Solution Manual for Trigonometry, 12th Edition

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’SSOLUTIONSMANUALBEVERLYFUSFIELDTRIGONOMETRYTWELFTHEDITIONMargaret L. LialAmerican River CollegeJohn HornsbyUniversity of New OrleansDavid I. SchneiderUniversity of MarylandCallie J. DanielsSt. Charles Community College

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CONTENTSRAlgebra ReviewR.1 Basic Concepts from Algebra ....................................................................................1R.2 Real Number Operations and Properties....................................................................3R.3 Exponents, Polynomials, and Factoring.....................................................................8R.4 Rational Expressions ................................................................................................14R.5 Radical Expressions .................................................................................................20R.6 Equations and Inequalities .......................................................................................27R.7 Rectangular Coordinates and Graphs.......................................................................34R.8 Functions ..................................................................................................................42R.9 Graphing Techniques ...............................................................................................47Chapter R Review Exercises...........................................................................................60Chapter R Test ................................................................................................................681Trigonometric Functions1.1 Angles .......................................................................................................................721.2 Angle Relationships and Similar Triangles ..............................................................79Chapter 1 Quiz (Sections 1.1−1.2)..................................................................................841.3 Trigonometric Functions...........................................................................................851.4 Using the Definitions of the Trigonometric Functions .............................................98Chapter 1 Review Exercises .........................................................................................106Chapter 1 Test ...............................................................................................................1122Acute Angles and Right Triangles2.1 Trigonometric Functions of Acute Angles .............................................................1152.2 Trigonometric Functions of Non-Acute Angles .....................................................1222.3 Approximations of Trigonometric Function Values...............................................131Chapter 2 Quiz (Sections 2.1−2.3)................................................................................1382.4 Solutions and Applications of Right Triangles.......................................................1392.5 Further Applications of Right Triangles .................................................................148Chapter 2 Review Exercises .........................................................................................158Chapter 2 Test ...............................................................................................................1643Radian Measure and the Unit Circle3.1 Radian Measure ......................................................................................................1683.2 Applications of Radian Measure.............................................................................1723.3 The Unit Circle and Circular Functions..................................................................181Chapter 3 Quiz (Sections 3.1−3.3)................................................................................1923.4 Linear and Angular Speed ......................................................................................192Chapter 3 Review Exercises .........................................................................................197Chapter 3 Test ...............................................................................................................202

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4Graphs of the Circular Functions4.1 Graphs of the Sine and Cosine Functions ...............................................................2054.2 Translations of the Graphs of the Sine and Cosine Functions ................................215Chapter 4 Quiz (Sections 4.1−4.2)................................................................................2304.3 Graphs of the Tangent and Cotangent Functions....................................................2334.4: Graphs of the Secant and Cosecant Functions.......................................................244Summary Exercises on Graphing Circular Functions...................................................2534.5 Harmonic Motion....................................................................................................256Chapter 4 Review Exercises .........................................................................................262Chapter 4 Test ...............................................................................................................2715Trigonometric Identities5.1 Fundamental Identities............................................................................................2765.2 Verifying Trigonometric Identities .........................................................................2845.3 Sum and Difference Identities for Cosine...............................................................2955.4 Sum and Difference Identities for Sine and Tangent..............................................302Chapter 5 Quiz (Sections 5.1−5.4)................................................................................3135.5 Double-Angle Identities..........................................................................................3145.6 Half-Angle Identities ..............................................................................................323Summary Exercises on Verifying Trigonometric Identities .........................................332Chapter 5 Review Exercises .........................................................................................337Chapter 5 Test ...............................................................................................................3496Inverse Circular Functions and Trigonometric Equations6.1 Inverse Circular Functions......................................................................................3526.2 Trigonometric Equations I ......................................................................................3656.3 Trigonometric Equations II.....................................................................................375Chapter 6 Quiz (Sections 6.1−6.3)................................................................................3856.4 Equations Involving Inverse Trigonometric Functions ..........................................387Chapter 6 Review Exercises .........................................................................................395Chapter 6 Test ...............................................................................................................4037Applications of Trigonometry and Vectors7.1 Oblique Triangles and the Law of Sines.................................................................4067.2 The Ambiguous Case of the Law of Sines..............................................................4147.3 The Law of Cosines ................................................................................................421Chapter 7 Quiz (Sections 7.1−7.3)................................................................................4337.4 Geometrically Defined Vectors and Applications ..................................................4347.5 Algebraically Defined Vectors and the Dot Product ..............................................443Summary Exercises on Applications of Trigonometry and Vectors ............................449Chapter 7 Review Exercises .........................................................................................451Chapter 7 Test ...............................................................................................................459

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8Complex Numbers, Polar Equations, and Parametric Equations8.1 Complex Numbers ..................................................................................................4628.2 Trigonometric (Polar) Form of Complex Numbers ................................................4678.3 The Product and Quotient Theorems ......................................................................4728.4 DeMoivre’s Theorem; Powers and Roots of Complex Numbers ...........................478Chapter 8 Quiz (Sections 8.1−8.4)................................................................................4908.5 Polar Equations and Graphs....................................................................................4928.6 Parametric Equations, Graphs, and Applications ...................................................507Chapter 8 Review Exercises .........................................................................................518Chapter 8 Test ...............................................................................................................525

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1Chapter RALGEBRA REVIEWSection R.1Basic Concepts fromAlgebra1.The set0, 1, 2, 3,describes the set ofwhole numbers.2.The set containing no elements is the empty(or null) set, symbolized.3.The opposite, or negative, of a number is itsadditive inverse.4.The distance on a number line from a numberto 0 is the absolute value of the number.5.If the real numberais to the left of the realnumberbon a number line, thena< (or is less than)b.6.(a)0 is a whole number. Therefore, it is alsoan integer, a rational number, and a realnumber. 0 belongs to B, C, D, F.(b)34 is a natural number. Therefore, it isalso a whole number, an integer, arational number, and a real number. 34belongs to A, B, C, D, F.(c)94is a rational number and a realnumber.94belongs to D, F.(d)366is a natural number. Therefore,it is also a whole number, an integer, arational number, and a real number.36belongs to A, B, C, D, F.(e)13 is an irrational number and a realnumber.13belongs to E, F.(f)21654100252.16is a rational number anda real number. 2.16 belongs to D, F.7.The set11139271,,,,isinfinite. No, 3 isnot an element of the set.8.Using set notation, the set {x|xis a naturalnumber less than 6} is {1, 2, 3, 4, 5}.9.11 10.(a)The additive inverse of 10 is –10.(b)The absolute value of 10 is 10.11.The elements in the set|is a whole number less than 6xxare0, 1, 2, 3, 4, 5 .12.The elements in the set|is a whole number less than 9m mare0, 1, 2, 3, 4, 5, 6, 7,8 .13.The elements in the set|is a natural number greater than 4zzare5, 6, 7,8,.14.The elements in the set|is a natural number greater than 8yyare9, 10, 11, 12,.15.The elements in the set|is an integer less than or equal to 4zzare,1, 0, 1, 2, 3, 4 .16.The elements in the set|is an integer less than 3ppare,2,1, 0, 1, 2, .17.The elements in the set|is an even integer greater than 8aaare10, 12, 14, 16,.18.The elements in the set|is an odd integer less than 1kkare,7,5,3,1 .19.The elements in the set|is a number whose absolute value is 4ppare4, 4 .20.The elements in the set|is a number whose absolute value is 7w ware7, 7 .

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2Chapter R Algebra Review21.There are no elements in the set{x|xis an irrational number that is alsorational}. Using set notation, this is.22.There are no elements in the set{r|ris a number that is both positive andnegative}. Using set notation, this is.For exercises 23–26, more than one description maybe possible.23.{2, 4, 6, 8} can be described as{x|xis an even natural number less than orequal to 8}.24.{11, 12, 13, 14} can be described as{x|xis an integer between 10 and 15}.25.{4, 8, 12, 16, …} can be described as{x|xis a positive multiple of 4}.26.{…, –6, –3, 0, 3, 6, …} can be described as{x|xis an integer multiple of 3}.For Exercises 27 and 28,A= {1, 2, 3, 4, 5, 6},B= {1, 3, 5},C= {1, 6}, andD= {4}.27.(a) 4 , orADD(b) 1BC(c)1, 3, 5 , orBAB(d)1, 6 , orCAC28.(a)1, 2, 3, 4, 5, 6 , orABA(b)1, 3, 4, 5BD(c)1, 3, 5, 6BC(d)1, 4, 6CDFor Exercises 29–34,51214846,,,3, 0,, 1, 2, 3,12.A29.1 and 3 are natural numbers.30.0, 1, and 3 are whole numbers.31.–6,124(or –3), 0, 1, and 3 are integers.32.51214846,(or3),, 0,, 1, and 3arerational numbers.33.3, 2and12are irrational numbers.34.All are real numbers.35.677521259,6,0.7, 0,,7, 4.6, 8,, 13,X(a)Natural numbers: 8, 13,755(or 15)(b)Whole numbers: 0, 8, 13,755(or 15)(c)Integers: –9, 0, 8, 13,755(or 15)(d)Rational numbers: –9, –0.7, 0,67, 4.6, 8,212,13,755(or 15)(e)Irrational numbers:6,7(f)Real numbers: All are real numbers36.341340228,5,0.6, 0,,3,, 5,, 17,X(a)Natural numbers: 5, 17,402(or 20)(b)Whole numbers: 0, 5, 17,402(or 20)(c)Integers: –8, 0, 5, 17,402(or 20)(d)Rational numbers: –8, –0.6, 0,34,5,132,17,402(or 20)(e)Irrational numbers:5,3,(f)Real numbers: All are real numbers37.False. Some are whole numbers, but negativeintegers are not.38.True39.False. No irrational number is an integer.40.True41.True42.False. No rational number is irrational.43.True44.True45.True46.True47.4xwhenx= –4 andx= 4.48.(a)A.44 (b)A.44(c)B.44  

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Section R.2 Real Number Operations and Properties3(d)B.44   49.(a)Additive inverse:6(b)Absolute value:6650.(a)Additive inverse:1212 (b)Absolute value:121251.(a)Additive inverse:6655(b)Absolute value:665552.(a)Additive inverse:0.16(b)Absolute value:0.160.1653.8854.191955.332256.334457.55 58.1212 59.22  60.66  61.4.54.5 62.12.412.4 63.Pacific Ocean, Indian Ocean, Caribbean Sea,South China Sea, Gulf of California64.Point Success, Rainier, Matlalcueyetl, Steele,Denali65.True.14, 04014, 040;12,80012,80014,040 > 12,80066.False.23752375;844884482375 is not greater than 8448.Use the following number line to answer exercises67–74.67.61 True68.42 True69.43 False70.31 False71.32 True72.63 True73.33 False74.55 False75.2676.1577.49 78.16 79.105 80.127 81.71 82.41083.5584.66 85.50086.5141987.05 False88.110False89.77True90.1010True91.673610 True92.74175 True93.2 5461010True94.873 51515True95.3333    True96.4444    True97.8686    False98.104104    FalseSection R.2Real Number Operationsand Properties1.The sum of two negative numbers is negative.2.The product of two negative numbers ispositive.3.The quotient formed by any nonzero numberdivided by 0 is undefined, and the quotientformed by 0 divided by any nonzero number is0.4.The commutative property is used to changethe order of two terms or factors, and theassociative property is used to change thegrouping of three terms or factors.5.Like terms are terms with the same variablesraised to the same powers.6.The numerical coefficient in the term27yzis–7.7.31010008.2 510210559.322162624    10.74728xyxy 11.6( 13)(613)19   

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4Chapter R Algebra Review12.8( 16)(816)24  13.156(156)9  14.179(179)8  15.13( 4)134916.19( 13)1913617.732891934121212  18.54158769181818  19.The difference between 4.5 and 2.8 is 1.7. Thenumber with the greater absolute value, 4.5 ispositive, so the answer is positive. Thus,2.84.51.7.20.3.86.26.23.82.421.494( 9)5  22.373( 7)4  23.656(5)(65)11    24.8178( 17)(817)25   25.8( 13)8132126.12(22)12223427.12.31( 2.13)12.312.13(12.312.13)10.18   28.15.88( 9.42)15.889.42(15.889.42)6.46   29.949427406710310330303030.33336212714414428282831.8614( 14)14   32.71522( 22)22   33.24242( 4)6     34.1613161316( 13)3 35.The product of two numbers with the samesign is positive,so8 (5)40.36.The product of two numbers with the samesign is positive,so20 (4)80.37.The product of two numbers with differentsigns is negative,so5 (7)35. 38.The product of two numbers with differentsigns is negative,so6 (9)54. 39.4(0)4 00The multiplication property of 0 states that theproduct of any real number and 0 is 0.40.0( 8)080 The multiplication property of 0 states that theproduct of any real number and 0 is 0.41.The product of two numbers with differentsigns is negative,so5125 2 66 .2252 5 55  42.The product of two numbers with differentsigns is negative,so9219 3 73 .7367 4 94  43.The product of two numbers with the samesign is positive,so3243 3 81.898 944.The product of two numbers with the samesign is positive,so2222 2 111.11411 2 245.The product of two numbers with differentsigns is negative,so0.06(0.4)0.024. 46.The product of two numbers with differentsigns is negative,so0.08(0.7)0.056. 47.The quotient of two nonzero real numbers withthe samesign is positive,so2411246 46.44448.The quotient of two nonzero real numbers withthe samesign is positive,so4511455 95.99949.The quotient of two nonzero real numbers withdifferentsigns is negative,so100111004 254.252525   

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Section R.2 Real Number Operations and Properties550.The quotient of two nonzero real numbers withdifferentsigns is negative,so150111505 305.303030   51.01008852.0100141453.Division by 0 is undefined, so50is undefined.54.Division by 0 is undefined, so130isundefined.55.The quotient of two nonzero real numbers withthe samesign is positive,so10121052 5 525 .175171217 2 610256.22332252 11 510235233323 3 116957.44345453555335  58.121251213121351313135513  59.1212412313413313433 4 3913 413  60.7727362636237 372 3 24  61.232322362.33261263.7.290.8 64.4.550.9 65.,141433dP Q   or,414133dP Q    66.,84841212dP R or,481212dP R  67.,818199d Q R or,1899d Q R  68.,1211313d Q S or,1121313d Q S  69.(a)2864(b)28(8 8)64  (c)2(8)(8) (8)64(d)2(8)(64)64   70.(a)3464(b)34(4 4 4)64  (c)3(4)(4) (4) (4)64 (d)3(4)(64)64   71.422 2 2 216  72.533 3 3 3 3243  73.   42222216   74.622222226475.    5333333243     76.5( 2)( 2)( 2)( 2)( 2)( 2)32 77.42 323 3 3 32 81162   

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6Chapter R Algebra Review78.34 545 5 54 125500   79.(a)Because91539514,+÷=+=thegrandson's answer was correct.(b)The reasoning was incorrect. Divisionmust be done first, and then the additionfollows. The grandson's “Order ofProcess rule” is not correct.It justhappens coincidentally in this problemthat he obtained the correct answer thewrong way.80.7777777149784975775081.123 412122482.155 215102583.6 3124181241831584.9 48236823643285.10302 31015 310455586.12243 2128 212162887.357 3( 2)57 3( 8)52181688    88.343 5( 3)43 5( 27)4152719278     89.Simplify within parentheses.2221845(37)1845( 4)184541816542547411−+−−=−+− −=−++=−++=++=+=90.Simplify within parentheses.2221029(18)1029( 7)102971049769715722 91.    334 98724 1724 17847845660          92. 4653265316303163048304818       93.846128241243438276677   94.155 4683 4686582654126828654146655  For Exercises 95–104,p= –4,q= 8, andr= –10.95.3234210122012208pr   96.56510645024502426rp    97.22222747 81047 8100167 810016561007210028pqr        98.22242 810162 810161610321042pqr          99.8( 10)218( 4)42qrqp   100.4( 10)1474842prpq  101.55( 10)50232( 4)3( 10)83050252211rpr 

Solution Manual for Trigonometry, 12th Edition - Page 12 preview image

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Section R.2 Real Number Operations and Properties7102.33 8243234210122102424243122012208qpr 103.2222342310228231064310430664302613663prq    104. 22262862444428484484182qpp 105.distributive property106.distributive property107.inverse property108.inverse property109.identity property110.identity property111.commutative property112.commutative property113.associative property114.associative property115.closure property116.closure property117.Using the distributive property,2()22.mpmp118.3()33abab119.Using the distributive property,12()12[()]12( )( 12)()1212 .xyxyxyxy     120.10()10[()]1010pqpqpq   121.(2)1(2)1(2)( 1) ()2dfdfdfdf    122.(3)1(3)1(3)( 1)()3mnmnmnmn    123.Use the second form of the distributiveproperty.53(53)8kkkk124.65(65)11aaaa125.797( 9 )[7( 9)]2rrrrrr   126.464( 6 )[4( 6)]2nnnnnn   127.Use the identity property, then the distributiveproperty.717(17)8aaaaaa128.919Identity property(19)10ssssss129.11(11)2xxxxxx130.11(11)2aaaaaa131.2(32 )22( 3 )2(2 )264xyzxyzxyz132.8(35 )8(3 )88( 5 )24840xyzxyzxyz133.31632408927931633234089827893163325898273245393yzyzyzyz134.1 20832411120832444111208324445+28=528myzmyzmyzmyzmyz   

Solution Manual for Trigonometry, 12th Edition - Page 13 preview image

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8Chapter R Algebra Review135.12432( 12432)3yyyyyy 136.5985( 5985)11rrrrrr 137.654611641156( 6411)11111 or11ppppppppp 138.812359835129( 835)3103xxxxxxxx  139.3(2)5633656335663(35)663215kkkkkkkk 140.5(3)624515624562154(562)11911rrrrrrrrrrr141.10(48)101(48)104842yyyy 142.6(95)61(95)69591yyyy 143.10 (3)()10 (3)3030xyxyx yxy144.8 (6)()8 (6)4848xyxyx yxy145.22(12)(7 )(12) (7 )338(7 )56wzwzwzwz  146.55(18)(5 )(18) (5 )6615(5 )75wzwzwzwz  147.3(4)2(1)312223212214mmmmmmm148.6(5)4(6)630424643024254aaaaaaa149.0.25(84)0.5(62)0.25(8)0.25(4)( 0.5)(6)( 0.5)(2)2323(11)23011pppppppppp   150.0.4(105 )0.8(510 )0.4(10)0.4( 5 )( 0.8)(5)( 0.8)(10 )424810xxxxxxx   Section R.3Exponents, Polynomials,and Factoring1.The polynomial524xxis a trinomial ofdegree 5.2.A polynomial containing exactly one term isa(n) monomial.3.A polynomial containing exactly two terms isa(n) binomial).4.When multiplying powers of like bases, as in23,yykeep the base and add the exponents.5.To raise a power to a power, as in32,amultiply the exponents.6.(a)B;222(5 )1025xyxxyy(b)C;222(5 )1025xyxxyy(c)A;22(5 )(5 )25xyxyxy(d)D;22(5)(5)25yxyxyx7.(a)B;23(23)(469)827xxxx.(b)C;23(23)(469)827xxxx(c)A;23(32 )(964)278xxxx8.D.26123423xxxx9.B:421(1)(1)(1)xxxx. Choice A isnot a complete factorization, because21xcan be factored as(1)(1).xxThe otherchoices are not correct factorizations of41x.

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Section R.3 Exponents, Polynomials, and Factoring910.C:328(2)(24)xxxx. Use thepattern for factoring the sum of cubes,3322()().xyxyxxyyWe have33382xx, so substitute 2 fory.11.5252527444 41616xxx xxx  12.434343736361818yyy yyy  13.6464 111nnnnn14.8585 114aaaaa15.35358999916.282810444417.  425425425113643647272mmmm m mmm  18.  3643 6 4364138258258080tttt t ttt  19.23423423145553531515x yx yx xyyxyx y  20.  3223123 134474 72828xyx yxxy yxyx y   21.222212123311882244mnm nmmnnmnm n22.4242411253223535771010m nmnm mnnmnm n  23.522 51022224.344 31266625. 33332266666216xxxx 26. 555555252522232xxxx 27.3022320223 20 226026266(4)4 () ()4441416m nmnmnm nmmm       28.0433034330 34 330121212(2)2 () ()228 18x yxyxyx yyy     29.38838 3242232 36()()rrrrssss30.24424 28222()ppppqqqq31.42424488244848424( 4) ()( 4)256mmmmtptptptp32.3434331212223665( 5) ()( 5)125()nnnnrrrr 33.0351x yz 34.02331p qr 35.(a)B;061(b)C;061 (c)B;061(d)C;061  36.(a)D;033 13p(b)E;033 13p  (c)B;031p(d)B;031p37.222225474355443751122xxxxxxxxxx     38.3232323232334263231461410410mmmmmmmmmm    

Solution Manual for Trigonometry, 12th Edition - Page 15 preview image

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10Chapter R Algebra Review39. 22222222 12864 3422 122 82 64 3444 224161212168124yyyyyyyyyyyyy40.22222223 855 3243 83 55 3525 424151510209520pppppppppppppp41.423224232243243243263254632546235114162746274mmmmmmmmmmmmmmmmmmmmmmmmmmmm   42.3322332232323283243183243182141331634316344xxxxxxxxxxxxxxxxxxxxx       43.23223222254324325143424541128204xxxxxxxxxxxxxxx44.32323354324322423286bbbbbbbbbbb45.Use the FOIL method.  2241724742171228872282rrrrrrrrrrr    46.Use the FOIL method.  225634535463641520182415224mmmmmmmmmmm    47.Use the FOIL method.  22312732371217621276197xxxxxxxxxxx  48.Use the FOIL method.  225323525332331015691099zzzzzzzzzzz  49.222(23)(23)(2)349mmmm50.2222(83 )(83 )(8 )(3 )649stststst51.2222242(45 )(45 )(4)(5 )1625xyxyxyxy52.3332262(2)(2)(2)4mnmnmnmn53.22222(42 )(4)2(4)(2 )(2 )16164mnmmnnmmnn54.22222(6 )2( )(6 )(6 )1236abaabbaabb55.222222224(53)(5 )2(5 )(3)(3)25309rtrrttrrtt56.424242842(23 )(2)2(2)(3 )(3 )4129zyzzyyzz yy57.222222432322421111111121212122112122422121xxxxxxxxxxxxxxxx xxxxxxxxxxxxxx  Alternatively,2242111111111121xxxxxxxxxxxx

Solution Manual for Trigonometry, 12th Edition - Page 16 preview image

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Section R.3 Exponents, Polynomials, and Factoring1158.22222244444444816816816881616816tttttttttttttttt tttt  432322428168641281612825632256ttttttttttAlternatively,224244444444161632256tttttttttttt59. 32232232222442442886128yyyyyyyyyyyyyy60. 32232232333693693182792727zzzzzzzzzzzzzz61.  3222222322322525252522 25525420252420255 420258405020100125860150125xxxxxxxxxxxxxxxxxxxxxx62. 3222222322324141414142 41141168141681116816432416816448121xxxxxxxxxxxxxxxxxxxxxx  63. 4222243232243222244444441616416168243216qqqqqqqqqqqqqqqqqqq64. 422224323224323336969696365495481125410881rrrrrrrrrrrrrrrrrrr65.The greatest common factor of both terms is8a.40168582852abaabaab66.The greatest common factor of both terms is5y.25155553553xyyyxyyx67.The greatest common factor of all three termsis34.xy3423432323322812364243494239x yx yxyxyx yxyxxyyxyx yxy68.The greatest common factor of all three termsis22.m n52332232223104182522292529m nm nm nm nmm nnm nmnm nmnmn69.The positive factors of 1 (the coefficient of thefirst term) are 1 and 1. The factors of –15 are–1 and 15, 1 and –15, –3 and 5, or 3 and –5.Try different combinations of these factorsuntil the correct one is found.221553xxxx70.The positive factors of 1 (the coefficient of thefirst term) are 1 and 1. The factors of –12 are–1 and 12, 1 and –12, –2 and 6, 2 and –6, –3and 4, or 3 and –4. Try different combinationsof these factors until the correct one is found.21243xxxx

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