only if ad 5 bc.
b
d
a
ax
• 5
b
bx
a c
ac
• ? 5
b d
bd
a
c
ad
• 4 5
b
d
bc
Algebraic Fractions
xamples
3x
6x
5
because 13x2 8 and 4 16x2 both equal 24x.
4
8
6x2
3?2?x?x
3
5
3 5
8x
2?4?x?x?x
4x
3p 2p
3p ? 2p
3p ? 2p
p2
?
5
5
5 2
2q 6q
2q ? 6q
2q ? 3 ? 2q
2q
2t
2t
2t 6s
2t ? 6s
2t ? 2 ? 3s
4
5
?
5
5
52
3s
6s
3s 2t
3s ? 2t
3s2t
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
Factoring out a common monomial:
E
Appendix Review A81
Definitions and Concepts
a
c
a1c
a
c
a2c
1 5
• 2 5
b
b
b
b
b
b
a
a
• a ? 1 5 a • 5 a • 5 1
1
a
a
2a
a
2a
• 5
52
52
b
2b
2b
b
a
a
2a
2a
5
52
• 2 5
b
2b
b
2b
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
• Simplifying rational expressions:
To simplify a rational expression, factor the numerator
and denominator, if possible, and divide out factors that
are common to the numerator and denominator.
Adding or subtracting rational expressions:
To add or subtract rational expressions with unlike
denominators, find the LCD of the expressions, write
each expression with a denominator that is the LCD,
add or subtract the expressions, and simplify the result,
if possible.
Examples
x
y
x1y
3p
p
3p 2 p
2p
p
1 5
2
5
5
5
4
4
4
2q
2q
2q
2q
q
7
7
7 ? 1 5 7 5 7 51
1
7
7
27
7
27
5
52
52
2
22
22
2
7
7
27
27
2 5
5
52
2
22
2
22
2 2 x
To simplify 2x
2 4 , factor 21 from the numerator and
2 from the denominator to get
21 122 1 x2
21 1x 2 22
21
1
22x
5
5
5
52
2x 2 4
2 1x 2 22
2 1x 2 22
2
2
2x
2x
2x 1x 2 32
2x 1x 1 22
2
5
2
1x 1 22 1x 2 32
1x 2 32 1x 1 22
x12
x23
2
1
1
2x x 2 3 2 2x x 1 22
5
1x 1 22 1x 2 32
2x2 2 6x 2 2x2 2 4x
1x 1 22 1x 2 32
210x
5
1x 1 22 1x 2 32
5
Complex fractions:
To simplify complex fractions, multiply the numerator
and denominator of the complex fraction by the LCD
of all the fractions.
y
y
2ya1 2 b
2
2
2y 2 y2
y 12 2 y2
5
5
5
1
1
1
1
21y
21y
1
2ya 1 b
y
2
y
2
12
Exercises
Simplify each rational expression.
22x
a2 2 9
127. 2
128. 2
x 2 4x 1 4
a 2 6a 1 9
Perform each operation and simplify. Assume that no
denominators are 0.
133.
x2 1 x 2 6 x2 2 x 2 6
x2 2 4
? 2
4 2
2
x 2x26 x 1x22
x 2 5x 1 6
2x 1 6
2x2 2 2x 2 4 x2 2 x 2 2
4
b 2
x15
x2 2 25
x 2 2x 2 15
2
3x
135.
1
x24
x15
134. a
129.
x2 2 4x 1 4 x2 1 5x 1 6
?
x12
x22
130.
2y2 2 11y 1 15 y2 2 2y 2 8
? 2
y2 2 6y 1 8
y 2y26
136.
5x
3x 1 7
2x 1 1
2
1
x22
x12
x12
131.
2t2 1 t 2 3
10t 1 15
4 2
2
3t 2 7t 1 4
3t 2 t 2 4
137.
x
x
x
1
1
x21
x22
x23
132.
p2 1 7p 1 12
p2 2 9
4
p3 1 8p2 1 4p
p2
138.
Not For Sale
x
3x 1 7
2x 1 1
2
1
x11
x12
x12
139.
140.
Not For Sale
Appendix
A Review of Basic Algebra
3 1x 1 12
5 1x2 1 32
x
2
1
2
x
x
x11
3x
x2 1 4x 1 3
x2 1 x 2 6
1 2
2
x11
x 1 3x 1 2
x2 2 4
Simplify each complex fraction. Assume that no denominators are 0.
5x
2
141. 2 3x
8
3x
y
142. 6x
y2
1
1
1
x
y
143.
x2y
144. x21 1 y21
y21 2 x21
APPENDIXpTest
Consider the set 5 27,223,0,1,3,"10,4 6 .
1. List the numbers in the set that are odd integers. 2. List the numbers in the set that are prime numbers.
Determine which property justifies each statement.
3. 1a 1 b2 1 c 5 1b 1 a2 1 c 4. a 1b 1 c2 5 ab 1 ac Graph each interval on a number line.
6. 12`, 232 c 3 6,` 2
5. 24 , x # 2 Write each expression without using absolute value symbols.
8. 0 x 2 7 0 , when x , 0
7. 0 217 0 Find the distance on a number line between points with the
following coordinates.
9. 24 and 12 10. 220 and 212 Simplify each expression. Assume that all variables represent positive numbers, and write all answers without using
negative exponents.
r2r3s
11. x4x5x2 12. 4 2 rs
1a21a22 22
x0x2 6
13.
14. a 22 b 23
a
x
Write each number in scientific notation.
15. 450,000 16. 0.000345 Write each number in standard notation.
17. 3.7 3 103 18. 1.2 3 1023 Simplify each expression. Assume that all variables represent positive numbers, and write all answers without using
negative exponents.
36 3/2
20. a b 19. 125a42 1/2 81
8t6 22/3
3
21. a 9 b
22. "27a6 27s
2 3. "12 1 "27 3
3
24. 2"
3x4 2 3x"
24x 25. Rationalize the denominator: 26. Rationalize the numerator: Perform each operation.
27. 1a2 1 32 2 12a2 2 42 28. 13a3b22 122a3b42 29. 13x 2 42 12x 1 72 30. 1an 1 22 1an 2 32 31. 1x2 1 42 1x2 2 42 32. 1x2 2 x 1 22 12x 2 32 33. x 2 3q6x2 1 x 2 23 34. 2x 2 1q2x3 1 3x2 2 1 Factor each polynomial.
35. 3x 1 6y 36. x2 2 100 37. 10t2 2 19tw 1 6w2 x
"x 2 2
"x 2 "y
"x 1 "y
. .
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
A82
AppendixTest A83
38. 3a3 2 648 39. x4 2 x2 2 12 40. 6x4 1 11x2 2 10 Perform each operation and simplify if possible. Assume
that no denominators are 0.
x
2
41.
1
x12
x12
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
42.
x
x
2
x11
x21
43.
x2 1 x 2 20 x2 2 25
?
x2 2 16
x25
44.
x12
x2 2 4
4
x 1 2x 1 1
x11
2
Simplify each complex fraction. Assume that no denominators are 0.
1
1
1
a
b
x21
45.
46. 21
1
x 1 y21
b
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Not For Sale