Practise Qixereisas

Practise Qixereisas
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Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers 17
Practice Exercises
In Exercises 1-16, evaluate each algebraic expression for the
given value or values of the variable(s).
1. 7 + 5x, for x = 10
2, 8 + 6x, for x = 5
3, 6x — y,forx = 3andy = 8
4. 8x — y, for x = 3 and y = 4
5, x? + Зх, Юг х = &
6. x? + Sx, for x = 6
7, х? — 6x + 3,forx = 7
8. x? — Tx + 4,forx = 8
9, 4 + 5(x — T°, for x = 9
10. 6 + 5(x — 6)°, forx = 8
11. x* — 3(x — y), for x = 8 and y = 2
12. x? — 4(x — y), for x = 8 and y = 3
S(x + 2)
13. 3x 014 ja > for x = 10
14. rn 2) for x =9
15, — > ,forx = —2andy = 4
x+1
2х + у
xy — 2x
The formula
16. ‚for x = —2andy = 4
C = SE 32)
expresses the relationship between Fahrenheit temperature, F,
and Celsius temperature, C. In Exercises 17-18, use the formula
to convert the given Fahrenheit temperature to its equivalent
temperature on the Celsius scale.
17. 50°F 18. 86°F
A football was kicked vertically upward from a height of 4 feet
with an initial speed of 60 feet per second. The formula
h = 4 + 60t — 162
describes the ball's height above the ground, h, in feet, t seconds
after it was kicked. Use this formula to solve Exercises 19-20.
19. What was the ball’s height 2 seconds after it was kicked?
20. What was the ball’s height 3 seconds after it was kicked?
In Exercises 21-28, find the intersection of the sets.
21. {1, 2, 3, 4} N {2, 4, 5} 22. {1,3,7} N {2, 3, 8)
23. {s,e,t} N {t, es] 24. {r,e,a, 1} N {L e, a, r}
25. (1,3, 5,7} N {2, 4, 6, 8, 10}
26. {0, 1,3, 5} N {-5, —3, —1}
27. {a, b,c, dy ND 28. {w,y,z) NY
In Exercises 29-34, find the union of the sets.
29. {1, 2,3, 4} U {2,4,5) 30. {1, 3, 7, 8} U {2, 3, 8}
31. {1,3, 5,7} U {2, 4, 6, 8, 10} 32. (0, 1, 3, 5} U (2, 4, 6}
33, la,e,i,o, ul UJ 34. lem, p, 1 y) UD
In Exercises 35-38, list all numbers from the given set that are
a. natural numbers, b. whole numbers, e. integers, d. rational
numbers, e. irrational numbers, E. real numbers.
35. {-9,-4,0,0.25, V3 ‚9.2, V100}
36. {-7,—0.6,0, 49, V50)
37. (-11,-2;0,0.75, V5, п, V64)
38. {-5,-03,0, V2, V4}
39. Give an example of a whole number that is not a natural
number. |
40. Give an example of a rational number that is not an integer.
41. Give an example of a number that is an integer, a whole
number, and a natural number.
42. Give an example of a number that is a rational number, an
integer, and a real number.
Determine whether each statement in Exercises 43-50 is true or false.
43. —13 = —2 44, —6 > 2
45. 4 = —7 46, —13 < —5
47. —п = —т 48. —3 > —13
49. 0 = —6 | 50. 0 = —13
In Exercises 51-60, rewrite each expression without absolute
value bars.
51. 1300| | 52. |-203|
53. [12 — al 54. |7 — |
55. | V2 — 5| 56. | VS — 13)
—3 —7
57, — 58, —
-3| -7|
59. |-3| — |--7| 60. 1-5] — |--13]
In Exercises 61-66, evaluate each algebraic expression for x = 2
andy = —5.
61. |x + у) 62. |x — yl
63. |x| + |y| 64. |x| — |y|
65. — ве. |+ М!
vl X
In Exercises 67-74, express the distance between the given
numbers using absolute value. Then find the distance by evaluating
the absolute value expression.
67. 2 and 17 68. 4 and 15
69. —2 and 5 70. —6 and 8
74. —19 and —4 | 72. —26 and —3
73. —3.6 and —1.4 74. —5.4 and —1.2
In Exercises 75-84, state the name of the property illustrated.
75. 6 + (+4) = —4) +6
76. 11-(7 + 4) = 11:7 + 11:4
77. 6 + (2 + 7) = (6 + 2) + 7
78. 6. (2:3) = 6. (3.2)
79. (2 + 3) + (4 + 5) = (4 + 5) + (2 + 3)
80. 7. (11:8) = (11: 8).7 |
81. 2(-8 + 6) = —16 + 12
82. —8(3 + 11) = —24 + (-88)
18 Chapter P Prerequisites: Fundamental Concepts of Algebra
1
. + 3) = 1,x = —3
3 (x + 3) (x ) X
84. (х + 4) + [-(х + 4)] = 0
In Exercises 85-96, simplify each algebraic expression,
85. 5(3x + 4) — 4 86. 2(5x + 4) — 3
87. 5(3x — 2) + 12x 88. 2(5x — 1) + 14x
89. 7(3y — 5) + 2(4y + 3)
90. 42y — 6) + 3(5y + 10)
91. 5(3y — 2) — (Ty + 2)
92. 4(5y — 3) — (6y + 3)
93. 7 — 4/3 — (4y - 3
95. 18x% + 4 — [6(x* — 2) + 5]
96. 14x* + 5 — [7(x* = 2) + 4]
94. 6 — 5/8 — (2y — 4)]
In Exercises 97-102, write each algebraic expression without
parentheses.
97. —(—14x)
99, —(2x — 3y — 6)
101. 3(3x) + [(4y) + (—4y)]
98. —(—17y)
100. —(5x — 13y — 1)
102. 5(2y) + [(-7x) + 7x]
Practice Plus
In Exercises 103-110, insert either <, >, or = in the shaded
area to make a true statement.
103. |-6| |-3| 104. |-20| |-50]
105. 2 0.6 106. Е |-2.5|
109. то |-1| 110. |-2| e
In Exercises 111-120, use the order of operations to simplify each
expression.
111. 82 — 16 + 22.4 —3 112. 10° — 100 = 52.2 — 3
.7 — 22 “= .
113, 92-3 114. Maria
[32 — (-2)F (12 — 3-2)
115. 8 — 3[-2(2 — 5) — 4(8 — 6)]
116. 8 — 3[-2(5 — 7) — 5(4 — 2)]
2(—2) — 4(=3) 6(-4) — 5(-3)
117. о 118. 9-10
(5 — 6)* — 213 — 7| 12 + 3.512? + 32|
119. —— 120. =
89 — 3:35 7+3-—6
In Exercises 121-128, write each English phrase as an algebraic
expression. Then simplify the expression. Let x represent the number.
121. A number decreased by the sum of the number and four
122. A number decreased by the difference between eight and
the number
123. Six times the product of negative five and a number
124. Ten times the product of negative four and a number
125. The difference between the product of five and a number
and twice the number
126. The difference between the product of six and a number
and negative two times the number
127. The difference between eight times a number and six more
than three times the number
128. Eight decreased by three times the sum of a number and six
Application Exercises
The maximum heart rate, in beats per minute, that you should
achieve during exercise is 220 minus your age:
220 — a.
— This algebraic expression gives maximum
heart rate in terms of age, a.
The following bar graph shows the target heart rate ranges
for four types of exercise goals. The lower and upper limits of
these ranges are fractions of the maximum heart rate, 220 — a.
Exercises 129-130 are based on the information in the graph.
Target Heart Rate Ranges for Exercise Goals
Exercise Goal
Boost performance ней уе! 1
| as a competitive athlete -
Improve cardiovascular -
...... Sonditioning
Lose weight |
Improve overall health and -
‚reduce risk of heart attack -
2 1 3 7 4 39
5 2 5 10 5 1 1
Fraction of Maximum Heart Rate, 220 — a
129. Ifyour exercise goalis to improve cardiovascular conditioning,
the graph shows the following range for target heart rate, H,
in beats per minute:
“ Lower limit of range > i,
= 15(220 — a)
ВЕ у À
: Upper limit of range > H = = (220 — a)
a. What is the lower limit of the heart range, in beats per
minute, for a 20-year-old with this exercise goal?
b. What is the upper limit of the heart range, in beats per
minute, for a 20-year-old with this exercise goal?
130. If your exercise goal is to improve overall health, the graph
shows the following range for target heart rate, H, in beats
per minute:
Upper limit of range. = H = =(220 — a)
a. What is the lower limit of the heart range, in beats per
minute, for a 30-year-old with this exercise goal?
b. What 1s the upper limit of the heart range, in beats per
minute, for a 30-year-old with this exercise goal?
The
privé
"Tuition and Fees
Sc
The
mot
coll
infc
131
132
13:
The bar graph shows the average cost of tuition and fees at
private four-year colleges in the United States.
Average Cost of Tuition and Fees at Private
Four-Year United States Colleges
27 pane ena 26273 eee
26 -
25
24
23 eee eas
22 --
2
20 +
19 ee a
18 EEE , №
17 Ш
16 3 HN
15]
Tuition and Fees
(in thousands of dollars)
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Ending Year in the School Year
Source: The College Board |
The formula
T = 26x* + 819x + 15,527
models the average cost of tuition and fees, T, at private U.S.
colleges for the school year ending x years after 2000. Use this
information to solve Exercises 131-132.
131. a. Use the formula to find the average cost of tuition and
fees at private U.S. colleges for the school year ending in
2010.
b. By how much does the formula underestimate or
overestimate the actual cost shown by the graph for the
school year ending in 2010?
Use the formula to project the average cost of tuition and
fees at private U.S. colleges for the school year ending in
2013. |
132. a. Use the formula to find the average cost of tuition and
fees at private U.S. colleges for the school year ending in
2009.
b. By how much does the formula underestimate or
overestimate the actual cost shown by the graph for the
school year ending in 2009? |
c. Use the formula to project the average cost of tuition and
fees at private U.S. colleges for the school year ending in
2012.
133. You had $10,000 to invest. You put x dollars in a safe,
government-insured certificate of deposit paying 5%
per year. You invested the remainder of the money in
noninsured corporate bonds paying 12% per year. Your
total interest earned at the end of the year is given by the
algebraic expression |
с
+
0.05x + 0.12(10,000 — x).
a. Simplify the algebraic expression.
b. Use each form of the algebraic expression to determine
your total interest earned at the end of the year if you
invested $6000 in the safe, government-insured certificate
of deposit.
Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers 19 |
134. It takes you 50 minutes to get to campus. You spend
t minutes walking to the bus stop and the rest of the time
riding the bus. Your walking rate is 0.06 mile per minute and
the bus travels at a rate of 0.5 mile per minute. The total
distance walking and traveling by bus is given by the
algebraic expression
0.067 + 0.5(50 — 7).
a. Simplify the algebraic expression.
b. Use each form of the algebraic expression to determine
the total distance that you travel if you spend 20 minutes
walking to the bus stop.
135. Read the Blitzer Bonus beginning on page 15. Use the formula
600n
BAC=—— ——
AC = TO.6n + 169)
and replace w with your body weight. Using this formula
and a calculator, compute your BAC for integers fromn = 1
to n = 10. Round to three decimal places. According to this
model, how many drinks can you consume in an hour without
exceeding the legal measure of drunk driving?
Writing in Mathematics
Writing about mathematics will help you learn mathematics.
For all writing exercises in this book, use complete sentences to
respond to the question. Some writing exercises can be answered
in a sentence, others require a paragraph or two. You can decide
how much you need to write as long as your writing clearly and
directly answers the question in the exercise. Standard references
such as a dictionary and a thesaurus should be helpful.
136. What is an algebraic expression? Give an example with
your explanation.
137. If n is a natural number, what does b” mean? Give an
example with your explanation.
138. What does it mean when we say that a formula models
real-world phenomena?
139. What is the intersection of sets À and В?
140. What is the union of sets A and B?
141. How do the whole numbers differ from the natural
numbers?
142. Can a real number be both rational and irrational? Explain
© your answer,
143. If you are given two real numbers, explain how to determine
~ which is the lesser. | |
Critical Thinking Exercises
Make Sense? In Exercises 144-1 47, determine whether each
statement makes sense or does not make sense, and explain your
reasoning.
144. My mathematical model describes the data for tuition
and fees at public four-year colleges for the past ten years
extremely well, so it will serve as an accurate prediction for
the cost of public colleges in 2050.
145. A model that describes the average cost of tuition and fees
at private U.S. colleges for the school year ending x years
after 2000 cannot be used to estimate the cost of private
education for the school year ending in 2000.
20 Chapter P Prerequisites: Fundamental Concepts of Algebra
146. The humor in this cartoon is based on the fact that the
football will never be hiked.
HUTT ONE...
HUTT Two...
HUTY THREE...
POINT 141592653589 79323846204
358327050288419710939937510582
OTTHIMEI230 7810 002562089980
280248253421170679821480B65132 ||
B23000470938U4G09550SB2231725 |!
35408128 48111745028410270193 lil
BEZ110855964462294,, OM... 4, UM... [1
WHOSE IDEA WAS
BALL ON PI, AGAIN?
м}
7 To HIKE THE YOURS,
Now KEEP
GOING.
Foxtrot © 2003, 2009 by Bill Amend/Used by permission of Universal Uclick.
All rights reserved.
147. Just as the commutative properties change groupings, the
associative properties change order.
In Exercises 148-155, determine whether each statement is true
or false. If the statement is false, make the necessary change(s) to
produce a true statement.
148. Every rational number is an integer.
149. Some whole number not int
150. Some rational numbers are not positive.
151. Irrational numbers cannot be negative.
152. The term x has no coefficient.
153. 5 + 3(x — 4) = 8(x — 4) = 8x - 32
154. —х — х = —х + (-x) = 0
155. x — 0.02(x + 200) = 0.98x — 4
In Exercises 156—158, insert either < or > in the shaded area
between the numbers to make the statement true.
156. V2 15 | 157. —
3.14 Tr
—3,5
Proview Exercises
Exercises 159-161 will help you prepare for the material covered
in the next section.
159. In parts (a) and (b), complete each statement.
a. b* b> = (b-b-b-bY(b-b-b) =
b. b> b> =(bb-b-b-b)b-b-b-b+b) =p’
c¢. Generalizing from parts (a) and (b), what should be
done with the exponents when multiplying exponential
expressions with the same base?
160. In parts (a) and (b), complete each statement.
Ь’ Be B-Wbbbb p?
“6 BU
DE BBb-bbbbb
bh. — ES — b’
bh? By
¢. Generalizing from parts (a) and (b), what should be
done with the exponents when dividing exponential
expressions with the same base?
161. If 6.2 is multiplied by 10°, what does this multiplication do
to the decimal point in 6.2?
Use the product rule.
Use the quotient rule. -
Use the zero-exponent
rule.
Use the negative-
exponent rule.
Use the power rule.
Find the power of a
product.
Find the power of a
quotient.
Simplify exponential
expressions.
Use scientific. notation.
x Bigger than the biggest thing ever and then some. Much bigger than that in fact,
| really amazingly immense, a totally stunning size, real wow, that's big”, time
Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of
concept we're trying to get across here.
Douglas Adams, The Restaurant at the End of the Universe
Although Adams’s description may not quite apply to this $15.2 trillion national
debt, exponents can be used to explore the meaning of this “staggeringly huge”
number. In this section, you will learn to use exponents to provide a way of putting
large and small numbers in perspective.
The Product and Quotient Rules
We have seen that exponents are used to indicate repeated multiplication. Now
consider the multiplication of two exponential expressions, such as b* - b*. We are
multiplying 4 factors of b and 3 factors of b. We have a total of 7 factors of b:
i"
practice Exercises
wa wg A
Bas Eh
fes Fa #
Evaluate each exponential expression in Exercises 1-22.
1. 5-2 2. 6-2
3. (-2)° 4. Gr
5, —2° 6. —2*
7. (-3)° 8. (-9)°
9, —3° 10. —9°
11. 47° 12. 27°
13, 2-7 14. 33.3?
15. (2) 16. (3%)
28 3%
17. 7 18. a
19. 372-3 20. 273-2
2° 34
21.75 22. 37
Simplify each exponential expression in Exercises 23-04.
23. x”y 24. ху”
25. x%° 26. x'y
27. xx’ 28. xx’
29. x >. x 30. xx”
31. (x) 32. (xy
33. (x7) 34. (>
x14 y
35. 7 36. i
14 30
X X
37. = 38. io
39. (83 40. (6x7
De I 6 3
a. (4) a. (-9)
X У
43. (-32Y 44. (-3х“уб)
45. (3х“)(2х”) 46. (11x°)(9x'%)
47. (-9x yX-2x y") 48.. (-Sx*y)(—6x'y'1)
8x” 20x%
49. — 50. ——
2x* 10х°
13,4 141.6
SL. 25a”b 82, Ba
—5a*b) —7a'b
53 14b” 54 205*9
764 " 105%
55. (43) 56. (10)
5 24x°y° 10x*y°
7. 12755 58. E
X y xy
3X-2 &\73
59, E 60. (=)
y y
— 41,2 \3 _ 147.8\3
а. ( 15% 6 ( 30a 2)
Sap 1007”
63.
Section P.2 Exponents and Scientific Notation 33
( 305b? Y
A)
6
120 >
In Exercises 65-76, write each number in decimal notation
without the use of exponents.
65.
67.
69.
71.
73.
75.
3.8 x 10°
6 x 1074
—7,16 x 10°
79x10"
—4,15 x 10°
—6.00001 x 10%
66.
68.
70.
72.
74.
76.
9.2 x 10?
7 x 105
—8.17 x 10%
6.8 x 107
—3.14 x 10%
—7.00001 x 109
In Exercises 77-86, write each number in scientific notation.
77.
79.
81.
83.
85.
32,000
638,000,000,000,000,000
—5716
0.6027
—0.00000000504
78.
80.
82.
84.
86.
64,000
579,000,000,000,000,000
—3829
0.6083
—0.00000000405
In Exercises 87-106, perform the indicated computations. Write
the answers in scientific notation. If necessary, round the decimal
factor in your scientific notation answer to two decimal places.
87.
89.
91.
93.
95.
97.
99.
101.
103.
105.
(3 xX 1092.1 x 10%
(1.6 x 105(4 x 10714)
(6.1 x 1082 x 10)
(4.3 x 10%)(6.2 x 10%
8.4 x 108
4 X 10°
3.6 x 10%
9 x 107
4.8 x 1072
2.4 x 106
24 x 107
4.8 X 10%
480,000,000,000
0.00012
0.00072 x 0.003
0.00024
Practice Plus
98.
100.
102.
104.
106.
. (2 X 104.1 x 10°)
. (14 x 105)(3 x 10715)
. (5.1 x 1053 x 1079
. (8.2 X 10%)(4.6 x 10%
6.9 x 10°
3 x 10°
1.2 x 10°
2 x 107
7.5 х 107
2,5 x 10°
1.5 x 1072
3 x 1076
282,000,000,000
0.00141
66,000 x 0.001
0.003 x 0.002
In Exercises 107-114, simplify each exponential expression.
Assume that variables represent nonzero real numbers.
107.
(xy)
(yy
109. (2х yz 5) (2x)
34,5 2
xy Z
111. | —————
(= ~4 =)
y ©
108.
(ay y”
(xy)
110. 3x*y7 3)”
x*y°7° —4
132. \ ~———
х yz
34 Chapter P Prerequisites: Fundamental Concepts of Algebra
1
13 (21х2у (2x*y)*(16x3
—3 42
(2x7y”)
1
4 CE 0
A AZ
(2x*y5)
Application Exercises
The bar graph shows the total amount Americans paid in federal
taxes, in trillions of dollars, and the U.S. population, in millions,
from 2007 through 2010. Exercises 115-116 are based on the
numbers displayed by the graph.
Federal Taxes and the United States Population
Federal Taxes Collected Population
$3.00 350
Federal Taxes Collected
(trillions of dollars)
Sources:
115. a.
с.
116. а.
с
$2.50 ----
$2.00
$1.50
$1.00
$0.50
2.57 303 306 308 309
250
200
1
150
Population (millions)
I
100
2007 2008 2009 2010
Year
Internal Revenue Service and U.S. Census Bureau
In 2010, the United States government collected
$2.17 trillion in taxes. Express this number in scientific
notation.
In 2010, the population of the United States was
approximately 309 million. Express this number in
scientific notation.
Use your scientific notation answers from parts (a)
and (b) to answer this question: If the total 2010 tax
collections were evenly divided among all Americans,
how much would each citizen pay? Express the answer
in decimal notation, rounded to the nearest dollar.
In 2009, the United States government collected
$2.20 trillion in taxes. Express this number in scientific
notation.
In 2009, the population of the United States was
approximately 308 million. Express this number in
scientific notation.
Use your scientific notation answers from parts (a)
and (b) to answer this question: If the total 2009 tax
collections were evenly divided among all Americans,
how much would each citizen pay? Express the answer in
decimal notation, rounded to the nearest dollar.
In the dramatic arts, ours is the era of the movies. As
individuals and as a nation, we've grown up with them. Our
images of love, war, family, country — even of things that
terrify us—owe much to what we’ve seen on screen. The bar
graph at the top of the next column quantifies our love for
movies by showing the number of tickets sold, in millions, and
the average price per ticket for five selected years. Exercises
117-118 are based on the numbers displayed by the graph.
United States Film Admissions
and Admission Charges
1600
1400
1190 1210
E 2
8 ? 1200 mg earns E e AA aE $6 =
Е © 1000 |- ren 40 0 29= = $5 5
Q “ Ë
chs |. mn 9
<< 800 $4 E
gg 600 еее ее $3 >
= 8 =
< = 400 FE $2 £
= >
= <
200
1990 1995 2000 2005 2010
Year
Source: Motion Picture Association of America
117. Use scientific notation to compute the amount of money that =
the motion picture industry made from box-office receipts -
in 2010. Express the answer in scientific notation.
118. Use scientific notation to compute the amount of money that
the motion picture industry made from box office receipts in
2005. Express the answer in scientific notation.
119. The mass of one oxygen molecule is 5.3 X 10% gram. Find ;
the mass of 20,000 molecules of oxygen. Express the answer
in scientific notation.
120. The mass of one hydrogen atom is 1.67 x 107% gram. Find -
the mass of 80,000 hydrogen atoms. Express the answer in | ;
scientific notation.
121. There are approximately 3.2 X 107 seconds in a year. 3
According to the United States Department of Agriculture,
Americans consume 127 chickens per second. How many
chickens are eaten per year in the United States? Express
the answer in scientific notation.
122
»
year. Express the answer in scientific notation.
Writing in Mathematics
123. Describe what it means to raise a number to a power. In …
your description, include a discussion of the difference
between —5* and (-5.
124. Explain the product rule for exponents. Use 2° - 2° in your
explanation.
125. Explain the power rule for exponents. Use (3° in your
explanation.
8
126. Explain the quotient rule for exponents. Use — in your Е
explanation.
127. Why is (—3x")(2x*) not simplified? What must be done to
simplify the expression?
128. How do you know if a number is written in scientific notation?
129. Explain how to convert from scientific to decimal notation
and give an example.
130. Explain how to convert from decimal to scientific notation ; |
and give an example.
131. Refer to the Blitzer Bonus on page 32. Use scientific notation | |
to verify any three of the bulleted items on ways to spend
$1 trillion.
Convert 365 days (one year) to hours, to minutes, and, finally,
to seconds, to determine how many seconds there are ina -
Ci
sia
ret
13
13
13
13
In
or
pr
13
13
14
14
14
14
14
14
Average Frice per licket
Critical Thinking Exercises
Make Sense? In Exercises 132-135, determine whether each
statement makes sense or does not make sense, and explain your
reasoning.
132. There are many exponential expressions that are equal
to 36x1, such as (6x, (6x>(6x), 3603), and A.
133. If 572 is raised to the third power, the result is a number
between 0 and 1.
134. The population of Colorado is approximately 4.6 x 10%,
135. I just finished reading a book that contained approximately
1.04 x 10° words.
In Exercises 136-143, determine whether each statement is true
or false. If the statement is false, make the necessary change(s) to
produce a true statement.
136. 472 < 4% 137. 5? > 25
138. (-2)* = 27° 139. 52.57? > 25.27
8 х 10%
140. 534.7 = 5.347 x 10° 141 = 2 x 10%
4 x 107
142. (7 x 10°) + (2 x 10°) = 9 x 10°
143. (4 x 10%) + (3 X 10%) = 4.3 x 10°
144. The mad Dr. Frankenstein has gathered enough bits and
pieces (so to speak) for 27 + 27 of his creature-to-be.
Write a fraction that represents the amount of his creature
that must still be obtained.
145. If b* = MN, b“ = M, and b? = N, what is the relationship
among À, C, and D?
146. Our hearts beat approximately 70 times per minute. Express
in scientific notation how many times the heart beats over
—a lifetime of 80 years. Round the decimal factor in your
scientific notation answer to two decimal places.
Evaluate square roots.
Simplify expressions of
the form Va.
Use the product rule to
simplify square roots.
Use the quotient rule to
simplify square roots.
Add and subtract square
Rationalize denominators.
Evaluate and perform
operations with higher roots.
Understand and use
rational exponents.
This photograph
mathematical models used by
Albert Einstein at a lecture on
relativity. Notice the radicals that
appear in many of the formulas.
Among these models, there 1s one
describing how an astronaut in a
moving spaceship ages more slowly
than friends who remain on Earth.
No description of your world can
be complete without roots and
— Tadicals. In this section, in addition
roots. to reviewing the basics of radical
— expressions and the use of rational
exponents to indicate radicals, you
will see how radicals model time
dilation for a futuristic high-
speed trip to a nearby star.
Section P.3 Radicals and Rational Exponents 35
Group Exercise
147. Putting Numbers into Perspective. A large number can
be put into perspective by comparing it with another
number. For example, we put the $15.2 trillion national debt
(Example 12) and the $2.17 trillion the government collected
in taxes (Exercise 115) by comparing these numbers to the
number of U.S. citizens.
For this project, each group member should consult an
almanac, a newspaper, or the Internet to find a number
greater than one million. Explain to other members of the
group the context in which the large number is used, Express
the number in scientific notation. Then put the number into
perspective by comparing it with another number.
Preview Exercises
Exercises 148-150 will help you prepare for the material covered
in the next section.
148. a. Find V16- V4.
b. Find V16-4.
с. Based on your answers to parts (a) and (b), what can you
conclude?
149. a. Use a calculator to approximate V300 to two decimal
places.
b. Use a calculator to approximate 10V3 to two decimal
places.
€. Based on your answers to parts (a) and (b), what can you
conclude?
150. a. Simplify: 21x + 10x.
b. Simplify: 21V2 + 10V2.
shows
48 Chapter P Prerequisites: Fundamental Concepts of Algebra
Fill in each blank so that the resulting statement is true.
1. The symbol V is used to denote the nonnegative,
or, Square root of a number.
2, \/64 = 8 because _.. = 64.
3 Va=
4. The product rule for square roots states that if à and b
are nonnegative, then Vab =
5. The quotient rule for square roots states that if à and b
- are nonnegative and b # 0,then „=
6. 8V3 + 10V3 =
7. V3 + V5 = V3 + V95-3 = V3+_ V3 =
8. The conjugate of 7 + V3is —.
a
San E=Xercisoes
Evaluate each expression in Exercises 1-12, or indicate that the
root is not a real number.
1. V36 2. V25
3, —V36 4 —V25
5. V-36 6. V-25
7. V25 — 16 8. \/144 + 25
9. V25 — V16 10. V144 + V25
11. V(-13Y 12. V(-17Y
Use the product rule to simplify the expressions in Exercises 13-22.
In Exercises 17-22, assume that variables represent nonnegative
real numbers.
13. V50 14. V27
15. V 45x? 16. \/125х?
17. \2х‹ \У/бх 18. V10x- V 8x
19. Vx3 20. Vy?
21. V2x?- V6r 22. V6x- Vx?
Use the quotient rule to simplify the expressions in Exercises 23-32.
Assume that x > O.
1 1
23. 31 24. 29
49 121
APTE 26. ./—
25 16 9
| 3 73
m. 48x 28. 72x
V3x V8x
4 4
29, 150x 30. 24x
V3x V3x
V200x° V500x°
3, —— 32. |
10x” V 10x”
In Exercises 33-44, add or subtract terms whenever possible.
33. 7V3 + 6V3 34. 8V5 + 11V5
35, 6V 17x — 8V 17x 36. 4V 13x — 6V 13x
9.
10.
11.
12
+
13.
14.
37,
39.
41.
43.
44.
We rationalize the denominator of by
5
vid - V2
multiplying the numerator and denominator
by :
In the expression V64, the number 3 is called
the and the number 64 1s called the
Y-32 = —2 because = —32,
If n is odd, Va” = ___.
ln iseven, Va? =__.
a" =
16 = (V16)* = (ys =
. V20+6V5
\/8 + 3\/2 38
V50x — V8x 40. V63x — V28x
3/18 + 5V50 42. 4V12 - 2V75
3V8 — V32 + 3V72 — V5
3V54 — 224 — V96 + 463
In Exercises 45-54, rationalize the denominator.
45.
47.
49.
51.
53.
1 2
—— 46. ——
V7 10
V2 48, L
V5 V3
3 50, — —
3 + V11 3 + V7
7 5
— 52, ———
V5 —2 V3 —1
6 sa 1
V5 + V3 V7 — V3
Evaluate each expression in Exercises 55—66, or indicate that the
root is not a real number.
55.
58.
61.
64.
V125 56. Y8 57. Y-8
Y-125 59. Y-16 60. Y-81
V(-3)* 62. V(-2)* 63. V(-3)
2) 65. \Y_ 66. A;
Simplify the radical expressions in Exercises 67-74 if possible.
I DS
al N
Ÿ32 68. Y150
Vt 70. Yx5
Vo. Ve - 72. V12-V4
Y 64x* \162х5
74.
In E
75.
77.
79.
80.
81.
mE
calc
83.
85.
67,
89,
In!
91
93
9¢
in.
101
10:
10:
10
Pr
In
10
11
In
va
Ц
11
uu
In Exercises 75-82, add or subtract terms whenever possible,
„в, 4N/2 + 3V/2 76. 6V/3 + 2V/3
77. 5/16 + №54 78. 3V/24 + V81
79. Ys4xy? — y V128x
во. Y24xy? — y V8lx
81. V2 + 8 82. V3 + V15
In Exercises 83-90, evaluate each expression without using a
calculator.
1 1
83. 36 84. 121?
1 1
85. 8° 86. 27°
2 2
87. 125° 88. 8°
4 LS
89. 32 ° 90. 16 ?
In Exercises 91-100, simplify using properties of exponents.
1 1 2 3
91. (7x3) (2x4) 92. (3x3) (4x*)
1 3
20x? 72x*
93. — 94, UT
Sx* Ox 3
Z\3 4 \ 5
0s. (x3) 96. (x5)
1 1
97. (25х“у“)? 98. (125x°y%)3
Log 14
зу“) (2y5)
99. 73 100. — 5
y? - 10
In Exercises 101-108, simplify by reducing the index of the radical.
101, №5? 102. №7?
103. Vx* 104. Vx
105, Vx* 106. Vx
107. Vx? 108. Vx%y®
Practice Plus
In Exercises 109-110, evaluate each expression.
109. V/16 + V625
110. VV V169 + V9 + V 1000 + Y216
In Exercises 111-114, simplify each expression. Assume that all
variables represent positive numbers.
Lol 3 34°
111. (49x 2y4) 2 xy 2) 112, (8x 6y3)3 (xy 75)
_5 175 78
x “у
1 1 _7
q 2
113. ’ xy 4
114.
_3 _2
x 4 y 4
Application Exercises
115, The popular comic strip Fox Trot follows the off-the-wall lives
of the Fox family. Youngest son Jason is forever obsessed
by his love of math. In the math-themed strip shown at the
top of the next column, Jason shares his opinion in a coded
message about the mathematical abilities of his sister Paige.
Section P.3 Radicals and Rational Exponents 49
by Bill Amend
t-{1-13-5-10-2-15-18-13-23-6-11-17-11-12-22-11-12-10
Key:
* HAL
De VUN R="
TE 5: (542x2)»
cla VE
= *Зсо5
= (9х+9х) + 3х Ves?
Mz (9xi1)-(7x11) № 2079
м Y400 X= 920% 512
THe +3
or ve + ++ т = 89 +9
= tf
a 7: АНН
SE 9x dx Te
La
Foxtrot O 2003, 2009 by Bill Amend/Used by permission of Universal
Uclick. All rights reserved.
Solve problems A through Z in the left panel. Then decode
Jason Fox’s message involving his opinion about the
mathematical abilities of his sister Paige shown on the first
line. |
Hints: Here 1s the solution for problem C and partial
solutions for problems Q and U.
These
are: = ain 7% — a o.
from С = sin > = sin 90° = 1
trigonometry. 2 2 |
= [9x2dx = 3 | = 3:22 — 3:0? =
This ВО 0 = -3 сов т = —3 сов 180° = -3(-1) =
rom
calculos.
116.
Note: “The comic strip FoxTrot is now printed in more
than one thousand newspapers. What made cartoonist Bill
Amend, a college physics major, put math in the comic? “I
always try to use math in the strip to make the joke accessible
to anyone,” he said. “But if you understand math, hopefully
youll like it that much more!” We highly recommend the
math humor in Amend's FoxTrot collection Math, Science,
and Unix Underpants (Andrews McMeel Publishing, 2009).
America is getting older. The graph shows the projected
elderly U.S. population for ages 65-84 and for ages 85 and
older.
Projected Elderly United States Population
Ages 65-84 Ages 85+
80
Projected Population (millions)
2010 2020 2030
Year
2040 2050
Source; U.S. Census Bureau
50 Chapter P Prerequisites: Fundamental Concepts of Algebra
117
”
118.
The formula Е = 5Vx + 34.1 models the projected
number of elderly Americans ages 65-84, E, in millions,
x years after 2010.
a. Use the formula to find the projected increase in the
number of Americans ages 65-84, in millions, from 2020
to 2050. Express this difference in simplified radical
form.
©. Use a calculator and write your answer in part (a) to the
nearest tenth. Does this rounded decimal overestimate or
underestimate the difference in the projected data shown
by the bar graph on the previous page? By how much?
The early Greeks believed that the most pleasing of all
rectangles were golden rectangles, whose ratio of width to
height 1s
W 2
h 5-1
The Parthenon at Athens fits into a golden rectangle once
the triangular pediment is reconstructed.
Rationalize the denominator of the golden ratio. Then use
a calculator and find the ratio of width to height, correct to
the nearest hundredth, in golden rectangles.
Use Einstein's special-relativity equation
2
R,= Rp /1 — (2).
described in the Blitzer Bonus on page 47, to solve this
exercise. You are moving at 90% of the speed of light.
Substitute 0.9c for v, your velocity, in the equation. What is
your aging rate, correct to two decimal places, relative to a
friend on Earth? If you are gone for 44 weeks, approximately
how many weeks have passed for your friend?
The perimeter, P, of a rectangle with length 1 and width w is
given by the formula P = 21 + Qw. The area, À, is given by the
formula À =
lw. In Exercises 119-120, use these formulas to
find the perimeter and area of each rectangle. Express answers in
simplified radical form. Remember that perimeter is measured in
linear units, such as feet or meters, and area is measured in square
units,
119.
T1
2
such as square feet, ft”, or square meters, m”.
120. |
175 feet 4/20 feet
| 2/20 feet рр
Writing in Mathematics
121. Explain how to simplify V10- V5.
122. Explain how to add V3 + V1.
123.
124.
125.
126.
127.
128.
Describe what it means to rationalize a denominator. Use
1 1
both 7 and 5 7
What difference is there in simplifying V(-5Y a
VS?
What does a” mean?
Describe the kinds of numbers that have rational fifth roots.
in your explanation.
Why must a and b represent nonnegative numbers when
we write Va- Vb =
restriction in the case of Ya- Yb = Vab? Explain.
Read the Blitzer Bonus on page 47. The future is now: You ê
have the opportunity to explore the cosmos in a starship
traveling near the speed of light. The experience will enable |
you to understand the mysteries of the universe in deeply
personal ways, transporting you to unimagined levels of
knowing and being. The downside: You return from your
two-year journey to a futuristic world in which friends and
loved ones are long gone. Do you explore space or stay here
on Earth? What are the reasons for your choice?
|
aT e i ay Fr pd + M à ge #
& цей LE Wat wing E y ea do pep ss Biar AP ott NT E ge 0 we
DUO HEHE ESSEN
- Ro .
Make Sense? In Exercises 129-132, determine whether each
statement makes sense or does not make sense, and explain your
reasoning.
129.
130.
131.
132.
The joke in this Peanuts cartoon would be more effective if …
Woodstock had rationalized the denominator correctly in
the last frame.
YOURE LUCKY, DO YOU | [YOU DON'T HAVE TO KNOW
KNOW THAT BIRD ? YOU'RE | | ABOUT RATIONALIZING THE
LUCKY BECAUSE YoU DONT | | DENOMINATOR AND DUMB
HAVE TO STUDY MATH! THINGS LIKE THAT
77
| Beau В
© 1979 United Feature Syndicate, Ine,
Peanuts © 1978 Peanuts Worldwide LLC, Used by permission
of Universal Uclick. All rights reserved.
Using my calculator, I determined that 67 = 279,936, so 6
must be a seventh root of 279,936.
I simplified the terms of 220 + 4\/75, and then I was
able to add the like radicals.
m
When I use the definition for a”, I usually prefer to first
raise a to the m power because smaller numbers are
involved.
In Exercises 133-136, determine whether each statement is true
or false. If the statement is false, make the necessary change(s) to
produce a true statement.
133.
1
1
72. 7 = = 49 134. 8 3 =—2
ab? Is it necessary to use this ;
135.
136.
ml
137
138
139
140
141
and f ;
ots. ;
hen -
this -
You ;
hip.
ble
ply
ur
nd
ère
La
Wd
Use. : |
of -
135. The cube root of —8 is not a real number.
vo V10
136.5 74° 4
In Exercises 137-1 38, fil in each box to make the statement true.
137. (5+ Y )5-V )=2
138, V x = 5x’
Section P.4 Polynomials 51
b. The birthday boy, excited by the inscription on the cake, tried
to wolf down the whole thing. Professor Mom, concerned
about the possible metamorphosis of her son into a blimp,
exclaimed, “Hold on! It is your birthday, so why
4
8 3 + 27° › ›
not take — —— of the cake? I'll eat half of what’s left
16 4 + 27!
over.” How much of the cake did the professor eat?
139. Find the exact value of J 13 + \/2 + — — —
the use of a calculator.
"140. Place the correct symbol, > or <, in the shaded area
between the given numbers. Do not use a calculator. Then
check your result with a calculator.
b. V7 + V18 V7 + 18
141. a. A mathematics professor recently purchased a birthday
cake for her son with the inscription
11
а. 32 33
5 3 1
Happy(22-24 + 24)th Birthday.
How old is the son?
Understand t the he vocabulary
of polynomials. о
Add and subtract
polynomials.
Multiply polynomials.
Use FOIL in polynomial
multiplication.
Use special products in
polynomial multiplication.
Perform operations with
~ polynomials in several
— variables.
Understand the vocabulary of
polynomials.
without
SR
Preview Praercisas
Exercises 142-144 will help you prepare for the material covered
In the next section.
142. Multiply: (2x*y")(5x*y7).
143. Use the distributive property to multiply:
2x*(8x* + 3x).
144. Simplify and express the answer in descending powers of x:
2x(x* + 4x + 5) + 3 + 4х + 5).
Can that be Axl, your author’s yellow lab, sharing a special
moment with a baby chick? And if it is (it is), what
possible relevance can this have to polynomials?
An answer 1s promised before you reach the
Exercise Set. For now, we open the section by
defining and describing polynomials.
How We Define Polynor Y:
More education results in a higher income.
The mathematical models
Old Dog...New Chicks
M = 0.6x> + 285x? — 2256x + 15,112
and W = —1.2x" + 367x° — 4900x + 26,561
describe the median, or middlemost, annual income for men, M, and women, W, who
have completed x years of education. Well be working with these models and the
data upon which they are based in the Exercise Set.
The algebraic expressions that appear on the right sides of the models are
examples of polynomials. A polynomial is a single term or the sum of two or
more terms containing variables with whole-number exponents. The polynomials
above each contain four terms. Equations containing polynomials are used in such
diverse areas as science, business, medicine, psychology, and sociology. In this section,
we review basic ideas about polynomials and their operations.
How We Describe Polynomials
Consider the polynomial
7х° — 9х? + 13х — 6.
Tited at ; ;
cessive,
lylabs
ts.
ng of ; |
of
=
we
al
a gra вы Ÿ A
Practice слег ее
In Exercises 1-4, is the algebraic expression a polynomial? If it is,
write the polynomial in standard form.
1. 2x + 3x* — 5 2. 2x + 3x * — 5
X .
‘In Exercises 5-8, find the degree of the polynomial.
6. — 4x) + 7x? — 11
8. x? — 8x7 + 15x* + 91
5. 3х? — 5х + 4
7, х? — 4x + 9х — 12х" + 63
In Exercises 9-14, perform the indicated operations. Write the
resulting polynomial in standard form and indicate its degree.
9. (—6x> + 5x? — 8x + 9) + (17x + 2x* — 4x — 13)
10. (—7x* + 6x? — 11x + 13) + (19x? — 11x* + 7x — 17)
1. (17x — 5x? + 4x — 3) — (5x? — 9x? — 8x + 11)
12. (18x* — 2x* — 7x + 8) — (9x* — 6x* — Sx + 7)
13. (5x? — 7x — 8) + (2x* — 3x + 7) — (x* — 4x — 3)
14. (8x? + Tx — 5) — (3x* — 4x) - (—6x? — 5х? + 3)
In Exercises 15-58, find each product.
15. (x + 1)(x* — x + 1) 16. (x + 5)(x* — 5x + 25)
17. (2x — 3)(x* — 3x + 5) 18. (2x — 1(x* — 4x + 3)
19. (x + 7)(x + 3) 20. (x + 8)(x + 5)
21. (x — 5)(x + 3) 22. (x — D(x + 2)
23. (3x + 5)(2x + 1) 24. (7х + 4)(3x + 1)
25. (2х — 3)(5х + 3) 26. (2х — 5)(7х + 2)
27. (5х — 4)(3x* — 7) 28. (7х? — 2)(3х? — 5)
29. (8x* + 3)(х” — 5) 30. (7x + 5)(x* — 2)
31. (x + 3)x — 3) | 32. (x + 5)(x — 5)
33. (3x + 2)(3x — 2) 34. (2x + 5)(2x — 5)
35. (5 — 7х)(5 + 7х) 36. (4 — 3x)(4 + 3x)
37. (4x? + 5xY(4x* — 5х) 38. (3х? + 4x)(3x* — 4x)
39. (1 — y)A + y”) 40. 2 — y) + y)
41, (x + 2)? 42. (x + 5)
43. (2x + 3)? 44. (3х + 2)*
45, (x — 3)? 46. (x — 4)“
47. (4х? — 1)“ 48. (5х? — 3)?
49. (7 — 2х)? 50. (9 — 5х)”
51, (x + 1)° 52. (x +2)”
53. (2x + 3)? 54. (3x + 4)
55. (x — 39 56. (x — 1}
57. (3x — 4° 58. (2x — 3)°
In Exercises 59-66, perform the indicated operations. Indicate the
degree of the resulting polynomial.
59, (5x?y — 3xy) + (2x*y — xy)
60. (—2x%y + xy) + (4x”y + 7xy)
Section P.4 Polynomials 61
61. (4xy + 8xy + 11) + (-2x”y + 5ху + 2)
62. (7x*y? — 5x%y? + 3xy) + (—18x%y? — 6x*y* — xy)
63. (х° + 7ху — 5y) — (6x>— xy + 4у”)
64. (x* — Txy — 5y*) — (6x* — 3xy + 4y”)
65. (3x1y? + Sx*y — 3y) — (2x'y” — 3x3y — 4y + 61)
66. (5x*y? +'6x*y — Ту) — (3x*y? — 5x"y — 6y + 8x)
In Exercises 67-82, find each product.
67. (x + S5yX7x + 3y — 68. (x + 9y)(6x + 7y)
69. (x — 3y)(2x + Ty) 70. (3x — у)(2х + 5у)
71. (3xy — 1)(Sxy + 2) 72. (Tx%y + 1)(2x%y — 3)
73. (Tx + Sy)? 74, (9x + Ty)”
75. (x2y? — 3) 76. (xy? - 5Y
77. (х — у)? + ху + у”) 78. (x + y)” — xy + y”
79. (3x + 5y)(3x — 5y) 80. (7x + 3y)(7x — 3y)
81. (7ху? — 10у)(7ху? + 10у) — 82. (3xy” — 4y)(3xy* + 4y)
Practice Plus
In Exercises 83-90, perform the indicated operation or operations.
83. (3x + 4yY — (3x — 4y)?
84. (5х + 2yY — (5x — 2yY
85. (5х — 7)(3х — 2) — (4x — 5(6x — 1)
86. (3x + 5)(2x — 9) — (7x — 2)(x — 1)
87. (2x + 5)(2x — 5)(4x* + 25)
88. (3x + 4)(3x — 4(9x* + 16)
(2х — 7)
89, ———— 90
(2x = 7)
(5х — 3)°
° (5х — 3)"
Application Exercises
As you complete more years of education, you can count
on a greater income. The bar graph shows the median,
or middlemost, annual income for Americans, by level of
education, in 2009.
Median Annual Income, by Level of Education, 2009
Men % Women
Median Annual Income
(thousands of dollars)
8 10 12 13 14 16 18 20
Years of School Completed
Source: Bureau of the Census
62 Chapter P Prerequisites: Fundamental Concepts of Algebra
Here are polynomial models that describe the median annual
income for men, M, and for women, W, who have completed x
years of education:
M = 312х? — 2615х + 16,615
W = 316x* — 4224x + 23,730
М = 0.6x* + 285x” — 2256x + 15,112
И’ = —1.2х° + 367х? — 4900х + 26,561
Exercises 91-92 are based on these models and the data displayed
by the graph at the bottom of the previous page.
91. a. Use the equation defined by a polynomial of degree 2 to
find the median annual income for a man with 16 years of
education. Does this underestimate or overestimate the
median income shown by the bar graph? By how much?
b. Use the equations defined by polynomials of degree 3 to
find a mathematical model for M — W.
According to the model in part (b), what is the difference,
rounded to the nearest dollar, in the median annual income
between men and women with 14 years of education?
Tao. +
d. According to the data displayed by the graph, what is the
actual difference in the median annual income between
men and women with 14 years of education? Did the result
of part (c) underestimate or overestimate this difference?
By how much?
Use the equation defined by a polynomial of degree 2 to
find the median annual income for a woman with 18 years
of education. Does this underestimate or overestimate
the median income shown by the bar graph? By how
much?
b. Use the equations defined by polynomials of degree 3 to
find a mathematical model for M — W.
ce. According to the model in part (b), what is the difference,
rounded to the nearest dollar, in the median annual income
between men and women with 16 years of education?
According to the data displayed by the graph, what is the
actual difference in the median annual income between
_ men and women with 16 years of education? Did the result
of part (¢) underestimate or overestimate this difference?
By how much? |
с
-
92. a
+
Se
The volume, V, of a rectangular solid with length 1, width w, and
height h is given by the formula V = Iwh. In Exercises 93-94, use
this formula to write a polynomial in standard form that models,
or represents, the volume of the open box.
In Exercises 95-96, write a polynomial in standard form that
models, or represents, the area of the shaded region.
95.
— x+9 я
+ 70
e —— x + +5 —
ff 0
96. ft x+4 mo
x+3| x+1 a
~ ul Li С
Writing in Mathematics
97. What is a polynomial in x?
98. Explain how to subtract polynomials.
99. Explain how to multiply two binomials using the FOIL
method. Give an example with your explanation.
100. Explain how to find the product of the sum and difference
of two terms. Give an example with your explanation.
101. Explain how to square a binomial difference. Give an
example with your explanation.
102. Explain how to find the degree of a polynomial in two variables.
Critical Thinking Exercises
Make Sense? In Exercises 103-106, determine whether each
statement makes sense or does not make sense, and explain your
reasoning.
103. Knowing the difference between factors and terms is
important: In (3x 2yY, I can distribute the exponent 2 on
each factor, but in (3x? + y, I cannot do the same thing
on each term.
104, 1 ‚used the FOIL method to find the product of x + 5 and
Xx + 2x +1.
105. Many English words have prefixes with meanings similar
to those used to describe polynomials, such as monologue,
binocular, and tricuspid. —
106. Special-product formulas have patterns that make their
multiplications quicker than using the FOIL method.
In Exercises 107-110, determine whether each statement is true
or false. If the statement is false, make the necessary change(s) to
produce a true statement.
107. (3x> + 2)Gx? — 2) = 9x? — 4
108. (х — 5)? = x? — 5х + 25
109. (x +12 =x +1
110. Suppose a square garden has an area represented by 9x”
“square feet. If one side is made 7 feet longer and the other
side is made 2 feet shorter, then the trinomial that models the
area of the larger garden is 9x* + 15x — 14 square feet.
In Exercises 111-113, perform the indicated operations.
111. [(7x + 5) + 4yl[(7x + 5) — 4y]
112. [(3x + у) + 1}?
113. (x" + 2)(x" — 2) — (x? — 3)
!
11:
IL Ë |
lice
dll
15
lg
114. Express the area of the plane figure shown as a polynomial
in standard form.
a
x+3
WHAT YOU KNOW: We defined the real numbers [{x|x is
rational} U {x|x is irrational}] and graphed them as points
on a number line. We reviewed the basic rules of algebra,
using these properties to simplify algebraic expressions.
We expanded our knowledge of exponents to include
exponents other than natural numbers:
1
b9 = 1; pr = +. = b = V b;
bh" bp”
m 1
pt = (Vp) = Y". 57 =.
b
We used properties of exponents to simplify exponential
expressions and properties of radicals to simplify radical
expressions. Finally, we performed operations with
polynomials. We used a number of fast methods for finding
products of polynomials, including the FOIL method
for multiplying binomials, a special-product formula
for the product of the sum and difference of two terms
[(А + В)(А — В) = A> — B%],andspecial-productformulas
for squaring binomials [(A + BY = А? + 2АВ + B*;
(А - В}? = А? — 2АВ + В”).
“In Exercises 1-25, simplify the given expression or perform the
indicated operation (and simplify, if possible), whichever is appropriate.
1. (3x + 5X4x — 7) 2. (3x + 5) - (4x - 7)
3. V6+9V6 4. 3V12 — V27
5. 7х + 3[9 — (2x — 6)! 6. (8x — 3)”
7 3 8. (2) _ 375
9. (2x — 5) — (х? — 3x +1)
10. (2x — 5(x? - 3x +1) 11.3 +x7-—.X
12. (9a — 10b)(2a + b) 13. (a, c, d, e) U (c, d, f, h)
14. (a, c, d, e) A fc, d, f, h)
15. (3xy> — ху + 4у?) — (-2Xy> — 3xy + 5y”)
24 2,13
16, 17. (Zu 5 y as y)
—2x° y? 3
18. Хх“
24 x 10°
19 “4 x 10 (Express the answer in scientific notation. )
2 X 10°
\ 32, 3 3
20. ; 21. (x° + 2)(х” — 2)
23. V50- V6
22. (x* + 2)
Mid-Chapter Check Point 63
TE as ge ар ER i, ©
AE plat uefa eh a ini) iL x us y Hi A
PSI qu PNG PE dg
1 i fs Yi e i if be Le El Hn Fo és ny
Exercises 115-117 will help you prepare for the material covered
in the next section, In each exercise, replace the boxed question
mark with an integer that results in the given product. Some trial
and error may be necessary.
115. (x + 3)(x + |? ]) = 22 + Tx + 12
116. (x — [2 ])(x — 12) = x? — 14x + 24
117. (4x + 1)@x — [2 |) = 8x2 — 10x — 3
7-3
26. List all the rational numbers in this set:
La
-5,0, 0.45, V23,V25 }
In Exercises 27-28, rewrite each expression without absolute value bars.
V13| 28. |x| if x<0
29. the population of the United States is approximately 3.0 x 10%
and each person produces about 4.6 pounds of garbage per day,
express the total number of pounds of garbage produced in the
United States in one day in scientific notation.
30. A human brain contains 3 X 10'° neurons and a gorilla brain
contains 7.5 X 10° neurons. How many times as many neurons
are in the brain of a human as in the brain of a gorilla?
31. TVs keep getting fancier and bigger, but prices do not. The
bar graph shows the average price of a TV in the United
States from 2007 through 2012.
Average Price of a TV
$1050
935
Average TV Price
2007 2008 2009 2010 2011 2012
Year
Source: Consumer Electronics Association
Here are two mathematical models for the data shown by the
graph. In each formula, P represents the average price of a
TV x years after 2007.
P = —86x + 890
- P = 18x% — 175x + 950
a. Which model better describes the data for 2007?
b. Does the polynomial model of degree 2 underestimate
or overestimate the average TV price for 20127 By how
much?
74 Chapter P Prerequisites: Fundamental Concepts of Algebra
Here is a list of the factoring techniques that we have
discussed.
| Factoring out the GCF
Factoring by grouping
Factoring trinomials by trial and error
Factoring the difference of two squares
А’ — В? = (А + В)(А - В)
e. Factoring perfect square trinomials
A* + 2AB + B* = (A + BY
A — 2AB + B” = (A — BY
“ff Factoring the sum of two cubes
A + B® = (A + B)(A> — AB + B”)
RO
g. Factoring the difference of two cubes
Æ — B° = (A — В)(А? + АВ + В?)
Ts 2 ea ve 2 Ls Час 3 sg e
Ff ОНО MARCIO
In Exercises 1-10, factor out the greatest common factor.
1. 18x + 27 2. 16x — 24
3. 3x" + 6x 4. 4х? — 8x
5, 9x* — 18 + 27x“ 6. 6x* — 18x? + 12x?
7. x(x + 5) + 3(x + 5) 8. x(2x + 1) + 4(2x + 1)
9. Xx(x — 3) + 12(x ~ 3) 10. x(2x + 5) + 17(2x + 5)
In Exercises 11-16, factor by grouping.
11. x — 2x + 5x — 10 12. x* — 3х? + 4х — 12
13. х° = х? + 2x — 2 14. х° + 6x7 — 2x — 12
15. 3х? — 2х? — 6х + 4 16. х° — х* — 5х + 5
In Exercises 17-38, factor each trinomial, or state that the
- trinomial is prime.
17. х? + 5х + 6 18. х? + 8x + 15
19. х* — 2х — 15 20. х* — 4х — 5
21. x? — 8x + 15 22. x? — 14x + 45
23. 3x* —x — 2 24. 2x* + 5x — 3
25. 3x* — 25x — 28 26. 3x“ — 2x — 5
27.6% —1x+4 — 28. 6x7 — 17x + 12
29. 4х? + 16х + 15 30. 8х? + 33x + 4
31. 9х” — 9х + 2 32. 9х? + 5х — 4
33. 20х? + 27x — 8 34, 15x“ — 19x + 6
35. 2x7 + 3xy + у? 36. 3x" + 4ху + у?
37. 6x* — 5ху — бу? 38. 6х? — 7ху — 5у?
In Exercises 39-48, factor the difference of two squares.
39, x“ — 100 40. x“ — 144
41. 36x? — 49 42. 64х? — 81
43. 9х? — 25)? 44. 36х? — 49у?
Fill in each blank by writing the letter of the technique
(a through g) for factoring the polynomial.
1. 16x? — 25
. 2x3 — 1
Xx + Tx + xy + 7y
4x” + 8x + 3
Ox? + 24x + 16
5x% + 10x
x* + 1000
»
о AMES
1 1 412
The algebraic expression (x + 1)2 — Fx + 1)2 can
be factored using as the greatest common
factor.
45. x* — 16 46. x* — 1
47. 16x* — 81 48. 81x* — 1
In Exercises 49-56, factor each perfect square trinomial.
49, x +2x +1 $0. х? + 4х + 4
51. x“ — 14x + 49 52. x* — 10x + 25
53, 4x* + 4x + 1 54. 25х? + 10x + 1
55. 9х” — 6х + 1 56. 64x* — 16x + 1
In Exercises 57-64, factor using the formula for the sum or
difference of two cubes.
57. х° + 27 58. х° + 64
59, x> — 64 60. x* — 27
61. 8x* — 1 62. 27x — 1
63. 64x) + 27 64. 8x) + 125
In Exercises 65-92, factor completely, or state that the polynomial
is prime.
65. 3x° — 3x 66. 5x3 — 45x
67. 4х? — 4х — 24 68. 6х? — 18x — 60
69. 2x* — 162 70. Tx* — 7
71. х° + 2х? — 9х — 18 72. х° + 3x" — 25x — 75
73. 2x" — 2x — 112 74. 6x7 — 6x — 12
75, х° — 4x 76. 9х” — 9х
77. х° + 64 78. х? + 36
79, х° + 2х? — 4х — 8 80. х° + 2х? — х — 2
81. y — 81y 82. y” — 16y
83. 20y* — 457? —. 84. 48y* — 37?
85, x” — 12x + 36 — 49y* 86. x — 10x + 25 — 36)?
87. 9b”x — 16y — 16x + 9b”y
88. 16а%х — 25y — 25x + 16a”y
Л
89.
91.
xy — 16y +32 - 2% 90. 12% — 27y — 4х + 9
2х3 — 8а?х + 24х° + 7дх
92. 2x7 — 98а?х + 28x” + 98x
Jn Exercises 93-102, factor and simplify each algebraic expression.
93.
93.
97.
99.
100.
101.
102.
3 1 3 1
2 1 _3 1
4х 3 + ar 96. 12x 4 + 6x4
3
(x + ” — (х + 3)?
98. (x + 4} + (x? + 4)
03-0457 |
2
(2 + y + (2 +3)%3
1 3
(4x — 1)? — M4x — 1)?
—8(4x + 3)7? + 10(5x + 1)(4x + 3)"
Practice Plus
In Exercises 103-114, factor completely.
103.
104.
105.
107.
109.
111.
113.
10x*(x + 1) — 7x(x + 1) — 6(x + 1)
12x*(x — 1) — 4x(x - 1) — 5(x — 1)
6x* + 35x* — 6 106. 7x* + 34x* — 5
y +y 108. (y + 1? +1
x* — 5x2? + 4y* 110. x* — 10x?y" + 9y*
(x — y)* — 4(x = у)? 112. (x + y)* — 100(x + y)”
2x? — Txy? + 3y* 114. 3х? + Sxy? + 2y*
Application Exercises
115.
116.
Your computer store is having an incredible sale. The price
on one model is reduced by 40%. Then the sale price is
reduced by another 40%. If x is the computer's original
price, the sale price can be modeled by
(x — 0.4x) — 0.4(x — 0.4x).
a. Factor out (x — 0.4x) from each term. Then simplify the
resulting expression.
b. Use the simplified expression from part (a) to answer
these questions. With a 40% reduction followed by a 40%
reduction, is the computer selling at 20% of its original
price? If not, at what percentage of the original price is it
selling?
Your local electronics store is having an end-of-the-year
sale. The price on a plasma television had been reduced
by 30%. Now the sale price is reduced by another 30%.
If x is the television’s original price, the sale price can be
modeled by
(x — 03x) — 0.3(x — 0.3x).
a. Factor out (x — 0.3x) from each term. Then simplify the
resulting expression.
b. Use the simplified expression from part (a) to answer
these questions. With a 30% reduction followed by a 30%
reduction, is the television selling at 40% of its original
price? If not, at what percentage of the original price 1s it
selling?
Section P.5 Factoring Polynomials 75
In Exercises 117-120,
117.
119.
a. Write an expression for the area of the shaded region.
b. Write the expression in factored form.
LIA 118.
I 3x >|
|) — 120. x
xy
In Exercises 121-122, find the formula for the volume of the
region outside the smaller rectangular solid and inside the larger
rectangular solid. Then express the volume in factored form.
121.
122.
Writing in Mathematics
123.
124.
125,
126.
127.
128.
129.
Using an example, explain how to factor out the greatest
common factor of a polynomial.
Suppose that a polynomial contains four terms. Explain how
to use factoring by grouping to factor the polynomial.
Explain how to factor 3x* + 10x + 8.
Explain how to factor the difference of two squares. Provide
an example with your explanation.
What is a perfect square trinomial and how is it factored?
Explain how to factor x* + 1.
What does it mean to factor completely?
132. I factored 4x* — 100
76 Chapter P Prerequisites: Fundamental Concepts of Algebra
Critical Thinking Exercises
Make Sense? In Exercises 130-133, determine whether each
statement makes sense or does not make sense, and explain your
reasoning.
130. Although 20x> appears in both 20x? + 8x* and 20x* + 10x,
Pil need to factor 20x” in different ways to obtain each
polynomial’s factorization.
131. You grouped the polynomial’s terms using different groupings
than I did, yet we both obtained the same factorization.
completely and obtained
(2x + 10)(2x — 10).
133. First factoring out the greatest common factor makes it
easier for me to determine how to factor the remaining
factor, assuming that it is not prime.
In Exercises 134-137, determine whether each statement is true
—or false. If the statement is false, make the necessary change(s) to
produce a true statement.
+ 4)(х? — 4).
e polynomial.
134. x* — 16 is factored completely as ©
135. The trinomial x* — 4х — 4 is a prim
136. x* + 36 = (x + 6)?
137. х° — 64 = (x + 4)(02 + 4x — 16)
Specify umbers tl that
-Must be excluded from
“the domain of a rational
expression.
, Sir mplify Y
expression
y ratio nal
expressions.
Миру г
‘ Divide rational”
expressions. В
Add and: su btract:
rational expressions.
Simplify complex |
rational expressions,
possible, reduce the answer to its lowest terms.
How do we describe the costs of reducing environmental
pollution? We often use algebraic expressions involving
quotients of polynomials. For example, the algebraic
250x
100 — x
describes the cost, in millions of dollars, to remove
Ш ation al | xpercent of the pollutants that are discharged into a
| river. Removing a modest percentage of pollutants,
say 40%, 1s far less costly than removing a
substantially greater percentage, such as 95%. We
see this by evaluating the algebraic expression
forx = 40 and x = 95.
250x \
Evaluating 100 — + for
100 — x
In Exercises 138—141, factor completely.
138. x2" + 6x" + 8 139, —
140. x* — y* — 2 y + 2xy*
141. (x — 5 + sy — (x + e — sy?
In Exercises 142-143, find all integers b so that the trinomial can
be factored.
142. x* + bx + 15
* — 4x +5
143. x? + 4x + b
LS
Preview Exercis
Exercises 144-146 will helo you prepare for the material covered
in the next section.
144. Factor the numerator and the denominator. Then simplify
by dividing out the common factor in the numerator and the
denominator.
x” + 6x + 5
? — 25
In Exercises 145-146, perform the indicated operation. Where
5 8 1 2
Cs
4. 15 16.373
= 40: x = 95:
el ‚ 250(40) ‚ 250(95)
What happens if you try Costis — = 167. Costis — —— = 4750.
substituting 100 for x in 100 — 40 100 — 95
250x , The cost increases from approximately $167 million to a possibly prohibitive
100 — x
What does this tell you about
the cost of cleaning up all of the
river’s pollutants?
$4750 million, or $4.75 billion. Costs spiral upward as the percentage of removed
pollutants increases.
86 Chapter P Prerequisites: Fundamental Concepts of Algebra
. 3x + .
8. An equivalent expression for with a
denominator of (3x + 4)(x — 5) can be obtained by
multiplying the numerator and denominator by
9. A rational expression whose numerator or
denominator or both contain rational expressions
is called a/an rational expression or
fraction.
a/an
In Exercises 1-6, find all numbers that must be excluded from the
domain of each rational expression.
7 13
1. - 2. -
x — 3 x + 9
х + 5 х + 7
Jo 2_ a 2_
X 25 X 49
x — 3
© x2 + Ах — 45
x — 1
SL STILL
x“ + 11x + 10
In Exercises 7-14, simplify each rational expression. Find all
numbers that must be excluded from the domain of the simplified
rational expression.
3x — 9 4x — 8
* 2 641 à 8. — II,
x“ — 6x + 9 x“ — 4х + 4
2 + 2 +
9. X 12x + 36 10. X 8x + 16
dx — 24 3x — 12
y? +7y — 18 2 — 4у — 5
п, —— 12. ——
у” — Зу + 2 у” + Sy +4
x + 12x + 2 — 14x +
qa, Y +12 +36 д 2 — 1x +49
x“ — 36 x“ — 49
In Exercises 15-32, multiply or divide as indicated.
‚| х— 2 2х + 6 16 6x +9 х- 5
° 3х + 9 2х — 4 TU 3x — 15 4х + 6
2 2 2 _ _
17, * —. X 3x 18. À 4 2x 4
x x“ +x — 12 x” — dx +4 x +2
2. 5x +6 x2 —
19. ©. 5х E 1
х” — 2х — 3 х“ — 4
+ 5х + 6х? — 9
20. © X —
X +x—-6 XxX —x—Ó
1, x 8 x 2 27, Ÿ 6x +9 1
Xx —4 3x x +27 x+3
Хх +] 3x + 3 x+5 4x +20
23. + 24, +
3 3 7 7 9
2— 4 +2 2—4 +
25, Кн Ля 26. —. 12
x x-2 x-2 4x — 8
1 1 (25-5)
10.3 x XX +3) +3 x/ ——(
3 x(x + 3) 3 3x(x + 3)
— 3x(x +3)
2 2
э7, ® + 10 | бх + 15
х — 3 х? — 9
2 4 2 1
28. © X X 1
x“ — 4 x“ + 5x +6
9 2-25, Y +10 + 25
"2-2 x +4x-5
x” — 4 х? + 5х + 6
30. — +
x” + 3x — 10 х“ + 8x + 15
х* + х — 12 х* + 5х + 6 x + 3
31. 5 5 +=
x* +x-— 30 х” — 2х = 3 x‘ +7xc +8
32 x — 25x 2x* — 2 „X“ + 5x
AX xX-6x+5 Tx +7
In Exercises 33-58, add or subtract as indicated.
+ +
33, Y 1 8x-+9 34, 3x +2 3x +6
6x +5 6x +5 3x +4 3x +4
2.7 2.
5 2 a
x“ + 3x Xx +3x
2-4 4х — 4
36, pg
x—-x-6 x -x-6 |
4x — 10 х- 4 2x + 3 3—x
7. — . —
3 x — 2 x — 2 38 3x -6 3x —6
2 +3 *— 12
39, — X —
х“ + Хх — 12 x +x- 12
Yo x* — 4x _ 2-6
x -x-6 x2-x-6
3 6 8 2
41. + 42.
x +4 x+5 x—2 x—3
43. 33 a 2 3
- x+1 x x x+3
2X x + 2 3x x + 4
Biri rec х+2
x +3 x-5 x+3 x—3
. + .
4. 3 x +5 * ya x + 3
3 2 5 7
49, + , +
2x + 4 3x + 6 > 2x +8 3x+12
4 4 3 5x
51. 52. +
x*+6x+9 x+3 5x +2 25х? — 4
La
р
6
6
Ti
Xx A
53.24 3x — 10 Xx +x—6
x X
4. 7 ax 24 x? 7x +6
x+3 x+2
5.271 x — 1
4 +x-6 3 3
12 + 3х + 2 х +1 x+2
6x7 + 17x — 40 | 3 _ Sx
21,20 x—-4 x+5
x+5 x+1
x —4 x-2
56.
57.
58. x? + x — 20
In Exercises 59-72, simplify each complex rational expression.
* 1 д
3 |
9. — 60. —
1
+7 8 + —
x 62 -
61. 1 al
3-7 x
1,1 1
xX y
64.
63. Fy ху
X
х —
x + 3 x — 3
65. — — —— 66.
65. х + 2 к — 3
x — 2
oh 5 73 +1
(7, 2 68. >
7 3
+ 1
х* — 4 х? — 4
|
х + 1
69,
1 1
Xx —2x-3 x-3
6 1
70 Xx +2x—-15 x-3
| ! + 1
х + 5
! LL x + h X
+4) x? x
ne ) x 7 2h x +1
Practice Plus
In Exercises 73-80, perform the indicated operations. Simplify the
result, if possible.
(ZEA ros __2
х + 1 2х% + х —3 x +2
m5— => ( 1 1 )
Cox =x -8 \x-4 x+2
(0-3
х + 1 2
Section P.6 Rational Expressions 87
3 5
76. | 4 —
+)
= (у +5)! 4-9 +21
7? (y +5) 78, ? (y + 2)
5 2
m. ( 1 Arad m ber ba) c—d
a? — 53 1 a’ + ab + b?
80 ab * ac — ad — bc + bd a> =P
a+ ab + Ъ? ac — ad + be — bd a +8
Application Exercises
81. The rational expression
130x
100 — x
describes the cost, in millions of dollars, to inoculate x percent
of the population against a particular strain of flu.
a. Evaluate the expression for x = 40, x = 80, and x = 90.
Describe the meaning of each evaluation in terms of
percentage inoculated and cost.
b. For what value of x is the expression undefined?
¢. What happens to the cost as x approaches 100%? How
can you interpret this observation?
82. The average rate on a round-trip commute having a one-way
distance d is given by the complex rational expression
2d
d а’
F1 Го
in which 7, and 7, are the average rates on the outgoing and
return trips, respectively. Simplify the expression. Then find
your average rate if you drive to campus averaging 40 miles per
hour and return home on the same route averaging 30 miles
per hour. Explain why the answer is not 35 miles per hour.
83. The bar graph shows the estimated number of calories per day
needed to maintain energy balance for various gender and age
groups for moderately active lifestyles. (Moderately active means
a lifestyle that includes physical activity equivalent to walking 1.5
to 3 miles per day at 3 to 4 miles per hour, in addition to the light
physical activity associated with typical day-to-day life.)
Calories Needed to Maintain Energy
Balance for Moderately Active Lifestyles
3200
2800 +
2400
2000 + -
1600
1200
800
400
- Gps
Group $ - Group 4
о | nan Group 6 …
Calories per Day
4-8 9-13 14-18 19-30 31-50 51+
Age Range
Source: US.D.A.
88 Chapter P Prerequisites: Fundamental Concepts of Algebra
(Be sure to refer to the graph at the bottom of the previous page.)
a. The mathematical model
W = —66x* + 526x + 1030
describes the number of calories needed per day, W, by
women in age group x with moderately active lifestyles.
According to the model, how many calories per day
are needed by women between the ages of 19 and 30,
inclusive, with this lifestyle? Does this underestimate or
overestimate the number shown by the graph? By how
much?
b. The mathematical model
М = —120x* + 998x + 590
‘describes the number of calories needed per day, M, by men
in age group x with moderately active lifestyles. According
to the model, how many calories per day are needed by men
between the ages of 19 and 30, inclusive, with this lifestyle?
Does this underestimate or overestimate the number
shown by the graph? By how much?
ce. Write a simplified rational expression that describes the ratio
of the number of calories needed per day by women in age
group x to the number of calories needed per day by men in
age group x for people with moderately active lifestyles.
In Exercises 84-85, express the perimeter of each rectangle as a
single rational expression.
84. X | 85. X
X +3 х +5
Writing in Mathematics
86. What is a rational expression?
87. Explain how to determine which numbers must be excluded
| from the domain of a rational expression.
88. Explain how to simplify a rational expression.
89. Explain how to multiply rational expressions.
90. Explain how to divide rational expressions.
91. Explain how to add or subtract rational expressions with the
same denominators.
92. Explain how to add rational expressions having no common
factors in their denominators. Use + in your
x+5 x+2
explanation.
93. Explain how to find the least common denominator for
denominators of x” — 100 and x? — 20x + 100.
3 2
— + metre
| | XX
Describe two ways to simplify 1 >
—_— + —
x* x
94
Explain the error in Exercises 95-97. Then rewrite the right side
of the equation to correct the error that now exists.
, 1 1 1 1 1
ST += 9. — + 7 =
9 a+b 6. 15777
а
b
` N | E
97: - +
a.
x + b
Preview E
Critical Thinking Exercises
Make Sense? In Exercises 98-101, determine whether each
statement makes sense or does not make sense, and explain your
reasoning.
3x — 3
4x(x — 1)
99. The rational expressions
7 7
14x and 14 + x E
can both be simplified by dividing each numerator and each _
denominator by 7.
98. I evaluated for x = 1 and obtained 0.
100. When performing the division Dis
Tx (x + 3)?
Ra R—5"
I began by dividing the numerator and the denominator by-
the common factor, x + 3.
101. Isubtracted 3x
X
x —3 ,
from Tq and obtained a constant.
— 1
In Exercises 102-105, determine whether each statement is true
or false. If the statement is false, make the necessary change(s) to
produce a true statement.
2
x“ — 25
102, ——— =x—3
x — 5
—Зу — 6
103. The expression — 5 simplifies to the consecutive .
integer that follows —4.
2х — 1 3x—1 5х — 2
° + — = С
104 х — 7 х — 7 x — 7 0 ; Pr
105. 6 + 1-7
x x
In Exercises 106-108, perform the indicated operations.
1 1 1
106 7 a] x” —1
1 1 1 1
— — 1 — re
197 (1 DU x + т) —)( +=)
108. (x — yY + (x — y)?
109. In one short sentence, five words or less, explain what
Y
2.
X X
Ti ;
4 5 х°
does to each number x.
HEF rcises
Exercises 110-112 will help you prepare for the material covered
in the first section of the next chapter.
110. If y = 4 — x? find the value of y that corresponds to values
of x for each integer starting with —3 and ending with 3.
111. Ify = 1 — x”, find the value of y that corresponds to values
of x for each integer starting with —3 and ending with 3.
112. В у = |x + 1], find the value of y that corresponds
to values of x for each integer starting with —4 and ending
with 2.
ur
Summary, Review, and Test 89
ach
Distance between Points a and b on a Number Line
‚|a—b| or |b-a
y Properties of Real Numbers
Commutative a+b=b-+a
ab = ba |
Associative (а + Б) + с = а + (Б + с)
| (ab)c = a(bc)
| Distributive a(b + c) = ab + ac
Identity а + 0 = а
а*1 = а
е Inverse а + (-а) = 0
a} = 1,a #0
a
Properties of Exponents
1
BR = Ь° = 1, pm pt = БР,
bp”
Me mn bo" mTR й == дб АЙ ay a"
(BY = pm, gr = В , (ab) - a, (4) DR
Product and Quotient Rules for nth Roots
Bf JT AR „Ja _ Va
Vab = Va Vb, b= о;
Rational Exponents
1 11 1
а” = Va, а "= — = ›
a” Va
n n A 1
a” = (Va) = Ма”, а " = —щ
#
Special Products
(A + B)(A — B) = À — B”
(А + В)? = А? + 2АВ + В?
(A — BP = À - 24B + B? |
(A + BY = 43 + ЗА?В + 3AB” + B*
(А — B) = A’ - 3A’B + 34B* — B®
Factoring Formulas
A — B* = (A + B)(A — B)
A + 2AB + B? = (A + BY
A — 2AB + B? = (A — BY
A? + B* = (A + BYIA — AB + BY)
A’ — B® = (A - B\(4 + AB + B”)
You can use these review exercises, like the review exercises at the
end of each chapter, to test your understanding of the chapter’s
topics. However, you can also use these exercises as a prerequisite
test to check your mastery of the fundamental algebra skills needed
in this book.
Р.1
In Exercises 1-2, evaluate each algebraic expression for the given
value or values of the variable(s).
L 34 6(x —2)°forx = 4
2. x* — 5(х — y) forx = 6and y = 2
3. You are riding along an expressway traveling x miles per
hour. The formula
S = 0.015x* + x + 10
models the recommended safe distance, $, in feet, between
your car and other cars on the expressway. What is the recom-
mended safe distance when your speed is 60 miles per hour?
In Exercises 4-7, let À = (а, |, с}, В = {а, с, а, е}, and
С = la, d, f, e). Find the indicated set.
4 AMB 5 AUB
6. AUC TL CNA
8. Consider the set:
[-17,—15,0,075,V2, 7, V81 ).
List all numbers from the set that are a. natural numbers,
b. whole numbers, e, integers, d. rational numbers,
e. irrational numbers, f. real numbers.
In Exercises 9-11, rewrite each expression without absolute value
bars.
V2 - 1 11. [3 — V17|
9. |—103| 10.
12. Express the distance between the numbers —17 and 4 using
absolute value. Then evaluate the absolute value.
80 Chapter P Prerequisites: Fundamental Concepts of Algebra
In Exercises 13-18, state the name of the property illustrated.
13. 3 + 17 = 17 + 3 14. (6:3)-9 = 6-(3-9)
15. V3(V5 + V3) = V15 + 3
16. (6-9)-2 = 2-(6-9)
17. Vil Vs + V3) = (V5 + УЗ) УЗ
18. (3-7) + (4-7) = (4-7) + (3-7)
In Exercises 19-22, simplify each algebraic expression.
19. 52x — 3) + 7x
20. $(5х) + [(Зу) + (-Зу) | - (-х)
21. 3(4y — 5) — (7у + 2)
22. 8— 2/3 — (5x — 1)
23. The diversity index, from 0 (no diversity) to 100, measures
the chance that two randomly selected people are a different
race or ethnicity. The diversity index in the United States
varies widely from region to region, from as high as 81 in
Hawaii to as low as 11 in Vermont. The bar graph shows the
national diversity index for the United States for four years
in the period from 1980 through 2010.
|
Chance That Two Randomly Selected
Americans Are a Different Race
or Ethnicity
47...
There: is isa 155% |
“ol chance that two
1 randomly selected
- Americans: differin:
| race or ethniolty.
50
AO ber
30
20 |-
(0-100 scale)
10-
Diversity Index: Chance of
Different Race or Ethnicity
2000 2010
Year
1980 1990
Source: USA Today
The data in the graph can be modeled by the formula
D = 0.005x* + 0.55x + 34,
where D is the national diversity index in the United States
x years after 1980. According to the formula, what was the
U.S. diversity index in 2010? How does this compare with
the index displayed by the bar graph?
P.2
Evaluate each exponential expression in Exercises 24-27.
24. (-3(-2Y 25. 27% + 47!
33
26. 57:5 27. —
3
Simplify each exponential expression in Exercises 28-31.
28. (—2x*y*) 29. (-5x*y")(-2x}y7?)
TX yS
28x15 y 2
In Exercises 32-33, write each number in decimal notation.
32. 3.74 x 10° 33. 7.45 x 10°
30. (2x°)* 31.
In Exercises 34-35, write each number in scientific notation.
34. 3,590,000 35. 0.00725
In Exercises 36-37, perform the indicated operation and write t
answer in decimal notation.
6.9 x 10°
3 x 10°
In 2009, the United States government spent more than it had
collected in taxes, resulting in a budget deficit of $1.35 trillion.
Exercises 38~40, you will use scientific notation to put a number
like 1.35 trillion in perspective. Use 102 for 1 trillion.
36. (3 x 101.3 x 103) 37.
38. Express 1.35 trillion in scientific notation.
39. There are approximately 32,000,000 seconds in a year.
Express this number in scientific notation.
40. Use your scientific notation answers from Exercises 38 and
39 to answer this question: How many years is 1.35 trillion
seconds? Round to the nearest year. (Note: 1.35 trillion
seconds would take us back in time to a period when
Neanderthals were using stones to make tools.)
P.3
Use the product rule to simplify the expressions in Exercises 41-44,
In Exercises 43-44, assume that variables represent nonnegative
real numbers.
V 300 42. V12x”
V10x- V2x 44. Vr°
Use the quotient rule to simplify the expressions in Exercises 45-46,
121 V 96x
45. |— — 46.
4 Vx
In Exercises 47-49, add or subtract terms whenever possible.
47. 7V5 + 13V5 48. 2V50 + 3V8
49. 4V72 — 2V48
In Exercises 50-53, rationalize the denominator.
30 V2
50. — 51, —
VS V3
5 s3.— 4
6 + V3 7-5
Evaluate each expression in Exercises 54-57 or indicate that the
root Is not a real number.
55, V—32
54. V125
/—125 57. Y (-5Y
Simplify the radical expressions in Exercises 58-62.
58. V/s1 59. Yy5
60. Ÿa- 10 61. 4V/16 + 5V2
4/25 .5
2
a (Assume that x > 0.)
V 16x
In Exercises 63-68, evaluate each expression.
(Assume that x > 0.)
52.
62.
1 À 1
63. 162 64. 25 2 65. 125°
1 2 4
66. 27 3 67. 645 68. 27 3
In Exercises 69-71, simplify using properties of exponents.
3
2 1 15x* 4
69. (5x3) (4x4) 70. >
su
2
71. (125х°)3
72. Simplify by reducing the index of the radical: V Y.
p4
In Exercises 73-74 I
resulting polynomial i
73. (6x7 + 7х2 — 9х + 3) + (14x) + 3x* — 11x — 7)
па, (13x* — 8? + 2х”) — (Sx* — 3x% + 2x“ — 6)
74, perform the indicated operations. Write the
n standard form and indicate its degree.
mn Exercises 75-81, find each product.
75, (3x — 2)(4х? + 3x — 5) 76. (3x — 5)(2x + 1)
T7, (4x + 5)(4x — 5) 78. (2x + 5Y
79, (3х — 4)? 80. (2х + 1Y
81. (5х — 2)
In Exercises 82-83, perform the indicated operations. Indicate the
degree of the resulting polynomial.
82, (7x” — 8xy + у?) + (-8х? — 9ху — 4у”)
83. (137 — 5х’ у — 9х”) — (~11x3y? — бх?у + 3x* — 4)
In Exercises 84-88, find each product.
84. (x + 7y)(3x — 5y) 85. (3x — Sy)”
86. (3x7 + 2y7 87. (7х + 4у)(7х — 4у)
88. (a — b)(a® + ab + b?)
Tn Exercises 89-105, factor completely, or state that the polynomial
“is prime.
89, 15x° + 3х” 90. х? — 11x + 28
91. 15x —x—2 92. 64 — x*
93. x? + 16 94. 3x* — 9x — 30x“
95, 20x” — 36x° 96. x — 3х? — 9х + 27
97, 16x* — 40х + 25 98. x* — 16
99. y — 8 100. x +64
27x? — 125
101. 3x* — 12x“ 102.
In Exercises 1-18, simplify the given expression or perform
the indicated operation (and simplify, if possible), whichever is
appropriate.
1. 5(2x? — 6x) — (4x2 — 3x)
2, 7 + 2[3(х + 1) — 23x — 1)]
3. [1,2, 5} п 15, а)
4. (1,2, 5} U {5, a}
5 (ху? - ху + y”) = 42) — 5xy — y”)
30x y*
6x9 y*
7. V6erV3r (Assume that r = 0.)
8.4/50- 3/18 о, ——
5 + V2
10. Yi6x* 11
. ~6
12. 5X10
20 x 108 (Express the answer in scientific notation.)
x? + 2x — 3
" х? = 3x +2
Summary, Review, and Test 91
103. x° — x 104. x* + 5x? — 2x — 10
105. x* + 18x + 81 — у?
In Exercises 106-108, factor and simplify each algebraic expression.
3 1
106. 16x 4 + 32x*
1 3
107. (x? — 40x? +3) — (x? — “x? +3)?
1 3
108. 12x 2 + 6x 2
P.6
In Exercises 109-111, simplify each rational expression. Also, list
all numbers that must be excluded from the domain.
x* + 2х? x? + 3x — 18 x? + 2x
109. — — — 110. e
х + 2 xXx — 36 Xx + 4х + 4
In Exercises 112-114, multiply or divide as indicated.
? + 6x + + +
112. X 6x 9 x 3 113. bx +2 3% X
х° = 4 x-2 x? = 1 x —1
2 _ 2
п © 5x 24 x 10x + 16
x“ —= x — 12 xr +x—6
In Exercises 115-118, add or subtract as indicated.
2x — 7 x —10 3x X
115 — 116 +
x* — 9 x” — 9 x+2 x-2
x — 1
117.
Xx —9 x—5x+6
Ax —
118. x — 1 x+3
27 +5x-3 6x2 +x —2
In Exercises 119-121, simplify each complex rational expression.
1 1 a 1
х 2 3+ > 7773
119. 120. X 11. ———
Lx | - © 34 À
3 6 x? x + 3
13. (2x — 5)(х? — 4x + 3) 14. (5x + Зу)?
2х + 8х? + 5х + 4 ох 5
15. + 16. +
x — 3 xs x+3 x-3
11
2x +3 2 x 3
17. — 18.
x*—"7x+12 x-3 1
X
In Exercises 19-24, factor completely, or state that the polynomial
is prime.
20. x + 2x? +3x +6
22. 36x” — 84x + 49
24. x? + 10x + 25 — 9y*
19. x? — 9x + 18
21. 25x* — 9
23. y? — 125
25. Factor and simplify:
3 2
x(x + 3) 5 + (x + 3%.
26. List all the rational numbers in this set:
(-7,-%,0,0.25,V3,V4,%,7).
P
Section P.1
Check Point Exercises |
1608 299209 33,7) 403456789 5aV9 b0,V9 e-9,0,V9 d-9,-13,0,03,V9 e z Y
-9,-13,0,03,7,V9, V10 8aV-1 bdar-3 el 79 8387+23x 9.42 — 4x
Concept and Vocabulary Check
1. expression 2. b to the nth power; base; exponent 3. formula; modeling; models 4. intersection; A NB 5. union; AUB
6. natural 7. whole 8. integers 9. rational 10. irrational 11. rational; irrational 12. absolute value; x; —x 13. b + a; ba
14. a + (b + 6); (ab) 15. ab + ac 16. 0; inverse; 0; identity 17. inverse; 1; identity 18. simplified 19. a
Exercise Set P.1
15 310 58 710 944 11.46 13.10 15. -8 17. 10°C 19.60ft 21. {2,4} 23 sen 25. 2
27. 3 29. {1,2,3,4,5} 31. [1,2,3,4,5,6,7,8,10) 33 (а, е,1,0,и} 3B a Vid 6. 0,V100 e —9,0,vi100
4 — 4 —
d. -9,-=,0,0.25, 92, V100 e Vi 1-9,-7,0,025, V3,92, V100 37.a V64 ».0,V64 e —11,0, V6
5 5 |
d. e 0, 0.75,V64 e. V5, т f. de 0, 0.75, V5, т, V64 39. 0 41. Answers may vary; an example is 2. 43, true
45. true — 47. true 49. true 51.300 53. 12 — п || 55. 5 — V2 57. —1 50.4 8.3 63.7 65-1 67.17- 2.15
69. 15 — (—2);7 71. 4 — (-19),15 73. |+1.4 — (-3.6)1;2.2 75. commutative property of addition 77. associative property of addition
79. commutative property of addition 81. distributive property of multiplication over addition 83. inverse property of multiplication
85. 15x + 16 87. 27x — 10 89. 29y — 29 91. 8у — 12 93, 16y — 25 95, 12x? + 11 97. 14x 99, —2x + 3y + 6 101. x
1 8 1
103. > 105. = 107. < 100. = 111.45 118.757 118. 14H Meo 12127 (x + 4); —4
123. 6(—5x); — 30x 125. 5х — 2x; 3x 127. 8x — (Bx + 6);5x — 6 129. a. 140 beats per minute b. 160 beats per minute
131. a. $26,317 b. overestimates by $44 c. $30,568 133. a. 1200 — 0.07х b. $780 145. does not make sense
147. does not make sense + 149. false + 151. false + 153. false 155. true 157. > 159. a. bb. bY °c Add the exponents.
160. a. b* b. bé c. Subtract the exponents. 161. It moves the decimal point 3 places to the right.
Section 2
Check Point Exercises
1 1 3y°
1. a 3°0r243 b 40x%y 2 a (-3°0r-27 b9x"y® Baz b= 26.16 @ 2 4 a 301729 bb. +
25 27 x6 y!
; 32 x? 12,24 3,8 y y |
ce. b2 5. -64х° 6. а. === bir 7a lx y” b-18y се. =~ @ 8. 8 —2,600,000,000 b. 0.000003017
y° 27 x 25х?
9. а. 5.21 х 10” hb. —6.893 x 10% 10. 4.1 х 10° 11. a. 3,55 x 10! b. 4 x 10° 12. $8300
Concept and Vocabulary Check
1
1, bp" add 2, b""" subtract 3.1 4. pa 5. false 6. b” 7. true 8. a number greater than or equal to 1 and less than 10; integer
9. true 10. false
Exercise Set P.2
1 1 1 y
1.50 3.64 85-64 7.1 9. —1 11. — 13. 32 15. 64 17. 16 19.— . + ‚у? xXx
6 я 5 27% 23. > 25. y 27. x
64
29. х° . 31. x 33 — 95. x7 | 37. 39,64% 41.—— 43.9 45.61! 47. 18% - 49.4x% 51.—5ab
xX X
2 1 3 14 2 2765 . |
5.5 8— 8. a. 59. — 61.2 6a 1 65380 67. 0.0006 69 —7,160,000 71.079 73. —0.00415
b 16x 4x 25x a
75. —60,000,100,000 77. 3.2 х 10* 79. 6.38 x 107 81. —5.716 x 10° 88. 2.7 x 1073 85. -504 x 107 87. 6,3 х 10’
89. 64 x 10% 91. 122 x 1071 93. 2.67 x 10** 95. 2.1 x 10° 97. 4 X 10° 90. 2 x 10° 101.5x 10? 103. 4 x 105
АА? Answers to Selected Exercises
y 1 x'8y5
105. 9 x 107 107.1 108. loss Maz 18 115. а. 2.17 X 10° b.3.09x10% с. $7023
хо Xx “y zZ
117. $1.0586 x 10'% 119. 1.06 x 10g 121. 4.064 x 10% — 193. makessense — 195. makessense — 137. true — 199. false 141. false |
143. true 145. A=C+D 148. а. 8 b8 ce V16-V4= V16-4 149. a 1732 b 1732 с. У/З00 = 10\/3 В
150. a. 31x b. 31V2 a
Section P.3 м
Check Point Exercises - С
1 5 1
a9 b-3 € 5 d 10 е 14 2a5V3 b5V7 3 a 7 + SxV3 4 a 17V13 b —19V17X 5 a 175 ов.
5V3 8(4 — V5) 32 — 8% 5 | в
b. 10V2x 6. a. b. V3 7. ( = ) or a v5 8. a. 2V5 - b 25 e. 3 9 5V3 10a5 b2 6-3 »
| 04
1 1 | | |
d-2 ед Mas b8 67 Mal b. 4x” 18 Vx
Concept and Vocabulary Check
1. principal 2. 8 3h 4va vb 5 6. 18V3 7.56V3 87-Vi 9 VIO + V2 10. index: radicand
11. (-2)° 12 ala) 18 Va 14 2:8
SIS
Exercise Set P.3
1.6 8-6 5 notarealnumber 7.3 9.1 11.13 1352 153xV5 17.273 19. xVE 22V/ 2 5
27. 4х 29.5xV2x 31.275 8313Vi 835. -2VI7x% 37, SVT 39. 3V0 437 14320V7-5vi
7
4 A
V7 V10 133 - VI
45. 47, > 49. BE = vib), 51. 7(V5 + 2) 53. 3(V5 — V3) 55. 5 57. —2 59. notarealnumber 61.3
1
63.3 65. — 67, 274 ex n37% ax BIO 711.1393 70. -yV2x 81. V2+2 86 85.2
— 9
87. 25 89. wo 14x"? 93, 4x4 95. x? 97, 5x|y|3 09.27 101. V5 408. x2 105 М? 107. Vx*y 109, 3 à
. . 5+1 1
11, — 113. = 115. Paige Fox is bad at math. 117. v5 ; 1.62101 119. Р = 18 \/5 ft; A = 100 ft? 129. does not make sense
77”
1 |
131. does not make sense 133. false 135. false 187. Let J = 3. 139. 4 141. a. 8 b. 142. 10x7y* 143. 16x° + 6x
144. 2x? + 11x2 + 22x + 15 4
Section P.A
Check Point Exercises
1. а. —х” + х* — &х — 20 b. 20x? — 11x? -2x-8 2. 15x? — 3x? + 30x = 8 3 28¢% — 41x + 15 4 a 49% ~64 b 4y6—25
5. a. x“ + 20x + 100 b. 25x? + 40x +16 6. а. х? - 18x +81 b. 49% -42x+9 7 2x’y + 5ху? — 2у° 8. a 21x? — 25xy + 6y“
b. 4x* + 16xy + 16y° | |
Concept and Vocabulary Check
1. whole 2. standard 3. monomial 4. binomial 5. trinomial 6.n 7. like 8. distributive: 4x5 — 8x2 + 6; 7x 9. 5x; 3; like
of the terms; plus 14. n +m | и ‘
Exercise Set P.A
1. уев; 3х” + 2х = 5 300 852 74 9. 1х3 + 7х? - 12х = 4;3 11. 12x? + 4x? + 12x — 14:3 18. 6x2 — 6x + 2;2
15. х° + 117. 2х3 — 9x* + 19x — 15 19. х? + 10х + 21 M 2 ~2x—15 28. 6х? + 13х + 5 25 10 - 9x — 9
27. 15х" — 47х° + 28 29. 8х° — 40х® + 3х? — 15° 31. х?2 = 9 33 9x2 —4 85 25 — 49x2 397. 16x* — 25х39, 1 — у“
41. x“ +4x +4 48. 4х? + 12x +9 45. х? - 6х + 9 47. 16x* — 8x2 +1 49. 4x? — 28x +49 51. x} + 3x2 + 3x + 1
53, 8x) + 36x? + 54x + 27 55, x? — 9x% + 27x — 27 57. 27x* — 108x? + 144x — 64 59. 7x"y — 4xy is of degree 3
61. 2x*y + 13xy + 13isof degree 3 63. —5x* + 8xy — 9y"isofdegree3 65, х“у? + 8х?у + у — 6xisofdegree6 67. Tx? + 38xy + 19%
69. 2x“ + xy — 21y* 71, 15x74? + xy —2 78. 49х? + 70xy + 25y? 75 4-6 +9 77. xP — y? 79. 9х? — 25)?
81. 49x°y* — 100y? 83. 48xy 85 —Ox2 + 3x +9 87. 16x*- 625 89. 4х? — 28х + 49 oi a $54,647; overestimates by $556
b. M — W = 1.8х° — 82х? + 2644х — 11,449 e. $14,434 d. $15,136; underestimates by $702 93. 4x? — 36x2 + 80x 95. 6x + 22
103. makes sense 105. makes sense 107. false 109. false 111. 49x% + 70x + 25 — 16y” 113. 6x" — 13 115.4 116.2 117.3
Mid-Chapter P Check Point
x2
1. 12х = х = 35 2-x+12 8.10V6 43V3 5. x+45 6 64x2 -48x+9 7.5 в
y
3 9. —х? + 5х — 6
10. 2х — 11x + 17x = 5 11. —x% + 2х3 12. 18a? — 11ab — 1062 13. (a,c,d,e,f,h} 14 {c,d} 15. 5) + 2wy - Y
Ajo
Answers to Selected Exercises AA3
12 15 6 3 +
8 4177 18 112x107 20.27 5-4 2x +4l+4 2.1073 2 7+1v3
; т ‘( 46
25. "va 26. 1 0,045 V25 27. V13-2 28. -x? 20. 1.38 x 10°pounds — 30. 4times
31. a. model2 b. overestimates by $5
Section P.5
Check Point Exercises
1. а. 2х*(5х — 2) b. (x — Dx + 3) 2. (х + 5)(х° — 2) 3 (x + 8)(x + 5) or (x + 5)(х + 8) 4, (x — Dx +2)or (x + Xx —7)
5 (3x — 1)(2х + 7) от (2х + 7)(3х — 1) 6. (3x — y)(x — 4y) or (x — 4y)(3x — y) 7. a. (x + 9)(x — 9) b. (6x + 5)(6x — 5)
8. (9х? + 4)(3х + 2)(3х - 2) 9 a (x +77? в. (4х — 7)? 10. а. (х + )(х — х + 1) b (5x — 2)(25x* + 10x + 4) 11. 3x(x — 5°
2x — 1 |
12. (x + 10 + 6a)(x + 10 — 6a) 13. a - DE
Concept and Vocabulary Check
1d 2g 3b 4c 5e 6a 7.Ё 8 (x+ 1)
Exercise Set P.§ | |
1. 9(2х + 3) 3 3x(x+2) |5. 9х%(х? — 2х + 3) T(x+SHx+3) %A-3IEÉ+1) 11.([email protected]?*+5) 134-167 +2)
15. (3x — 2)(х? — 2) 17. (x + 2)(x + 3) 19. (x — 5)(х + 3) 21. (x — 5)(x — 3) 23. (3x + 2Xx — 1) 25. (3х — 28)(х + 1)
27. (2х — 1)(3х — 4) 29. (2х + 3)(2х + 5) 231. 3x- 2)3x— 1) 133. (5x + 8)(4х — 1) 386 (2х + у)(х + у) 37. (3x + 2y)(2x — 3y)
39. (x + 10)(x — 10) 41. (6х + 7)(бх — 7) 48. (3x + 5у)(3х — 5у) 45. (x“ + (x + (x — 2) | 47. (4х? + 9)(2х + 3)(2х — 3)
49. (х +1? 51. (к — 7)? 58. (2х + 1)? 86 (3х = 1)? 57. (x + 3)(x* — 3x +9) 59 (x — 4)(х? + 4х + 16)
61. (2х — 14x? + 2х + 1) 68. (4x + 3)(16х? — 12х + 9) 66 3x(x + (x — 1) 67. 4(x + 2)(x — 3) 69. 2(x* + 9)(х + 3)(х — 3)
71. (x — 3)(x + 3)(x +2) 78 20x — 8)(х +7) | ТБ. х(х = 2)(х + 2) 77 prime 78 (x — 2)(х + 2)? 81. yO” + 9 + 3 — 3)
83. 5y(2y + 3)(2y — 3) 85. (x —6 + 7у)(х — 6 — 7y) 87. (x + y)(3b + 4)(3b — 4) — 89. (y — 2)(x + 4)(x — 4)
05 4(1 + 2x) x+4 4(4x — D(x — 1)
TB (x + 5% vi 3
103. (x + 1)(5x — 6)(2x +1) 105, (x? + 6)(6x? — 1) 107. y(Y? + 19 — y? +1) 109. (1 + 2» — ду)(х + y) — y)
111. (x — y)*(x —y +2)x — y —2) 113. (2x - y) — 37) 115. а. (x — 0.4х)(1 — 0.4) = (0.6x)(0.6) = 0.36x b: no;36%
117. а. 9х? — 16 b. (3x + 4)(3x — 4) 119, a. x(x + у) — у(х + y) b. (x + y)(x — y) 121. 40% — 4ab” = 4a(a + bY(a — b)
10
(x — (x + 592
91. 2x(x + 6 + 2a)(x + 6 — 2a) — 93. x*(x — 1) 97. —(x + 3Yx +2) 9.
131. makes sense 133. makes sense 135. true 137. false 139. —(x + 5)(x — 1) 141. —
à ni (к + 5) + 1) _x+1 2 7 |
. = — + > > . = 145, — Co
143. b = 0,3,4,—c(c + 4), where c > 0 is an integer 144 +5 -5) x—5 45 3 146 6 |
section P.6 р
Check Point Exercises ;
1. а. —5 b 6-6 —2,7 2 2x #3 b. rx q — 23 # 2,x% # —2,x # —3 |
. à . 6, c. —2, . A. XX IT 3 &TDa+3p" x E |
=D orn 8-2x2-1 в ЕЙ 41551 7 (em 3) — Be + Bore 3 + 3 |
a+ ‚X E . —2, Xx «+ DE -D?* „X . (Xx — 3)(x — 3)(x + 3) or (x — 3)“(x + 3) |
2 — ;
—x* + 11x — 20 2(2 — 3x) 4 1
an —— == A, XA р
2 — 5) ‚х * 5 9 1 + ar x # 0,x 7 3 Xe + 7) х É 0,x # —7 |
Concept and Vocabulary Check :
x 3 x" = х + 4
1. polynomials 2. domain;0 8. factoring; common factors 4. 15 5. 5 T3 7. х + 3andx — 2; x + 3andx +1;
x — (x + 3) —3 1
_— + , + . . 1 10. = ==
(x + 3)(x — 2)(x + 1) 8 3x +4 9. complex; complex 30x + 3) 3x(x + 3) Xx + 3)
Exercise Set P.6
13 85-5 5-1-10 ‚лез 0°" %:x6 2, yx12 10 F066 185x223 es
x — 3 4 y —1 x — 6 3 A
x — 3)(x +3 — 1 2.47 + 4 — 92
EME, 4 0,—4,3 19. x ,X * —2,-1,2,3 21. xo # —2,0,2 23. 1, #-i 25. ) ‚х # 0,—2,2
x(x + 4) x + 2 3x 9
2(x +3 — х + 2)(х + 4 —
27. +3). # 3,—3 29, x > e 1,—5 31. rx + 4) % —6,—3,—1,3,5 33. 2,x # _2 35. 2x Ly # 0, —3 ЗЕ
9х + 39 3 3х? + 4 à
A Zak A A-5 48 ——— x #—-1,0 AX A, |
(x + dx +35)" x(x + 1) Da 77 22
37. 3,x # 2 a0. — x # 3,4 mM
3°
AA4 Answers to Selected Exercises
2x* + 50 13 4x + 16 Xx — x
4. ————————,X £ -5,5 AQ, — = Am ‚ ==, хо # =3 B3, ‚х * —5,2,-3
(x = 5)(x + 5)” 6(x +2)” (x + 372° (x + 5)(x — 2)(x + 3)"
2
Xx —2x +1 x—1 1 x +1 1 1
Le x 5 —1,1 87 x # —2, —1 > 61. ‚х + 0,= Lx #0, ‚х # —
5 Da 1? ух 59. 3x #3 я 0.3 63. Xx 7 0,y 7 0% % y
— 14 — 2x + h 24+ 5х +
6. ——x 22,3 6x2 22 6 yw 2-13 NA AO UN pu
x + 3 7 x +2 x*(x + hY | (x + 2)(x + 1)
1 2d ;
75. 2 77. — 79. 7% 1. a. 86. 2 586 i 40% i inst this strai
O +5) ab = be 81. a. 86.67, 520, 1170; It costs $86,670,000 to inoculate 6 of the population against this strain of
flu, $520,000,000 to inoculate 80% of the population, and $1,170,000,000 to inoculate 90% of the population. b. x = 100 e. The cost increases
rapidly; it is impossible to inoculate 100% of the population. — 83. a. 2078; underestimates by 22 calories — b. 2662; underestimates by 38 calories
—33х° + 263х + 515 4x” + 22x x—1
с. ‚ щие 99. does not make sense 101. makes sense 103. true 105. false 107.
—60x” + 499x + 295 (x + 5)(x + 6) х + 3
109. It cubes x. 110. —3; 0; 3; 4; 3; 0; —5 111. —8; —3; 0; 1; 0; —3; —8 112. 3; 2; 1; 0; 1; 2; 3
Chapter P Review Exercises |
1.5 216 312 4 fac) 5. fa,b,c,d,e) 6 {a,b,c,d fg} 7 (a) 8a VEL b0v8 e —-17,0V8
d. —17, == 0,0.75,V81 e V2,m ff 7, 0,0.75,V2,7,V81 9.103 10. VZ-1 ня. VT7-3 12 |4- (-17)|; 21
13
13. commutative property of addition . 14. associative property of multiplication 15. distributive property of multiplication over addition
16. commutative property of multiplication 17. commutative property of multiplication 18. commutative property of addition
1 1
19. 17x — 15 20. 2x 21. 5y — 17 22 10x 23. 55; It’s the same. 24. —108 28. = 2.55 2.77 & —8x*y”
0 1 5 |
29. 10 30. —— 31. — 32. 37,400 33. 0.0000745 34. 3.59 x 10° 35. 7.25 x 107 36. 390,000 37. 0.023
x8 16x12 4x8 | ;
38. 135 x 10” 39. 3.2 X 107 40, 42,188 years 41. 10V3 42. 2|х| УЗ 4325 M VF 4, > 46. 4xV3
6 5(6 — V3 | |
47. 20\/5 48. 16V2 49.24V2-8V3 50.6V5 81. V6 52. 3) 53. 7(V7 + V5) 54.5 55. —2
33
1
56. not areal number 57,5 58. 33 | 59. у)? 60.245 61.137 62 xV2 63.4 Ga. = 65 6.
1
68. = 69. 20x12 70. 3x 71. 25x* 720 Vy 78 8x? + 10x? — 20x — 4;degree3 74. 8x* — 5x3 + 6; degree 4
75. 120° + x* — 21x +10 76. 6x* — Tx — 5 77. 16x32 ~ 25 78 4x? +20x +25 79. 9x2 — 24x +16 80. 8x3 + 12x2 + 6x +1
81. 125х° — 150х° + 60x — 8 82. —x* — 17xy — 3y*; degree 2 83. 24x)? + xP — 124% + 4; degree S 84. 3x? + 16xy — 35y*
85. 9x“ — 30xy + 25y* 86. 9x* + 12x’y + 4y? 87. 49x? ~ 16y> 88. a3-b? 89. 3х?(5х + 1) 90. (x — 4)(x — 7)
91. (3x + 1)(5x — 2) 92 (8 —x)(8 + x) 93. prime 94. 3x'(x — 5)(x +2) 95. 4x°(5x* - 9) 96. (x + 3)(x — 3% 97 (4x — 5
98. (х° + 4)(x + 2)(x —2) 99. (y—2)(y? +2y +4) 100. (x + 4)(x? — 4x +16) 101. 3xX(x — 2)(x + 2)
67. 16
о
| | 16(1 + 2х)
102. (3x — 5)(9x* + 15х + 25) 103. x(x ~ D(x + Dx? + 1) 104. (x? — 2)(x + 5) 105, (x + 9 + y)(x + 9 — y) 06. er
X
| | 6(2x + 1 — 3 |
107. (x + 2)(x — 2)(x* + 3) 4{—x* + x? +13) 108. ex +1) 109. x2 x # —2 110. x 5-66 A111, —— x 5% ~2
| x ; x—6 x +2
x + 3 3 2 1 + ; 1 .
. a 7 2,—2 18, ———— x #.0,1,~1, — 114, X =. # —3,4,2,8 118, ——, x # 3,—3
(x — 2) (x + 2) x(x + 1) 3 x — 4 x — 3”.
4х(х — 1) 2х? — 3 Mx ~ x — 11 1 2 3 |
116. —————mx € 2,-2 47. ‚х * 3, — 118. XxX E -,—-3,—= 419, — x # 0,2
| (x + 2)(x — 2) * (x — 3)(x + 3)(x — 2) 7 32 (2x — D(x + 3)(3x + 2) "72 3 X
3x 3x + 8 10
120. ZZ 7 = 0,4,—4 121. 3 + 10* = 7
Chapter P Test
5y* 36 — V2
1, 6x” — 27x 2. -6x +17 3 {5} 4 (1,2,5,4) 5. бх?у3 + 4ху + 2у? 6 2 7.3rV2 , 811V2 9. >
X
+ 3 ` 2{x + 3
10. 2xW2x 11. ha £21 12.25 x 100 13 2x? — 13x% + 26x — 15 14 256% + 30xy + 992 45. etd) £ 3,-1,-4,73
x — 2 х +1
x“ + 2х + 15 11 3-х
. ‚ —3 17. с ‚4 (x — 20. (x* + +
16 (x + 3)}x - 3)" 73 (x — 3)(x — 4) x #3 18 354 = 0 19. (х — 3) — 6) {x“ + 3)(х + 2)
+
21. (5х — 3)(5х + 3) 22. (6x -7)* 28. (у - 5)()? + 5) + 25) 24 (х + 5 + Зу)(х + 5 — Зу) 25 с: 7
| X
4 22 . " en dinar ны A
26. —7,——, 0, 0.25, V4, 7 27. commutative property of addition 28. distributive property of multiplication over addition 29. 7.6 x 10
5
; —0.28n + 4
30. 743 31, 1.32 x 101° 32. a. 43.08%; overestimates by 0.08% b. R = 0.28n 7
degrees for every two men.; It describes the projections exactly.
2
3 i helor's
0.281 + 53 e = Three women will receive bac
A y
dr
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