INSTRUCTOR’S SOLUTIONS MANUAL
BEVERLY FUSFIELD
P RECALCULUS
A R IGHT T RIANGLE A PPROACH
FIFTH EDITION
J.
S. Ratti
University of South Florida
Marcus McWaters
University of South Florida
Leslaw A. Skrzypek
University of South Florida
Jessica Bernards
Portland Community College
Wendy Fresh
Portland Community College
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2023, 2019, 2015 by Pearson Education, Inc. 221 River Street, Hoboken, NJ 07030. All rights reserved.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-13-751980-4
ISBN-10: 0-13-751980-X
CONTENTS
Chapter P Basic Concepts of Algebra
P.1TheRealNumbersandTheirProperties......................................................1
P.2IntegerExponentsandScientificNotation...................................................8
P.3Polynomials................................................................................................13
P.4FactoringPolynomials ................................................................................20
P.5RationalExpressions..................................................................................28
P.6RationalExponentsandRadicals...............................................................38
ChapterPReviewExercises....................................................................................48
ChapterPPracticeTest...........................................................................................53
Chapter 1 Equations and Inequalities
1.1LinearEquationsinOneVariable..............................................................54
1.2ApplicationsofLinearEquations:Modeling.............................................64
1.3QuadraticEquations...................................................................................76
1.4ComplexNumbers:QuadraticEquationswithComplexSolutions...........93
1.5SolvingOtherTypesofEquations...........................................................101
1.6Inequalities...............................................................................................122
1.7EquationsandInequalitiesInvolvingAbsoluteValue.............................138
Chapter1KeyIdeasataGlance...........................................................................155
Chapter1ReviewExercises..................................................................................156
Chapter1PracticeTestA......................................................................................167
Chapter1PracticeTestB......................................................................................169
Chapter 2 Graphs and Functions
2.1TheCoordinatePlane...............................................................................171
2.2GraphsofEquations.................................................................................181
2.3Lines.........................................................................................................194
2.4Functions..................................................................................................209
2.5PropertiesofFunctions .............................................................................219
2.6ALibraryofFunctions.............................................................................229
2.7TransformationsofFunctions..................................................................240
2.8CombiningFunctions;CompositeFunctions...........................................257
2.9InverseFunctions.....................................................................................275
Chapter2KeyIdeasataGlance...........................................................................286
Chapter2ReviewExercises..................................................................................288
Chapter2PracticeTestA......................................................................................298
Chapter2PracticeTestB......................................................................................300
CumulativeReviewExercises(ChaptersP2)......................................................300
Chapter 3 Polynomial and Rational Functions
3.1QuadraticFunctions.................................................................................305
3.2PolynomialFunctions...............................................................................326
3.3DividingPolynomials...............................................................................342
3.4TheRealZerosofaPolynomialFunction ................................................353
3.5TheComplexZerosofaPolynomialFunction........................................376
3.6RationalFunctions ....................................................................................386
3.7Variation...................................................................................................406
Chapter3KeyIdeasataGlance...........................................................................411
Chapter3ReviewExercises..................................................................................414
Chapter3PracticeTestA......................................................................................431
Chapter3PracticeTestB......................................................................................432
CumulativeReviewExercises(ChaptersP3)......................................................433
Chapter 4 Exponential and Logarithmic Functions
4.1ExponentialFunctions..............................................................................436
4.2LogarithmicFunctions.............................................................................449
4.3RulesofLogarithms.................................................................................463
4.4ExponentialandLogarithmicEquationsandInequalities........................475
4.5LogarithmicScales;Modeling.................................................................490
Chapter4KeyIdeasataGlance...........................................................................501
Chapter4ReviewExercises..................................................................................501
Chapter4PracticeTestA......................................................................................510
Chapter4PracticeTestB......................................................................................511
CumulativeReviewExercises(ChaptersP4)......................................................512
Chapter 5 Trigonometric Functions
5.1AnglesandTheirMeasure.......................................................................516
5.2Right-TriangleTrigonometry...................................................................523
5.3TrigonometricFunctionsofAnyAngle;TheUnitCircle........................535
5.4GraphsoftheSineandCosineFunctions.................................................548
5.5GraphsoftheOtherTrigonometricFunctions.........................................567
5.6InverseTrigonometricFunctions.............................................................582
Chapter5KeyIdeasataGlance...........................................................................590
Chapter5ReviewExercises..................................................................................592
Chapter5PracticeTestA......................................................................................596
Chapter5PracticeTestB......................................................................................597
CumulativeReviewExercises(ChaptersP5)......................................................598
Chapter 6 Trigonometric Identities and Equations
6.1VerifyingIdentities..................................................................................601
6.2SumandDifferenceFormulas..................................................................615
6.3Double-AngleandHalf-AngleFormulas.................................................629
6.4Product-to-SumandSum-to-ProductFormulas.......................................645
6.5TrigonometricEquationsI........................................................................654
6.6TrigonometricEquationsII......................................................................664
Chapter6KeyIdeasataGlance...........................................................................684
Chapter6ReviewExercises..................................................................................685
Chapter6PracticeTestA......................................................................................691
Chapter6PracticeTestB......................................................................................693
CumulativeReviewExercises(ChaptersP6)......................................................694
Chapter 7 Applications of Trigonometric Functions
7.1TheLawofSines ......................................................................................698
7.2TheLawofCosines ..................................................................................713
7.3AreasofPolygonsUsingTrigonometry...................................................726
7.4Vectors.....................................................................................................741
7.5TheDotProduct.......................................................................................751
7.6PolarCoordinates.....................................................................................761
7.7PolarFormofComplexNumbers;DeMoivre’sTheorem.......................778
Chapter7KeyIdeasataGlance...........................................................................792
Chapter7ReviewExercises..................................................................................793
Chapter7PracticeTestA......................................................................................802
Chapter7PracticeTestB......................................................................................803 CumulativeReviewExercises(ChaptersP–7)......................................................805
Chapter 8 Systems of Equations and Inequalities
8.1SystemsofLinearEquationsinTwoVariables.......................................808
8.2SystemsofLinearEquationsinThreeVariables.....................................828
8.3Partial-FractionDecomposition...............................................................851
8.4SystemsofNonlinearEquations..............................................................875
8.5SystemsofInequalities .............................................................................889
8.6LinearProgramming.................................................................................905
Chapter8KeyIdeasataGlance...........................................................................918
Chapter8ReviewExercises..................................................................................918
Chapter8PracticeTestA......................................................................................931
Chapter8PracticeTestB......................................................................................934
CumulativeReviewExercises(ChaptersP8)......................................................937
Chapter 9 Matrices and Determinants
9.1MatricesandSystemsofEquations..........................................................940
9.2MatrixAlgebra.........................................................................................963
9.3TheMatrixInverse...................................................................................983
9.4DeterminantsandCramer’sRule...........................................................1004
Chapter9KeyIdeasataGlance.........................................................................1015
Chapter9ReviewExercises................................................................................1018
Chapter9PracticeTestA....................................................................................1029
Chapter9PracticeTestB....................................................................................1030
CumulativeReviewExercises(ChaptersP9)....................................................1032
Chapter 10 The Conic Sections
10.2TheParabola...........................................................................................1034
10.3TheEllipse..............................................................................................1050
10.4TheHyperbola........................................................................................1069
Chapter10KeyIdeasataGlance.......................................................................1092
Chapter10ReviewExercises..............................................................................1093
Chapter10PracticeTestA..................................................................................1102
Chapter10PracticeTestB..................................................................................1104
CumulativeReviewExercises(ChaptersP10)..................................................1105
Chapter 11 Further Topics in Algebra
11.1SequencesandSeries.............................................................................1108
11.2ArithmeticSequences;PartialSums......................................................1120
11.3GeometricSequencesandSeries............................................................1127
11.4MathematicalInduction ..........................................................................1138
11.5TheBinomialTheorem..........................................................................1149
11.6CountingPrinciples................................................................................1157
11.7Probability..............................................................................................1164
Chapter11KeyIdeasataGlance.......................................................................1168
Chapter11ReviewExercises..............................................................................1169
Chapter11PracticeTestA..................................................................................1173
Chapter11PracticeTestB..................................................................................1173
CumulativeReviewExercises(ChaptersP11)..................................................1174
Chapter P Basic Concepts of Algebra
P.1 The Real Numbers and Their Properties
Practice Problems
1. Let x =2.132132132…. Then,1000x =2132.132132… 10002132.132132 2.132132 9992130 2130710 999333 x x x x
2. a. Naturalnumbers:2,7
b. Wholenumbers:0,2,7
c. Integers: 21 6,3,0,2,7 7
d. Rationalnumbers: 2114 6,3,,0,,2,7 723
e. Irrationalnumbers:3,17
f. Realnumbers:thesetB
3. a. 3 22228
b. 223339 aaaa
c. 4 111111 2222216
4. a. 20 truebecause2istotheleftof0 onthenumberline.
b. 57 truebecause5istotheleftof7on thenumberline.
c. 41 falsebecause4istotheleftof 1onthenumberline.
5. 3,1,0,1,3,4,2,0,2,4 AB
6. a.
7. a. 1010
b. 3411
c.
2376711
9. a.
352015205
b. 51262522541
c. 918 575723533
Concepts and Vocabulary
1. Wholenumbersareformedbyaddingthe numberzerotothesetofnaturalnumbers.
2. Thenumber3isaninteger,butitisalsoa rationalnumberandarealnumber.
3. If a < b,then a istotheleftof b onthe numberline.
4. Ifarealnumberisnotarationalnumber,itis anirrationalnumber.
5. True
6. False.51 2 22
7. True
8. False.Anexampleis
22224, whichisrational.
Building Skills
9. 0.3,repeating
10. 0.6,repeating
11. 0.8,terminating
12. 0.25,terminating
13. 0.27,repeating
14. 0.3,repeating
15. 3.16,repeating
16. 2.73,repeating 17. 37515 3.75 1004 18. 23547 2.35 10020
24. 100142.3535 1.4235 99140.93 140.9314,093 999900 x x x
25. Rational 26. Rational
27. Rational 28. Rational
29. Rational 30. Rational
31. Irrational 32. Irrational Exercises33-38refertotheset 1217 A19,,3,0,2,10,,11 34
33. Naturalnumbers:2,11
34. Wholenumbers:0,2,11
35. Integers: 12 19,4,0,2,11 3
36. Rationalnumbers:121719,,0,2,,11 34
37. Irrationalnumbers:3,10
38. Realnumbers:Allnumbersinset A arereal numbers.
39. 3 10
base:10;exponent:3 3 101010101000
40. 54
base:5;exponent:4 4 55555625
41. 23 3
base: 2 3;exponent:3 3 22228 333327
42. 54 2
base:5;2exponent:4 4 55555625 2222216
43.
45.
48.
325
5 32322222 33296
50. 532
base:–3;exponent:2
2 535335945
55. 52 x 56. 12 x
57. 0 x 58. 0 x
59. 2714 x 60. 235 x
61. 24 4 6 62. 52
63. 40 64. 91 4 22
65. 4,3,2,0,1,2,3,4 AB
66. 0,2,4 AB
67. 4,2,0,2 AC
68. 4,3,2,1,0,1,2,3,4 BC
BCAA
69. 3,0,2 4,3,2,0,2,4
70. 4,3,2,1,0,2,4 3,0,2,4 ACBB
71. 4,3,2,0,1,2,3,4 4,3,2,0,2
ABCC
ABCC
72. 4,3,2,0,1,2,3,4 4,3,2,1,0,1,2,3,4
73. 12122,5;1,3IIII
74. 12121,7;3,5IIII
75. 1212 6,10; IIII
76. 1212 ,; IIII
77. 1212,;2,5IIII
78. 12122,;0,IIII
79. 1212,;1,35,7IIII
80. 1212,26,;3,0IIII
81. 44 82. 1717 83. 55 77 84. 33 55
85. 5252 86. 2552
87. 3223 88. 33 89. 88 1 88 90. 88 1 88
91. 575722
92. 474733
93.
(3,8)3855
94.
(2,14)2141212
95.
(6,9)691515
(12,3)1231515
97.
(20,6)20(6)1414
98.
(14,1)14(1)1313
100. 1631631919 , 555555
x 107. 39 44 x 108. 1 3 2 x 109. 4(1)44 xx 110. (3)(2)63 xx 111. 5(1)555 xyxy 112. 2(35)6102 xyxy
113. Additiveinverse:5;reciprocal: 1 5
114. Additiveinverse: 2 3;reciprocal:3 2 115. Additiveinverse:0;noreciprocal 116. Additiveinverse:1.7;reciprocal:10 17 117. Additiveinverse 118. Additiveinverse 119. Multiplicativeidentity 120. Multiplicativeidentity 121. Associativepropertyofmultiplication
122. Associativepropertyofmultiplication
123. Multiplicativeinverse
124. Multiplicativeinverse
125. Additiveidentity
126. Additiveidentity
127. Associativepropertyofaddition
128. Commutativeandassociativepropertiesof addition
129. 34334592029 5353351515
130. 737235141529 104102452020
131. 6755422567 57753535
132. 9596554559 21226121212
133. 5355332593417 61065103303015
134. 828325241034 159153954545
135. 595594253611 810851044040
136. 71751835827 8585584040
137. 575117955638 9119111199999
138. 57537215141 812831222424
139. 212215451 5252251010
140. 111312321 4643621212
38383 427427
142. 9149149 727727
7314311 2 22222
213213 3 1531153
121 5353 2315651 5335151515
150. 5310320911 2 3232666
151. 2()32(3(5))3(5) 2(2)(15)41511
xyy
xyy
152. 2()52(3(5))5(5) 2(2)(25) 4(25)21
153. 3233253(3)2(5) 9101 xy
154. 773(5)787(8)56 xy
xy xy
155. 333(5)3(5) 22 3(15)(15) 2 18 2(15)9(15)6
156. 3533(5) 3 243 (15) 33
157. 2(12)2(12(3))()(3)(5) 5 2(5)1521513 5 x xy y
158. 3(2)3(23)(1)(13(5)) 5 3(1)(1(15)) 5 377 16 55
170. 1 1 11 xyy xx y
171. 2 ,222 2 ,222 ababaab dPMa abbabba dQMb
M is themidpointofthelinesegment PQ 172. Answersmayvary.Usingthehint,wehave 440 13130 and770. 13130 Therefore, 403130 0 130130130 1106970 . 130130130130
Applying the Concepts
173. a. peoplewhoowneitherMP3playersor peoplewhoownDVDplayers.
b. peoplewhoownbothMP3playersand DVDplayers.
174. a. A ={FordBronco,HondaAccord}
b. B ={FordBronco}
c. C ={FordBronco,HondaAccord, AudiA4}
d. AB ={FordBronco}
e. BC ={FordBronco}
f. AB ={FordBronco,HondaAccord}
g. AC ={FordBronco,HondaAccord, AudiA4}
175. 119.5134.5 x
176. 4995 x
177. a. 1241204
b. 13712017
c. 11412066
178. a. 725616 MBperhour
b. 700380320 MBperhour
179. 42 C414418C 3
180. 2241854 33 cc c
IftheCelsiustemperatureis22°,then54 chirpswillbecounted.
181. Let x =thenumberofcaloriesfrombroccoli. Thenwehave 522.5550522.5559.5 xxx Thenumberofgramsofbroccoliis 9.5×100=950grams.
182. Let x =thenumberofordersoffrenchfries. Thenumberofcalorieslostfrombroccoliis 6×55=330.Thenwehave 16533001653302 xxx So,Carmenwillhavetoeat2ordersoffrench fries.
Beyond the Basics
183. True 184. True
185. a. False.Forexample,020. b. Theproductsofanonzerorationalnumber andanirrationalnumberisanirrational number.
186. False.Forexample,222.
187. False.Forexample,
23634.
188. False.Forexample, 12 42. 3
189. True 190. True Forexercises191–192,usethefollowingdefinition: Aninteger P is even if p =2n forsomeinteger n.An integer q isoddif q =2k +1forsomeinteger k
191. a. If a isodd,then a =2m +1forsome integer m
22 2 212121 441411 2211, ammm mmmm mm
whichisoftheformat2k +1.Therefore, 2 a isanoddinteger.
b. Notethatthisstatementisthe contrapositiveofthestatementinpart(a). Thatis,ifthestatementis“if p,then q”,the contrapositiveis“ifnot q,thennot p.”
Thecontrapositiveof“Ifaninteger a is odd,then2 a isalsoanoddinteger”is“If 2 a isnotanoddinteger(i.e.,aneven integer),then a isalsonotanoddinteger (i.e.,aneveninteger.”Logically,thesetwo statementsaretrue.Thatmeansthatthe originalstatementanditscontrapositiveare eitherbothtrueorbothfalse.Wealready knowfrompart(a)thattheoriginal statementistrue.Therefore,the contrapositiveisalsotrue.Thus,if2 b is even,then b isalsoaneveninteger.
192. 22 q isevenbecauseitisoftheform2n,soit followsthat2 p iseven.Then,fromexercise 191(b), p isalsoeven.Therefore, p =2n,for someinteger n.Substituting,wehave 2 2222 22222 qpnqn
222,qn and,thus, q isevenusingexercise 191(b).
Getting Ready for the Next Section
GR1. a. 235 aaa
b. 4711 aaa
c. mnmn aaa
GR2. a. 3 2 b b b b. 7 4 3 b b b
c. m mn n b b b
GR3. a. 236 aa
b. 428 aa
c. n mmnmn aaa
GR4. a. 222abab
b. 444abab
c. nnnabab
Copyright©2023PearsonEducationInc.
P.2 Integer Exponents and Scientific Notation
Practice Exercises
1.
2333333 0 244416 1616116 xxxxx x
3. a. 4 404 0 3 3381 3
b.
7771
5 0(0)(5)0 7771
c. 8 1(1)(8)8 8 1 xxx x
d. 5 2(2)(5)10 xxx
b. 22 1212(1)(2) 2 2 555 25 25 xxx x x
c. 3 232(3)36 xyxyxy
d. 362(2)(3)3 3 x xyxy y
6. a. 22 2 111 39 3
b. 222 2 107749 710100 10
7. a. 422 48 11 2 24
x xx
b. 2323 23363 1 xyxy xyxyxy
8. 5 732,0007.3210
9. 1010 2 88 2 2.1102.1102.10 10 3.53.53.51010 0.6010$60perperson
Concepts and Vocabulary
1. Intheexpression27,thenumber2iscalled theexponent.
2. Intheexpression73,thebaseis3.
3. Thenumber2 1 4 simplifiestobethepositive integer16.
4. Thepower-of-a-productruleallowsusto rewrite 53 a as 33 5. a
5. False. 10101111
6. False.When 23 x issimplified,the expressionbecomes6 x
7. True
8. False
xx xx
5. a. 11 1 1112 2 22
3 3 11 2 28
2 2 11 3 39
2 2 11 3 39
3 3 11 2 28
33. 11 (1110)1 10 2 222 2 34. 6 (68)2 82 311 33 339
35. 3412 1212 55 1 55 36. 5210 (108)2 88 99 9981 99 37. 52 (54)(2(3))11 43 23 23236 23
38. 23 (2(3))(31) 3 12 45 45 45 45425100
39. 2 1 12 5112 22 252525
40. 2 1 12 7113 33 374949
41. 1 23 32
42. 11 5 5
43. 222 2 2339 324 2
44. 222 2 3224 239 3
45. 2 2 111149 121 7121 11 749
46. 2 2 131125 169 5169 13 525
47. 4044 1 xyxx
48. 1011 1 xy xx
49. 11
y xyy xx
50. 2 222 22 1 x xyx yy
51. 12 2 1 xy xy 52. 32 32 1 xy xy
53.
4 312 12 1 xx x 54.
2 510 10 1 xx x
55. 3 11(11)(3)33 xxx
56. 12 4(4)(12)48 xxx
57. 5553()3 xyxy
58. 6668()8 xyxy
59. 22122 2 4 44 x xyxy y
60. 33133 3 6 66 y xyxy x
61. 551(1)(5)5 5 3 33 x xyxy y
62. 6616(1)(6) 6 5 55 y xyxy x
63. 32326 6104 2525104 1 xxx xx xxxx
64. 222 21210 34341210 1 xxx xx xxxx 65. 333333 363 2236 3 33 3 228 8 8 8
xyxyxy xy xxx y xy x
444444 4124 33412 4 84 8 55625 625 625 625
5 25255105 55 105555 3(3)243 243243
69. 22222 22 111 3(3)25 99 51159 2525 xxxx x
444444 4 4 4 111 5(5)5625
3 23236 5353315915 446464 xxx xyxyxyxy
5 2525510510 3351510 33243243
223 2(1)2234 124 xyx xyxy xyy
35 3(4)5712 472 273 33 9
xyx xyxy xyy
15105 15105 16 16 xyz xyz
abcabc abc abcabc abca bc
3 323(3)(3)(2)(3) 2432(3)(4)(3)3(3) 396 6129 3(6)9126(9) 315 3315 3
xyzxyz xyzxyz xyz xyz xyz xz xyz y
86. 1 211(2)(1)(1)(1) 58(5)(1)1(8)(1) 121 152(1)18 518 3 637 67 xyzxyz xyzxyz xyz xyz xyz y xyz xz
87. 1251.25102
88. 2472.47102
89. 5 850,0008.510
90. 5 205,0002.0510
91. 0.0077.0103
92. 0.00191.9103
93. 0.000002752.75106
94. 0.00000383.8106
Applying the Concepts
95. 135333 ft21080ft
96. 3 33 1 675in.25in. 3
97. a. 222 (2)442 xxAA
b. 222 (3)993 xxAA
98. a. 22 222 42 22
ddd A
99. 2225,000(0.25)1562.5lb
102. 10 8 56billion56,000,000,0005.610 330million330,000,0003.310
103. 7 24hr60min60sec 365.25daysdayhrmin
365.25246060sec 31,557,600sec 3.1557610sec
104. 7 24hr60min60sec 366daysdayhrmin
366246060sec
31,622,400sec 3.1622410sec
105. Celestial Body Equatorial Diameter (km) Scientific Notation Earth12,7001.27×104km Moon34803.48×103km Sun1,390,0001.39×106km Jupiter134,0001.34×105km Mercury48004.8×103km
Beyond the Basics
Getting Ready for the Next Section
GR1. a. 25257 xxxx
b.
c.
232510 xxx
23523510 2342424 yyyyy
GR2. a. 222257 xxx
b. 22234xxx
c. 333335119 xxxx
GR3. 2 252525 xxxxxxx
GR4. 27327231462 xxxxxxx
GR5. 22 3 51551 55 xxxxx xx
GR6. True.Thisisanexampleofthecommutative propertyofmultiplication.
GR7. False.2 5 x and3 3 x arenotliketerms,sothey cannotbecombined.
GR8. 2323212 532 25341068 1068 xxxxxx xxx
P.3 Polynomials
Practice Problems
1. 2 167157889ft
2. Usingthehorizontalmethod:
3. Tousethecolumnmethod,firstchangethe signineachterminthesecondpolynomial andthenadd.
4.
5.
32 3233 543 2425 242225
22 222 43232 432 5227 527227 105354214 103314
6. a. 2 2 4174287 4277 xxxxx xx
b. 2 2 3225615410 61910 xxxxx xx
7. 222 2 3232322 9124 xxx xx
8. 2 2212121214 xxxx
9. a. 2 2222224 xyxyxyxy b.
3 3223 3223 3223 2 23232 8346 8126 xy xxyxyy xxyxyy xxyxyy
Concepts and Vocabulary
1. Thepolynomial723294 xxx has leadingcoefficient3anddegree7.
2. Whenapolynomialiswrittensothatthe exponentsineachtermdecreasefromleftto right,itissaidtobeinstandardform.
3. Whenapolynomialin x ofdegree3isaddedto apolynomialin x ofdegree4,theresulting polynomialhasdegree4.
4. Whenapolynomialin x ofdegree3is multipliedbyapolynomialin x ofdegree4,the resultingpolynomialhasdegree7.
5. True
6. True.Thisistrueif A or B orbotharezero.
7. False.Thisisnotapolynomialbecausetheterm 3 x doesnothaveanexponentthatiseithera positiveintegerorzero.
8. True
Building Skills
9. Polynomial;221 xx
10. Notapolynomial
11. Notapolynomial
12. Apolynomial;7543321 xxxx
13. Notapolynomial
14. Notapolynomial
15. Apolynomial;instandardform
16. Apolynomial;instandardform
17. Degree:1;terms:7x,3
18. Degree:2;terms:23,7 x
19. Degree:4;terms:42,,2,9xxx
20. Degree:7;terms:739,2,,21 xxx 21.
27.
22 22 2 2316322 622181212 241414
28.
29.
30.
332 332 32 342214 342214 221 yyyyy yyyyy yyy
222 222 2 5312325 5312325 661 yyyyyy yyyyyy yy
31. 2 6(23)1218 xxxx
32. 2 7(34)2128 xxxx
33.
2 22 322 32 122 22122 2222 342 xxx xxxxx xxxxx xxx
34.
2 22 322 32 5231 2315231 2310155 213165 xxx xxxxx xxxxx xxx
2 22 322 32 321 3121 333222 3552 xxx xxxxx xxxxx xxx
2 22 322 32 2134 234134 26834 2554 xxx xxxxx xxxxx xxx
37. 22 (1)(2)(2)1(2) 2232 xxxxx xxxxx
38. 2 2 (2)(3)(3)2(3) 326 56 xxxxx xxx xx
39. 2 2 (32)(31)9362 992 xxxxx xx
40. 2 2 (3)(25)25615 21115 xxxxx xx
41. 2 2 (45)(3)412515 4715 xxxxx xx
42. 2 2 (21)(5)2105 2115 xxxxx xx
43. 2 2 (32)(21)6342 672 xxxxx xx
44. 2 2 (1)(53)5353 583 xxxxx xx
45. 22 22 (23)(25)410615 4415 xaxaxaxaxa xaxa
46. 22 22 (52)(5)525210 52310 xaxaxaxaxa xaxa
47. 2222 (2)4444 xxxxxx
48. 2222 (3)6969 xxxxxx
49. 22 (41)1681 xxx
50. 22 (32)9124 xxx
51. 22 2 2 333 2 444 39 216 xxx
52.
53.
54.
55.
58.
222 22 2522255 42025 xyxxyy
60. 3323 32 2123213211 81261 xxxx xxx
61. 333233 2 333333 92727 xxxxxx xx
62. 333233 2 232322 6128 xxxxxx xx
63. 32323 322 232322 6128 xyxxyxy xxyxy
64. 3 3223 3223 23 23233233 8365427 xy xxyxyy xxyxyy
65. 2 22525252254 xxxx
66. 2 22343434916 xxxx
67. 22 22 232323 49 xyxyxy xy
68. 22 22 525252 254 xyxyxy xy
2 22 2 1111 xxxx xxxx
22 2 2 222 222 4 4 yyy yyy y y
2 333262224xxxx
2 22 322 32 2335 235335 26103915 291915 xxx xxxxx xxxxx xxx
2 22 322 32 243 43243 43286 656 xxx xxxxx xxxxx xxx
2 22 2233 11 111 11 yyy yyyyy yyyyyy
2 22 322 3 4416 4164416 41641664 64 yyy yyyyy yyyyy
77.
2 22 322 3 6636 6366636 636636216 216 xxx xxxxx xxxxx x
2 22 3223 11 111 11 xxx xxxxx xxxxxx
79. 22 22 (2)(35)35610 31110 xyxyxxyxyy xxyy
80. 22 22 (2)(72)14472 14112 xyxyxxyxyy xxyy
81. 22 22 (2)(37)61437 6117 xyxyxxyxyy xxyy
82. 22 22 (3)(25)25615 215 xyxyxxyxyy xxyy
3 3223 3223 3223 3223 22 32322 2 61282 6128 26128 xyxy xxyxyy xy xxyxyyxy xxxyxyy yxxyxyy
43223 32234 4334 6128 2122416 41616 xxyxyxy xyxyxyy xxyxyy
3 3223 3223 22 23232 2 81262 xyxy xxyxyy xy xxyxyyxy
83.
22 2222 4224 2 xyxyxyxyxyxy xyxyxyxy xyxy xxyy
84.
22 2222 4224 22 2222 2222 44 168 xyxy xyxyxyxy xyxyxyxy xyxy xxyy
85.
222 2222 322223 323 244 4444 4444 34 xyxyxyxxyy xxxyyyxxyy xxyxyxyxyy xxyy
222 2222 322223 323 244 4444 4444 34 xyxyxyxxyy xxxyyyxxyy xxyxyxyxyy xxyy
3223 3223 43223 32234 4334 28126 8126 1624122 8126 16164 xxxyxyy yxxyxyy xxyxyxy xyxyxyy xxyxyy
Forexercises8994,use 222222. ababababab
89.
90.
222 2 4,3 2 42316610 xyxy xyxyxy
222 2 3,2 2 3229413 xyxy xyxyxy
91. a. 1 3 x x
92.
b. Let2 ax and2 1 b x Usingtheresult fromparta,wehave 2
93. Let a =3x and b =2y.Then ab =6xy
222 2 943226 122661447272 xyxyxy
94. Let a =3x and b =7y.Then ab =21xy
222 2 94937221 102211 1004258 xyxyxy
Beyond the Basics
Applying the Concepts
95. 2 0.1150.0255.57$8.425 in2015 (fiveyearsafter2010)
96. 2 0.03540.1545.176.33 In2012(fouryearsafter2008),theatergrosses were6.33billiondollars.
97. 2 0.1404050$250.00
98. 22 1510515575 2575$100.00
99. 2 1652051625100500feet d
100. 2 1621021642084feet d
101. a. 22.50 x b.
2 2 2 301022.50 3022.503022510 67519510 10195675 xx xxx xx xx
102. a. 10250 n b. 2 2 (502)(10250) 50(10)50(250)2(10)2(250) 50012,50020500 2012,500 nn nnnn nnn n
(continued)
22222222 2244244424 88 xyzxyzxyzxyyzxzxyzxyyzxz xyxz
Alternatively,recognizethedifferenceoftwosquares.
22 222222 42288 xyzxyzxyzxyzxyzxyz xyzxyxz
106. 8;12abcabbcac 2222222 2222 222 222 2222 8212 6424 40 abcabcabbcacabcabbcac abc abc abc
107. 222 12;44xyzxyz
2222222 2 2222 12442 144442 1002 50 xyzxyzxyyzxzxyzxyyzxz xyyzxz xyyzxz xyyzxz xyyzxz
108. 22264;18 xyzxyyzxz
2222222 2 2 2222 64218 100 10010 xyzxyzxyyzxzxyzxyyzxz xyz xyz xyz
109.
3333 abcabcabbcac aabcabbcacbabcabbcaccabcabbcac aabacababcacabbbcabbcabcacbccabcbcac abcabc
222 222222222 322222322222322
110. Fromexercise109, 222333333333 03033 abcabbcacabcabcabcabcabcabc
111. Let a = x – y, b = y – z,and c = z – x.Then
0. abcxyyzzx
Fromexercise110,if0, abc then3333. abcabc So, 3333. xyyzzxxyyzzx
112. Let a =2x –3y, b =3y –5z,and c =5z –3x.Then
2335530.abcxyyzzx Fromexercise110,if0, abc then3333. abcabc So, 333 2335523233552. xyyzzxxyyzzx
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113. Fromexercise109, 2223333. abcabcabbcacabcabc
If8 abc and19, abbcac wehave
222222333 222333 3 8193(1)
abcabcabbcacabcabcabbcacabcabc abcabcabc
Fromexercise103, 2222222 2222222222 2222 8219643826
abcabcabbcacabcabbcac abcabcabc
Substitutingintoequation(1)givesus 333333 826193563. abcabcabcabc
114. Fromexercise109, 2223333. abcabcabbcacabcabc If9 abc and11, abbcca wehave
222222333 222333 3 9113(1) abcabcabbcacabcabcabbcacabcabc abcabcabc
Fromexercise103,
2222222 2222222222 2222 9211812259 abcabcabbcacabcabbcac abcabcabc
Substitutingintoequation(1)givesus 333333 9591134323. abcabcabcabc
Getting Ready for the Next Section
GR1. 2 252525 xxxxxxx
GR2. 27327231462 xxxxxxx
GR3.
223 5155155 xxxxxxx
GR4. True.Thisisanexampleofthecommutative propertyformultiplication.
a b a + b ab
GR5. 34712
GR6. –352–15
GR7. 4268
GR8. 3–5–2–15
GR9. 25710
GR10. 35815
GR11. –572–35
GR12. 5–7–2–35
GR13. –2–3–56
GR14. 2356
P.4 Factoring Polynomials
Practice Problems
1. a. 5332 614237 xxxx
b. 542232 72135735 xxxxxx
c. 22 5252 xxyxyxyx
2. a 26842 xxxx
b. 231052 xxxx
3. a. 22 442xxx
b. 22 96131 xxx
4. a. 21644 xxx
b. 2 4252525 xxx
5. 422 2 8199 339 xxx xxx
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6.
7. FollowingthereasoninginExample7,infour years,thecompanywillhaveinvestedthe initial12milliondollars,plusanadditional4 milliondollars.Thus,thetotalinvestmentis 16milliondollars.Tofindtheprofitorloss with16milliondollarsinvested,let x =16in theprofit-losspolynomial:
Thecompanywillhavemadeaprofitof 16.224milliondollarsinfouryears.
8.
42422 422 2 22 22 22 5969 69 3 33 33 xxxxx xxx xx
Concepts and Vocabulary
1. Thepolynomials x +2and x 2arecalled factorsofthepolynomial24. x
2. Thepolynomial3y isthegreatestcommon monomialfactorofthepolynomial236. yy
3. TheGCFofthepolynomial32 1030 xx is 2 10. x
4. Apolynomialthatcannotbefactoredasa productoftwopolynomials(excluding constantpolynomials1) issaidtobe irreducible.
5. True
6. True
7. False.Thepolynomial24 x canbefactored as 22.xx Therefore,itisnot irreducible.
8. True
Building Skills
9. 8248(3) xx
10. 5255(5) xx
11. 2 6126(2) xxxx
12. 22 3213(7) xx
13. 232 7147(12) xxxx
14. 343 9189(12) xxxx
15. 43222221xxxxxx
16. 432225757xxxxxx
17. 322 3(31) xxxx
18. 322 222(1) xxxx
19. 322 844(21) axaxaxx
20. 423221axaxaxaxxx
21. 33xxyxyxyx
22. 2131123 aaaaa
37. 2712(3)(4) xxxx
38. 2815(3)(5) xxxx
39. 268(2)(4) xxxx
40. 2914(2)(7) xxxx 41. 234(1)(4) xxxx
42. 256(1)(6) xxxx
43. Irreducible 44. Irreducible
45. 2 236(29)(4) xxxx
46. 2 2327(29)(3) xxxx
47. 2 61712(23)(34) xxxx
48. 2 8103(23)(41) xxxx
49. 2 3114(31)(4) xxxx
50. 2 572(52)(1) xxxx
51. Irreducible 52. Irreducible
53. 222045 xxyyxyxy
54. 22 82154523 ppqqpqpq
55. 22 15281528 3457 xxxx xx
56. 22 2435442435 2725 yyyy yy
57. 22 69(3)xxx
58. 22 816(4)xxx
59. 22 961(31) xxx
60. 22 36121(61) xxx
61. 22 25204(52) xxx
62. 22 2 643244(1681) 4(41) xxxx x
63. 22 49429(73) xxx
64. 22 92416(34) xxx
65. 264(8)(8) xxx
66. 2121(11)(11) xxx
67. 2 41(21)(21) xxx
68. 2 91(31)(31) xxx
69. 2 169(43)(43) xxx
70. 2 2549(57)(57) xxx
71.
72.
4222 111111xxxxxx
422 2 8199 339 xxx xxx 73.
74.
4422 20554152121 xxxx
4422 1275342532525 xxxx
75.
76.
77.
78.
33326444416xxxxx
333212555525xxxxx
3332273339xxxxx
333221666636xxxxx
79. 333282242xxxxx
80.
3332273393xxxxx
81.
3332 827(2)323469 xxxxx
42422 422 2 22 22 22 121 21 1 11 11 xxxxx xxx xx xxxx xxxx
422 2 22 22 22 7969 69 3 33 33 xxxxx xxx xx xxxx xxxx
42422 422 2 22 22 22 168916 8169 49 4343 3434 xxxxx xxx xx xxxx xxxx
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90.
91.
42422 422 2 22 22 22 163612436 12364 64 6262 2626 xxxxx xxx xx xxxx xxxx
4422 422 2 22 22 22 4444 444 24 2222 2222 xxxx xxx xx xxxx xxxx
92.
4422 422 2 22 22 22 64161664 166416 816 8484 4848 xxxx xxx xx xxxx xxxx
93. 2 116(14)(14) xxx
94. 2 425(25)(25) xxx
95. 22 69(3)xxx
96. 22 816(4)xxx
97. 22 441(21) xxx
98. 22 1681(41) xxx
99.
100.
22 2810245 2(1)(5) xxxx xx
22 51040528 524 xxxx xx
101. 2 2320(25)(4) xxxx
102. 2 2730(25)(6) xxxx
103. 22436xx isirreducible.
104. 22025xx isirreducible.
105. 54332 32 312123(44) 3(2) xxxxxx xx
106. 54332 32 216322(816) 2(4) xxxxxx xx
107. 2 91(31)(31) xxx
108. 2 1625(45)(45) xxx
109. 22 16249(43) xxx
110. 22 42025(25) xxx
111. 215 x isirreducible.
112. 224 x isirreducible.
113.
114.
322 45844584 5292 xxxxxx xxx
322 2540925409 5951 xxxxxx xxx 115.
22322 7878 8 axaxaaxaxa axaxa 116.
22322 10241024 212 axaxaaxaxa axaxa
2222 22 1681681616 416 4444 xaxxxa xa xaxa
2222 22 36696936 336 3636 xaxxxa xa xaxa
5432322 32 31212344 32 xxyxyxxxyy xxy
5432322 32 42436469 43 xxyxyxxxyy xxy
2222 22 25444425 225 2525 xaxxxa xa xaxa
122.
123.
Applying the Concepts
125.
Ifonesideofthegardenis x feet,andthe perimeteris16feet,then1628 2 x x givestheotherdimensionoftherectangle.So, theareaofthegardenis(8). xx
126.
Ifonesideofthetrayis x inches,andthe perimeteris42inches,then 422 21 2 x x givestheotherdimensionofthetray.So,the areaofthetrayis(21). xx
127. a. Mateowillbreakevenwhentheprofitor lossis0,sowemustdetermineinhow manymonthsthatwilloccur.Evaluatethe polynomialthatrepresentsMateo’sprofit orlossfordifferentvaluesuntilthevalue ofthepolynomialis0.Thenumberof monthsisrepresentedby x x 32 10309050 xxx Polynomial value
32 1003009005050
32 10130190150160
x 32 10309050 xxx Polynomial value
32 10230290250270
32 10330390350320 4
32 10430490450250
32 105305905500
Mateowillbreakevenin5months.
b. Evaluatetheprofitorlossafter10months byletting x =10. 32 101030109010506050 After10months,thereisaprofitof$6050.
128. a. Thecompanywillbreakevenwhenthe profitorlossis0,sowemustdeterminein howmanymonthsthatwilloccur.Evaluate thepolynomialthatrepresentsthe company’sprofitorlossfordifferent valuesuntilthevalueofthepolynomialis 0.Thenumberofmonthsisrepresented by x x 32 903606302520 xxx Polynomial value 0
32 900360063002520 2520 1 32 901360163012520 2160 2 32 902360263022520 1980
3 32 903360363032520 1440 4
32 904360463042520 0 Thecompanywillbreakevenin4months.
b. Evaluatetheprofitorlossafter10months byletting x =10.
32 901036010630102520 57,780 After10months,thereisaprofitof $57,780.
Usethefigurebelowforexercises129and130.
129. If x =thelengthofthecutcorner,then36–2x =thelengthofthebox,and16–2x =thewidthofthebox. Theheightoftheboxis x.So,(362)(162)(2(18))(2(8))4(18)(8). vxxxxxxxxx
130. 222 22 ..(362)(162)2(362)2(162)5761044724324 57644(144)4(12)(12) SAxxxxxxxxxxxx xxxx
131. Theareaoftheoutsidecircle=2 4cm. Theareaoftheinsidecircle=22cm. x So,theareaofthedisk= areaofoutsidecircle–areaofinsidecircle=22 4(4) xx 2 (2)(2)cm. xx
132. Thevolumeoftheinsidecylinderis238 72ft3 ,andthevolumeof theoutsidecylinderis223 88ft.xx Sothevolumebetweenthecylindersis 22 8728(9) xx 3 8(3)(3)ft. xx
133. Ifonesideofthefenceis x feet,andtherancherneedsatotalof2800feetof fencing,thenthewidthofthefenceis2800–2x.So,theareaofthepenis (28002)280022 xxxx
2 2(1400)ft. xx
.Theperimeterofthefigureis48,sothelength
134. Theareaofthefigure=theareaoftherectangleplustheareaofthecircle.Findthelengthoftherectangle asfollows:Thecircumferenceofthecircle=2 2 x x
andtheareaoftherectangleis
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143. Fromexercise109inSectionP.3,wehave
Then,
144. Seeexercise111insectionP.3.
145. Let
333 3333 zxyxyzyzxxyzxyyzzx
146.
Inthenumerator,let2222 ,,axybyz and22. czx Then a + b + c =0,andwecanapply SectionP.3exercise111.Usetheresultfromexercise144aboveforthedenominator.
Getting Ready for the Next Section
GR1.
Notethat2.
3. Notethat0,4,1
Notethat6,6.
P.5 Rational Expressions
Practice Problems
1.
Thelikelihoodthatastudentwhotests positiveisanonuserwhenthetestthatisused is95%accurateisabout32%.
2.
5. a. Notethat2,5. xx
b. Notethat4,3. xx
6. a.
22 222 341 , 2232 xxx xxxxx
222LCD322
b. 2 22 2137 , 2545 xx
2 2 2555 4515 LCD155 xxx xxxx xxx
7. a. Notethat2,2. xx
22 2 22 22 22 4 444 4 222 422 2222
b. Notethat0,5. xx
TheLCDis 2 65. xx
222 22 22 418901490 6565 2745745 6535 xxxxx xxxx xxx xxx
8. Notethat0,5,5. xxx 22 2 51513 33335 2525 25 3 33 5 x xxx xx x x xx x
9. Notethat 4 5
Concepts and Vocabulary
1. Theleastcommondenominatoroftwo rationalexpressionsisthepolynomialofleast degreethatcontainseachdenominatorasa factor.
2. ThefirststepinfindingtheLCDoftwo rationalexpressionsistofactorthe denominatorscompletely.
3. Ifthedenominatorsoftworational expressionsare22xx and22, xx then theLCDis 21.xxx
4. Arationalexpressionthatcontainsanother rationalexpressioninitsnumeratoror denominatoriscalledacomplexrational expression.
5. False 6. True 7. False 8. True Building Skills
9. 22 222(1)2,1 21(1)1 xx x xxxx
10. 22 363(2)3,2 44(2)2 xx x xxxx
11. 2 333(1)3,1,1 1(1)(1)1 xx xx xxxx
12. 2 1055(2)5,2,2 4(2)(2)2 xx xx xxxx
13. 2 262(3)2(3) 9(3)(3)(3)(3) 2 3,3,3 xxx xxxxx xx x
14. 2 1533(5)3,5,5 25(5)(5)5 xx xx xxxx
15. 211(12)1 1, 12122 xx x xx
16. 251(52)2 1, 52525 xx x xx
17. 2269(3)3,34124(3)4 xxxx x xx
18. 221025(5)5,5 3153(5)3 xxxx x xx
19. 2 22 777(1)7,1 21(1)1 xxxxx x xxxx
20. 2 22 4124(3)4,3 69(3)3 xxxxx x xxxx
21. 2 2 1110(10)(1) 67(7)(1) 10 7,7,1 xxxx xxxx x xx x
22. 2 2 215(5)(3) 712(4)(3) 5 4,3,4 xxxx xxxx x xx x
43222 43222 2 2 61442372 61042352 2312 2312 21 ,,0,223 xxxxxx xxxxxx
322 43222 2 2 331 31143114 31 314 11 ,,0,443 xxxx xxxxxx xx xxx xxx x
31020310(2)1 245152(2)5(3)
xxxx xxxx 26. 6442(32)41 28962(4)3(32)3
xxxx xxxx
27. 2 2 2662(3)(3)(2) 484(2)(3)(3) 9 3 2(3) xxxxxx xxxx x x x
28. 222594 42106 (53)(53)(2)(2) 2(2)2(53) (53)(2) 4 xx xx xxxx xx xx
29. 22 22 2 71 67 (7)(1)(1)1 (1)(7) xxx xxx xxxxx xxx x
30. 2 2 9515(3)(3)5(3) 693(3)(3)3 5 xxxxx xxxxxx
31. 22 22 61 329 (3)(2)(1)(1)1 (1)(2)(3)(3)3 xxx xxx xxxxx xxxxx
32. 22 22 2816(2)(4)(4)(4)(2)(4) 2054(4)(5)(1)(4)(5)(1) xxxxxxxxx xxxxxxxxxx
33. 2 2 2321(2)(1)(2)1 11(2)(2) 4 xxxxxx xxxx x
34. 2 2 38151(3)(3)(5)1 55(3)(3) 9 xxxxxx xxxx x
35. 248293 6964(2)8
xxx x
36. 3412399 209204(3)80
37. 22 2 926(3)(3)55(3) 52(3)2 xxxxxxx xxx x
38. 22 2 177(1)(1)(1)(1) 337(1)21 xxxxxxx xxx
39. 2 22 231(1)(3)3(4)3(3) 31214816(4) xxxxxxx xxxxxx
40. 22 22 5632(2)(3)(3)(4)4 69712(3)(3)(1)(2)1 xxxxxxxxx xxxxxxxxx
41. 22 3322222 931(3)(3)(1)(2)13 3 8221(2)(24)124 xxxxxxx x xxxxxxxxxxx
42. 22 22 253102(5)(5)(1)(1)21 3415(4)(1)(5)(2)54 xxxxxxxxxx xxxxxxxxxx
43. 33 555 xx
44. 77 444 xx
45. 44 212121 xx xxx
46. 23 737373 xxx
47. 22221(1)1 1111 xxxx
48. 272(27)(2)9 32323232 xxxxx
49. 4242 331(3)3 422(2) 33 xx xxxx xx
50. 2222 1111(1) 224 111 44 or 11 xx xxxx xx xxx xx
51. 222 527 111 xxx xxx
52. 2222 342 2(1)2(1)2(1)(1) xxxx xxxx
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53. 784 2(3)2(3)2(3)3 xxxx xxxx
54. 2222 48123 4(5)4(5)4(5)(5) xxxx xxxx
55. 22 221 44(2)(2)2 xx
56. 22 55555(1)5 11(1)(1)(1)(1)1 xxx xxxxxxx
57. 22 2(2)(21)(21)(252)(2) 2121(21)(21)(21)(21)(21)(21) 622(13) (21)(21)(21)(21)
58. 22 212(21)(41)2(41)(861)(82)18 4141(41)(41)(41)(41)(41)(41)(41)(41)
59.
222 323232 3232 2(2)(2)(2)(2) 22(2)(2)(2)(2)(2)(2) (2)(248)(2) (2)(2)(2)(2)(2)(2) 248248 (2)(2)(2)(2)
60. 222 32323232 3123(1)(1)(1)2(1) 11(1)(1)(1)(1)(1)(1) 33122261 (1)(1)(1)(1)(1)(1)(1)(1) xxxxxxxxx xxxxxxxxxxxx xxxxxxxxxx
Inexercises61–68,tofindtheLCD,firstfactoreachdenominatorcompletely,formaproductofthedistinct irreduciblefactorsofeachpolynomial,andthenattachtoeachfactorinthisproductthegreatestexponentthat appearsonthisfactorinanyofthefactoreddenominators.
61. Thedenominatorsare363(2) xx and484(2) xx LCD=34(2)12(2) xx
62. Thedenominatorsare7217(13)and393(13) xxxx LCD=73(13)21(13) xx
63. Thedenominatorsare241(21)(21) xxx and2 (21) x (21)(21) xx LCD=2 (21)(21) xx
64. Thedenominatorsare2(31)(31)(31) xxx and291 x (31)(31) xx LCD=2(31)(31) xx
65. Thedenominatorsare232(1)(2) xxxx and21 x (1)(1)LCD(1)(1)(2) xxxxx
66. Thedenominatorsare26(3)(2) xxxx and29 x (3)(3)LCD(3)(3)(2) xxxxx
67. Thedenominatorsare254(1)(4) xxxx and 22xx (1)(2)LCD(1)(4)(2) xxxxx
68. Thedenominatorsare223(3)(1) xxxx and 232xx (1)(2) xx LCD(3)(1)(2) xxx Forexercises69–84,remembertofindtheLCDfirst,andthenrewriteeachrationalexpressionasarational expressionwiththeLCDasthedenominator.Nextaddorsubtractnumerators,followingtherulesfororderof operations.Finallysimplifythefinalrationalexpression.Leaveyouranswerinfactoredform.
69. 2 52525(3)25152715 33(3)(3)(3)(3)(3)(3)(3)(3)(3)(3) 9
xxxxxxxxxxxx x
70. 22 2 333(1)3334 11(1)(1)(1)(1)(1)(1)(1)(1)(1)(1) 1 (34) (1)(1) xxxxxxxxxxxx xxxxxxxxxxxx x xx xx
71. 2 22 222(2) 42(2)(2)2(2)(2)(2)(2) 2(2)4(4) (2)(2)(2)(2)(2)(2) xxxxxxx xxxxxxxxx xxxxxxx xxxxxx
72. 2 222 3121312131(21)(4) 164(4)(4)4(4)(4)(4)(4) 31(294)265265 (4)(4)(4)(4)(4)(4) xxxxxxx
73. 22 232311 3106(2)(5)(2)(3)52 2523 (5)(2)(5)(2)(2)(5) xxxx xxxxxxxxxx xxx xxxxxx
74. 22 222 2 3131(3)(1)(1)(2) 221(2)(1)(1)(1)(2)(1)(1)(1)(1)(2) (43)(32)25 (2)(1)(1)(2)(1) xxxxxxxx xxxxxxxxxxxxxx xxxxxx xxxxx
75. 22222 2222 222 23412341(23)(31)(41)(31) (31)(31) 91(31)(31)(31)(31)(31)(31) (6113)(121)181022(951) (31)(31)(31)(31)(31)(31) xxxxxxxx xx xxxxxxx xxxxxxxx xxxxxx
76. 22222 2222 222 313313(31)(21)(3)(21) (21)(21) (21)41(21)(21)(21)(21)(21) (61)(273)8622(431) (21)(21)(21)(21)(21)(21) xxxxxxxx xx xxxxxxx xxxxxxxx xxxxxx
77. 22 22 3333(3)(4)(3)(5) 25920(5)(5)(4)(5)(5)(5)(4)(4)(5)(5) (12)(815)9279(3) (5)(5)(4)(5)(5)(4)(5)(5)(4) xxxxxxxx xxxxxxxxxxxxx xxxxxx xxxxxxxxx
78. 22 22 2272272(3)(27)(4) 16712(4)(4)(4)(3)(4)(4)(3)(4)(4)(3) (26)(228)728 (4)(4)(3)(4)(4)(3) 7(4)7 (4)(4)(3)(4)(3) xxxxxxxx xxxxxxxxxxxxx
79. 22 311311311 2222(2)(2)2244 31(2)1(2)3(2)(2)23 (2)(2)(2)(2)(2)(2)(2)(2)(2)(2) xxxxxxxx
80. 22 257257257 55555(5)(5)5 2525 2(5)57(5) (5)(5)(5)(5)(5)(5) (102)5(357)5405(8) (5)(5)(5)(5)(5)(5) xxxxxxxx xx xx xxxxxx xxxx xxxxxx
81. 22222 35(3)(3)(5)(5)(9)(25)16 53(5)(3)(3)(5)(5)(3)(5)(3) xaxaxaxaxaxaxaxaa xaxaxaxaxaxaxaxaxaxa
82.
22222 32(3)(3)(2)(2)(9)(4)5 23(2)(3)(3)(2)(2)(3)(2)(3) xaxaxaxaxaxaxaxax xaxaxaxaxaxaxaxaxaxa
84. 22 222222 22 22 222 22 2 2222 111() ()()() () () (2) () 2( 2) ()() xxh xhxxxhxxh xxh xxh xxxhh xxh xhhhxh xxhxxh
87. UsingMethod1: 111 1 11111 1 1,0,1 xxxx xxxx xxxx x x
88. UsingMethod1: 2222 2222 2222 2 111 1 1111 1 11 1(1)(1),1,0,1 xxxx xxxx xxxx x xxx
89. UsingMethod2: 11111 1,0,1 111 1 x xxx xx x x xx
