Solution Manual for College Algebra and Trigonometry, 4th Edition

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SOLUTIONMANUALBEVERLYFUSFIELDCOLLEGEALGEBRA ANDTRIGONOMETRYANDPRECALCULUSARIGHTTRIANGLEAPPROACHFOURTHEDITIONJ. S. RattiUniversity of South FloridaMarcus McWatersUniversity of South FloridaLeslaw A. SkrzypekUniversity of South Florida

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CONTENTSChapter PBasic Concepts of AlgebraP.1The Real Numbers and Their Properties ......................................................1P.2Integer Exponents and Scientific Notation...................................................7P.3Polynomials................................................................................................13P.4Factoring Polynomials................................................................................20P.5Rational Expressions ..................................................................................27P.6Rational Exponents and Radicals ...............................................................37Chapter P Review Exercises....................................................................................46Chapter P Practice Test ...........................................................................................51Chapter 1Equations and Inequalities1.1Linear Equations in One Variable ..............................................................531.2Applications of Linear Equations: Modeling .............................................631.3Quadratic Equations ...................................................................................741.4Complex Numbers: Quadratic Equations with Complex Solutions ...........891.5Solving Other Types of Equations .............................................................961.6Inequalities ...............................................................................................1161.7Equations and Inequalities Involving Absolute Value .............................131Chapter 1 Review Exercises..................................................................................148Chapter 1 Practice Test A......................................................................................159Chapter 1 Practice Test B......................................................................................160Chapter 2Graphs and Functions2.1The Coordinate Plane ...............................................................................1632.2Graphs of Equations .................................................................................1722.3Lines .........................................................................................................1852.4Functions ..................................................................................................2002.5Properties of Functions.............................................................................2102.6A Library of Functions .............................................................................2192.7Transformations of Functions ..................................................................2302.8Combining Functions; Composite Functions ...........................................2482.9Inverse Functions .....................................................................................263Chapter 2 Review Exercises..................................................................................275Chapter 2 Practice Test A......................................................................................285Chapter 2 Practice Test B......................................................................................286Cumulative Review Exercises (Chapters P−2)......................................................287Chapter 3Polynomial and Rational Functions3.1Quadratic Functions .................................................................................2913.2Polynomial Functions...............................................................................3103.3Dividing Polynomials...............................................................................3233.4The Real Zeros of a Polynomial Function................................................3343.5The Complex Zeros of a Polynomial Function ........................................3533.6Rational Functions....................................................................................3633.7Variation...................................................................................................382Chapter 3 Review Exercises..................................................................................387Chapter 3 Practice Test A......................................................................................403Chapter 3 Practice Test B......................................................................................404Cumulative Review Exercises (Chapters P−3)......................................................405

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Chapter 4Exponential and Logarithmic Functions4.1Exponential Functions..............................................................................4084.2Logarithmic Functions .............................................................................4204.3Rules of Logarithms .................................................................................4334.4Exponential and Logarithmic Equations and Inequalities........................4464.5Logarithmic Scales; Modeling .................................................................459Chapter 4 Review Exercises..................................................................................470Chapter 4 Practice Test A......................................................................................478Chapter 4 Practice Test B......................................................................................479Cumulative Review Exercises (Chapters P−4)......................................................480Chapter 5Trigonometric Functions5.1Angles and Their Measure .......................................................................4835.2Right-Triangle Trigonometry ...................................................................4905.3Trigonometric Functions of Any Angle; The Unit Circle ........................5025.4Graphs of the Sine and Cosine Functions.................................................5145.5Graphs of the Other Trigonometric Functions .........................................5265.6Inverse Trigonometric Functions .............................................................538Chapter 5 Review Exercises..................................................................................546Chapter 5 Practice Test A......................................................................................551Chapter 5 Practice Test B......................................................................................552Cumulative Review Exercises (Chapters P−5)......................................................552Chapter 6Trigonometric Identities and Equations6.1Verifying Identities ..................................................................................5556.2Sum and Difference Formulas..................................................................5686.3Double-Angle and Half-Angle Formulas .................................................5826.4Product-to-Sum and Sum-to-Product Formulas .......................................5986.5Trigonometric Equations I........................................................................6066.6Trigonometric Equations II ......................................................................616Chapter 6 Review Exercises..................................................................................637Chapter 6 Practice Test A......................................................................................644Chapter 6 Practice Test B......................................................................................645Cumulative Review Exercises (Chapters 1−6)......................................................647Chapter 7Applications of Trigonometric Functions7.1The Law of Sines......................................................................................6507.2The Law of Cosines..................................................................................6647.3Areas of Polygons Using Trigonometry...................................................6787.4Vectors .....................................................................................................6937.5The Dot Product .......................................................................................7037.6Polar Coordinates .....................................................................................7137.7Polar Form of Complex Numbers; DeMoivre’s Theorem .......................729Chapter 7 Review Exercises..................................................................................744Chapter 7 Practice Test A......................................................................................752Chapter 7 Practice Test B......................................................................................754Cumulative Review Exercises (Chapters 1–7) ......................................................756

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Chapter 8Systems of Equations and Inequalities8.1Systems of Linear Equations in Two Variables .......................................7598.2Systems of Linear Equations in Three Variables .....................................7778.3Partial-Fraction Decomposition ...............................................................7988.4Systems of Nonlinear Equations ..............................................................8218.5Systems of Inequalities.............................................................................8358.6Linear Programming.................................................................................848Chapter 8 Review Exercises..................................................................................863Chapter 8 Practice Test A......................................................................................876Chapter 8 Practice Test B......................................................................................879Cumulative Review Exercises (Chapters P−8)......................................................881Chapter 9Matrices and Determinants9.1Matrices and Systems of Equations..........................................................8849.2Matrix Algebra .........................................................................................9059.3The Matrix Inverse ...................................................................................9259.4Determinants and Cramer’s Rule .............................................................946Chapter 9 Review Exercises..................................................................................959Chapter 9 Practice Test A......................................................................................970Chapter 9 Practice Test B......................................................................................971Cumulative Review Exercises (Chapters P−9)......................................................972Chapter 10The Conic Sections10.2The Parabola.............................................................................................97510.3The Ellipse................................................................................................99010.4The Hyperbola........................................................................................1008Chapter 10 Review Exercises..............................................................................1032Chapter 10 Practice Test A..................................................................................1041Chapter 10 Practice Test B ..................................................................................1043Cumulative Review Exercises (Chapters P−10)..................................................1044Chapter 11Further Topics in Algebra11.1Sequences and Series .............................................................................104711.2Arithmetic Sequences; Partial Sums ......................................................105811.3Geometric Sequences and Series............................................................106611.4Mathematical Induction..........................................................................107511.5The Binomial Theorem ..........................................................................108611.6Counting Principles ................................................................................109411.7Probability ..............................................................................................1101Chapter 11 Review Exercises..............................................................................1105Chapter 11 Practice Test A..................................................................................1109Chapter 11 Practice Test B ..................................................................................1109Cumulative Review Exercises (Chapters P−11)..................................................1110

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1Chapter P Basic Concepts of AlgebraP.1The Real Numbers and TheirPropertiesP.1 Practice Problems1.Letx= 2.132132132….Then, 1000x= 2132.132132…10002132.1321322.13213299921302130710999333xxxx2. a.Natural numbers: 2, 7b.Whole numbers: 0, 2, 7c.Integers:216,3, 0, 2, 77 d.Rational numbers:21146,3,, 0,, 2, 7723 e.Irrational numbers:3,17f.Real numbers: the set B3. a.328b.223339aaaac.411111122222164. a.Trueb.Truec.False5.3,1, 0, 1, 3 ,4,2, 0, 2, 4AB 0 ,4,3,2,1, 0, 1, 2, 3, 4ABAB6. a.12,II b.122, 5II 7. a.1010b.3411 c.237118.7, 27299  d9. a.352015205 b.5126 252 2541c.9185 75 72353344 d.234xforx= 62236432341  10. a.731431748888b.824063435151515c.9791433142713132d.55155 1658158168 15161811621532311. a.233233110133333xxb.1122777333xy12.39C413417 C3P.1 Concepts and Vocabulary1.Whole numbers are formed by adding thenumber zero to the set of natural numbers.2.The number−3 is an integer, but it is also arational number and a real number.3.Ifa<b, thenais to the left ofbon thenumber line.4.If a real number is not a rational number, it isan irrational number.

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2Chapter PBasic Concepts of Algebra5.True6.False.51222 7.True8.False. An example is22224,which is rational.P.1 Building Skills9.0.3, repeating10.0.6, repeating11.0.8, terminating12.0.25, terminating13.0.27, repeating14.0.3, repeating15.3.16, repeating16.2.73, repeating17.375153.75100418.235472.3510020  19.1053.35.3948481693165.33xxxx 20.1096.69.6987872993xxxx21.100213.132.139921121199xxxx22.100323.233.239932032099xxxx23.100452.3234.52399447.8447.84478223999990495xxxx24.100142.35351.423599140.93140.9314, 093999900xxxx25.Rational26.Rational27.Rational28.Rational29.Rational30.Rational31.Irrational32.IrrationalExercises 33-38 refer to the set1217A19,,3, 0, 2,10,, 113433.Natural numbers: 2, 1134.Whole numbers: 0, 2, 1135.Integers:1219,4, 0, 2, 113 36.Rational numbers:121719,, 0, 2,, 113437.Irrational numbers:3,1038.Real numbers: All numbers in setAare realnumbers.39.310base: 10; exponent: 331010 10 10100040.45base: 5; exponent: 4455 5 5 562541.323base:2 ;3exponent: 332222833332742.452base:5 ;2exponent: 44555556252222216

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Section P.1The Real Numbers and Their Properties343.32base: –2; exponent: 3  322228    44.412base:1 ;2exponent: 441111112222216         45.42 3base: 3; exponent: 442 32 3 3 3 32 8116246.3153 base:1 ;3exponent: 331111155553333272747.42 3base: 3; exponent: 442 32 3 3 3 32 81162   48.432 base: –2; exponent: 4432322223 1648    49.532 base: –2; exponent: 553232222233296    50.253 base: –3; exponent: 22535335 945    51.32 52.32 53.112254.1xx55.52x56.12x57.0x58.0x59.2714x60.235x61.244662.52 63.4064.91422 65.4,3,2, 0, 1, 2, 3, 4AB66.0, 2, 4AB67.4,2, 0, 2AC68.4,3,2,1, 0, 1, 2, 3, 4BC69.3, 0, 24,3,2, 0, 2, 4BCAA70.4,3,2,1, 0, 2, 43, 0, 2, 4ACBB71.4,3,2, 0, 1, 2, 3, 44,3,2, 0, 2ABCC72.4,3,2, 0, 1, 2, 3, 44,3,2,1, 0, 1, 2, 3, 4ABCC73.12122, 5 ;1, 3IIII74.12121, 7 ;3, 5IIII75.12126, 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  83.557784.335585.525286.255287.322388.3389.8818890.88188 

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4Chapter PBasic Concepts of Algebra91.575722  92.474733 93.(3,8)3855 d94.(2,14)2141212 d95.( 6, 9)691515  d96.( 12, 3)1231515  d97.( 20,6)20( 6)1414   d98.( 14,1)14( 1)1313   d99.2242242626,777777d100.1631631919,555555d101.31x102.62 x103.3 x104.0x105.5x106.1 x107.3944x108.132 x109.4(1)44xx110.( 3)(2)63 xx111.5(1)555xyxy112.2(35)6102xyxy113.Additive inverse:5; reciprocal:15114.Additive inverse:23; reciprocal:32115.Additive inverse: 0; no reciprocal116.Additive inverse:1.7; reciprocal:1017117.Additive inverse118.Additive inverse119.Multiplicative identity120.Multiplicative identity121.Associative property of multiplication

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Section P.1The Real Numbers and Their Properties5122.Associative property of multiplication123.Multiplicative inverse124.Multiplicative inverse125.Additive identity126.Additive identity127.Associative property of addition128.Commutative and associative properties ofaddition129.349202953151515130.73141529104202020131.6542256757353535132.9554559212121212133.53259341761030303015134.82241034159454545135.59253611810404040 136.713582785404040137.5755638911999999 138.5715141812242424139.2145152101010 140.1132146121212141.3824279142.91427273143.88168 1535165155 16215144.5515561615666 1536145.77217162821816821316146.337315910710151071415147.3131251102222148.7314311222222149.21216513 15353151515150.53103209112 3232666151.2()32(3( 5))3( 5)2( 2)( 15)41511   xyy152.2()52(3( 5))5( 5)2( 2)( 25)4( 25)21      xyy153.323 3253(3)2(5)9101 xy154.77 3( 5)7 87(8)56 xy155.333( 5)3( 5)223( 15)( 15)218( 15)9( 15)62     xyxy156.3533( 5)3243( 15)33 yxyx157.2(12 )2(12(3))()( 3)( 5)52( 5)15215135xx yy   

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6Chapter PBasic Concepts of Algebra158.3(2)3(23)(1)(13( 5))53( 1)(1( 15))53771655  xxyy159.14114131 462232( 5)651544 xy160.4848482( 5)355352252210415575       yxy161.333xxxxyyy162.222332656xxxxxxx163.5355 3515xxx.164.2254100xxx165.3232xyxy166.245245xyxy167.1xyxyyxxxx168.xyxyxzxzxz169.111xyxyxy170.111111yyxxyxxy171.2,2222,222ababaabdPMaabbabbad QMbSince,,,22abbadPMd QMMisthe midpoint of the line segmentPQ.172.Answers may vary. Using the hint, we have44013130 and770 .13130Therefore,40313001301301301106970 .130130130130  P.1 Applying the Concepts173. a.people who own either MP3 players orpeople who own DVD players.b.people who own both MP3 players andDVD players.174. a.A= {Lexus LS 460}b.B= {Lexus LS 460, Lincoln Continental}c.C= {Lexus LS 460, Lincoln Continental,Audi A4}d.AB= {Lexus LS 460}e.BC= {Lexus LS 460, LincolnContinental }f.AB= {Lexus LS 460, LincolnContinental }g.AC= {Lexus LS 460, LincolnContinental, Audi A4}175.119.5134.5x176.30107x177. a.1241204b.13712017c.11412066 178. a.15141b.1517.52.5c.1596179.Letx= the number of calories from broccoli.Then we have522.5550522.5559.5xxxThe number of grams of broccoli is9.5 × 100 = 950 grams.

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Solution Manual for College Algebra and Trigonometry, 4th Edition - Page 12 preview image

Section P.2Integer Exponents and Scientific Notation7180.Letx= the number of orders of french fries.The number of calories lost from broccoli is6 × 55 = 330. Then we have16533001653302xxxSo, Carmen will have to eat 2 orders of frenchfries.P.1 Beyond the Basics181.True182.True183. a.False. For example,020.b.The products of a nonzero rational numberand an irrational number is an irrationalnumber.184.False. For example,222.185.False. For example,23634. 186.False. For example,1242.3187.True188.TrueFor exercises 189–190, use the following definition:An integerPisevenifp= 2nfor some integern. Anintegerqis odd ifq= 2k+ 1 for some integerk.189. a.Ifais odd, thena= 2m+ 1 for someintegerm.2222121214414112 211,ammmmmm mm mwhich is of the format 2k+1. Therefore,2ais an odd integer.b.Assume thatbis odd. Then, from part (a),2bis also odd. However,2bis an eveninteger, so, by contradiction,bmust be aneven integer.190.22qis even because it is of the form 2n, so itfollows that2pis even. Then, from exercise189(b),pis also even. Therefore,p= 2n, forsome integern. Substituting, we have222222222 2qpnqn222,qnand, thus,qis even using exercise189 (b).P.1 Getting Ready for the Next Section191.a.523aaab.4711aaac.mnmnaaa192.a.32bbbb.743bbbc.mmnnbbb193. a.362aab.284aac.nmmnaa194. a.222ababb.444ababc.nnnababP.2Integer Exponents and ScientificNotationP.2 Practice Exercises1. a.1122b.0415c.223242392. a.27933xxxb.332333302444161616 116xxxxxx 3. a.44040333813

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Solution Manual for College Algebra and Trigonometry, 4th Edition - Page 13 preview image

8Chapter PBasic Concepts of Algebrab.1(2)325551255 c.33(4)74222333 xxxx4. a.05( 5)(0)07771b.50(0)(5)07771c.81(1)(8)881xxxxd.52(2)(5)10xxx5. a. 1111112222xxxxb.221212(1)(2)225552525xxxxxc.3232(3)36xyx yx yd.632(2)(3)33xxyxyy6. a.222111393b.2222107749710100107. a.2244811224xxxb.3223336321 xyxyx yxyxy8.5732, 0007.32109.102282.910290100.8910$893253.2510per personP.2 Concepts and Vocabulary1.In the expression27 ,the number 2 is calledthe exponent.2.In the expression73 ,the base is 3.3.The number214simplifies to be the positiveinteger 16.4.The power-of-a-product rule allows us torewrite35aas335.a5.False.101011116.False. When32xis simplified, theexpression becomes6.x7.True8.FalseP.2 Building Skills9.22111133339   10.3311111222228      11.441216212.221242 13.07114.08115.071  16.02117.233 262226418.322 3633372919.2222441133381320.12122211777497

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Solution Manual for College Algebra and Trigonometry, 4th Edition - Page 14 preview image

Section P.2Integer Exponents and Scientific Notation921.3332211152515, 625522.333111155512523.35(35)2444416 24.23(23)177777 25.0031011226.005911027.2222111112333399928.2222111112555525252529.3311282  30.2211393  31.221139332.3311282 33.11(11 10)1102222234.6(68)2823113393335.431212125515536.2510(108)2889999819937.52(54)(2(3))1143232323623  38.23(2(3))(3 1)312454545454 25100  39.2112511222252525   40.2112711333494937   41.1233242.115543.22222339324244.22223224239345.2211114912171211149746.2213112516951691325547.40441xyxx48.10111xyxx49.11yxyyxx50.2222221xxyxyy51.1221xyxy52.32321xyx y53.4312121xxx54.2510101xxx55.311(11) (3)33xxx 56.124(4) ( 12)48xxx 

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Solution Manual for College Algebra and Trigonometry, 4th Edition - Page 15 preview image

10Chapter PBasic Concepts of Algebra57.5553()3 xyx y58.6668()8 xyx y59.221222444xxyxyy60.331333666yxyxyx61.551(1) ( 5)55333xxyxyy 62.6616( 1) (6)6555yxyxyx   63.233 26610452 510421xxxxxxxxx64.2222121043 4121031xxxxxxxxx65.33333336322 363333228888xyx yx yxyxxxyxyx66.444444412433 412484855625625625625xyx yxyxyxxxyxyx67.5252 55105551055553( 3)243243243  xyxyxyxxxxyx y68.32332 33633363332( 2)888  xyx yx yyyyx yx y69.22222221113( 3)2599115592525xxxxx70.4444444441115( 5)5625113381381625yyyyy71.3232 36535 3315915446464xxxxyx yx yx y 72.5252 551051033 5151033243243xyxyxyxyyyy73.3353(2)3 15424  xyxxyx yxyy74.2232( 1)2234124  xyxxyx yxyy75.353(4)5712472273339  xyxxyx yxyy76.52572(3)21732155553  xyyxyxyxyx77.32432 343643364111xxxxxxxxxxxx 78.1212124343441243124121212341241212121212121241212121212412001616162288222222212aaa babbbababababaabbaba bbb 

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Solution Manual for College Algebra and Trigonometry, 4th Edition - Page 16 preview image

Section P.2Integer Exponents and Scientific Notation1179.  32342224223324442232 3362 42 488386852522221616116111616xyxyxyxyxyxyxyx yxyx yxyxyx y   80.223242 334555261552651522266126222122xyyxy yx yxyx yx yxyxyxyyxyxyx  81.23227323273222232277327444x y zx y zx y zx y zxyzxyz3 22 226423 72 77211476214142715105151051616161616xyzx y zxyzxyzxyzxyzxyz82.2222222223 2211621113222212522521330332244444xyzxy zxy zxy xzx y xzx yxzxy zxyzx y zzxy zx 83.22241( 3)22432464655551  abcabca bcbaba84.252522522324323423433122( 3)()998( 2)89988   abca b cabca b ca b caa bcbc85.3323(3)( 3)(2)(3)2432( 3)(4)(3)3(3)39661293(6)9126(9)31533153   xyzxyzxyzxyzxy zxyzxyzx zx yzy86.1211(2)(1)(1)( 1)58(5)(1)1(8)( 1)121152(1)1 8518363767xyzxyzxyzxyzxy zxyzx yzyxy zx z  87.21251.251088.22472.471089.5850, 0008.51090.5205, 0002.051091.30.0077.01092.30.00191.91093.60.000002752.751094.60.00000383.810P.2 Applying the Concepts95.333135 ft21080 ft96.3331675 in.25 in.397. a.222(2 )442xxAAb.222(3 )993xxAA

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