Finite Mathematics And Applied Calculus, 6th Edition Solution Manual

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Complete Solutions ManualFinite Mathematics &Applied CalculusSIXTH EDITIONStefan WanerHofstra UniversitySteven R. CostenobleHofstra University

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Chapter 0 – Precalculus Review1Table of ContentsChapter 1 – Functions and Applications17Chapter 2 – The Mathematics of Finance121Chapter 3 – Systems of Linear Equations and Matrices172Chapter 4 – Matrix Algebra and Applications266Chapter 5 – Linear Programming367Chapter 6 – Sets and Counting579Chapter 7 – Probability632Chapter 8 – Random Variables and Statistics734Chapter 9 – Nonlinear Functions and Models827Chapter 10 – Introduction to the Derivative883Chapter 11 – Techniques of Differentiation with Applications962Chapter 12 – Further Applications of the Derivative1048Chapter 13 – The Integral1167Chapter 14 – Further Integration Techniques and Applications of the Integral1233Chapter 15 – Functions of Several Variables1337Chapter 16 – Trigonometric Models1406

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Solutions Section 0.1Section 0.11.2(4 + (-1))(2·-4)= 2(3)(-8) = (6)(-8) = -482.3 + ([4 - 2]·9)= 3 + (2×9) = 3 + 18 = 213.20/(3*4)-1= 2012- 1 = 53 - 1 = 234.2-(3*4)/10= 2 - 1210= 2 - 65 = 455.3 + ([3 + (-5)])3 - 2¿2= 3 + (-2)3 - 4=1-1 = -16.12 - (1 - 4)2(5 - 1)·2 - 1 = 12-(-3)16 - 1= 1515 = 17.(2-5*(-1))/1-2*(-1)= 2 - 5·(-1)1- 2·(-1)= 2+51+ 2 = 7 + 2 = 98.2-5*(-1)/(1-2*(-1))= 2 -5·(-1)1 - 2·(-1)= 2 --51 + 2 = 2 + 53 = 1139.2·(-1)2/ 2 = 2¿(-1)22= 2¿12= 22 = 110.2 + 4·32= 2 + 4×9 = 2 + 36 = 3811.2·42+ 1 = 2¿16 + 1 = 32 + 1 = 3312. 1-3·(-2)2¿2 = 1 - 3¿4¿2= 1 - 24 =-2313.3^2+2^2+1= 32+ 22+ 1 = 9 + 4 + 1 =1414.2^(2^2-2)= 2()22-2= 24-2= 22= 415.3 - 2(-3)2-6(4 - 1)2= 3 - 2¿9-6(3)2= 3 - 186×9= -1554=51816.1 - 2(1 - 4)22(5 - 1)2·2=1 - 2(-3)22(4)2·2= 1 - 2¿92¿16¿2= 1-1864= -176417.10*(1+1/10)^3= 10⎝⎛⎠⎞1 +1103= 10(1.1)3= 10×1.331 = 13.3118.121/(1+1/10)^2=121⎝⎛⎠⎞1 +1102= 1211.12= 1211.21= 10019.3⎣⎢⎡⎦⎥⎤- 2·32-(4 - 1)2= 3⎣⎢⎡⎦⎥⎤-2×9-32= 3⎣⎡⎦⎤-18-9= 3×2 = 620.-⎣⎢⎡⎦⎥⎤8(1 - 4)2-9(5 - 1)2= -⎣⎢⎡⎦⎥⎤8(-3)2-9(4)2= -⎣⎡⎦⎤8×9-9×16= -⎝⎛⎠⎞72-144= -⎝⎛⎠⎞-12=1221.3⎣⎡⎦⎤1 -⎝⎛⎠⎞- 1222+ 1 = 3⎣⎡⎦⎤1 - 142+ 1= 3⎣⎡ ⎦⎤342= 3⎣⎡⎦⎤916+ 1 = 2716 + 1 = 431622.3⎣⎡⎦⎤19 -⎝⎛ ⎠⎞2322+ 1 = 3⎣⎡⎦⎤19 - 492+ 1= 3⎣⎡⎦⎤-392+ 1 = 3⎣⎡⎦⎤-132+ 1= 3⎣⎡ ⎦⎤19+ 1 = 39+ 1 = 4323.(1/2)^2-1/2^21

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Solutions Section 0.1=⎣⎡ ⎦⎤122- 122= 14 - 14 = 024.2/(1^2)-(2/1)^2= 212-⎣⎡ ⎦⎤212= 21- 41 = -225.3¿(2-5) =3*(2-5)26.4 + 59 =4+5/9or4+(5/9)27.32-5 =3/(2-5)Note3/2-5is wrong, since it corresponds to32 - 5.28.4-13=(4-1)/329.3-18+6 =(3-1)/(8+6)Note3-1/8-6is wrong, since it corresponds to3 -18- 6.30.3 +32-9=3+3/(2-9)31.3 - 4+78=3-(4+7)/832.4¿2[ ]23=4*2/(2/3)or(4*2)/(2/3)33.23+x - xy2=2/(3+x)-x*y^234.3+3+xxy=3+(3+x)/(x*y)35.3.1x3- 4x-2-60x2-1=3.1x^3-4x^(-2)-60/(x^2-1)36.2.1x-3- x-1+ x2-32=2.1x^(-3)-x^(-1)+(x^2-3)/237.⎣⎡ ⎦⎤235=(2/3)/5Note that we use only (round) parentheses intechnology formulas, and not brackets.38.2⎣⎡ ⎦⎤35=2/(3/5)39.34-5¿6 = 3^(4-5)*6Note that the entire exponent is in parentheses.40.23+57-9=2/(3+5^(7-9))41.3⎣⎡⎦⎤1 +4100-3=3*(1+4/100)^(-3)Note that we use only (round) parentheses intechnology formulas, and not brackets.42.3⎣⎡⎦⎤1 + 4100-3=3((1+4)/100)^(-3)43.32x-1+ 4x- 1 =3^(2*x-1)+4^x-1Note that the entire exponent of 3 is in parentheses.44.2x2- (22x)2= 2^(x^2)-(2^(2*x))^245.22x2-x+1=2^(2x^2-x+1)Note that the entire exponent is in parentheses.46.22x2-x+1 = 2^(2x^2-x)+147.4e-2x2-3e-2x=4*e^(-2*x)/(2-3e^(-2*x))or4(*e^(-2*x))/(2-3e^(-2*x))or(4*e^(-2*x))/(2-3e^(-2*x))48.e2x+ e-2xe2x- e-2x=(e^(2*x)+e^(-2*x))/e^(2*x)-e^(-2*x))49.3[]1 -()-1222+ 1 =3(1-(-1/2)^2)^2+1Note that we use only (round) parentheses intechnology formulas, and not brackets.2

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Solutions Section 0.2Section 0.21.33=272.(-2)3= -83.-(2·3)2= -(22·32) = -(4·9) = -36or-(2·3)2= -(62) = -364.(4·2)2= 42·22= 16·4 = 64or(4·2)2= 82= 645.⎝⎛⎠⎞-232= (-2)232= 496.⎝⎛ ⎠⎞323= 3323= 2787.(-2)-3=1(-2)3=1-8 = - 188.-2-3= - 123= - 189.⎝⎛ ⎠⎞14-2=1(1/4)2=11/42=11/16 =1610.⎝⎛⎠⎞-23-2=1(-2/3)2=14/9 =9/411.2·30= 2·1 = 212.3·(-2)0= 3·1 = 313.2322= 23+2= 25=32 or2322= 8·4 = 3214.323 = 32+1= 33=27 or323 = 9·3 = 2715.222-1242-4= 22-1+4-4= 21=216.525-3525-2= 52-3+2-2= 5-1= 1517.x3x2= x3+2=x518.x4x-1= x4-1= x319.-x2x-3y = -x2-3y = -x-1y =- yx20.-xy-1x-1= -x1-1y-1= -x0y-1= - 1y21.x3x4= x3-4= x-1= 1x22.y5y3= y5-3=y223.x2y2x-1y= x2-(-1)y2-1= x3y24.x-1yx2y2= x-1-2y1-2= x-3y-1=1x3y25.(xy-1z3)2x2yz2= x2(y-1)2(z3)2x2yz2= x2-2y-2-1z6-2=y-3z4= z4y326.x2yz2(xyz-1)-1=x2yz2x-1y-1z = x2-(-1)y1-(-1)z2-1= x3y2z27.⎝⎛⎠⎞xy-2zx-1z3= (xy-2z)3(x-1z)3= x3y-6z3x-3z3= x3-(-3)y-6z3-3=x6y-6=x6y628.⎝⎛⎠⎞x2y-1z0xyz2= x4y-2x2y2z2= x4-2y-2-2z-2= x2y-4z-2=x2y4z229.⎝⎛⎠⎞x-1y-2z2xy-2= (x-1-1y-2-1z2)-2= (x-2y-3z2)-2=x4y6z-4= x4y6z430.⎝⎛⎠⎞xy-2x2y-1z-3= (x1-2y-2+1z-1)-3= (x-1y-1z-1)-3=x3y3z331.3x-4=3x43

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Solutions Section 0.232.12x-4=12x433.34 x-2/3=34x2/334.45 y-3/4=45y3/435.1 - 0.3x-2- 65x-1=1- 0.3x2- 65x36.13x-4+ 0.1x-23=x43+ 0.13x237.4=238.5‡2.23639.14=14=1240.19=19=1341.169=169=4342.94=94=3243.45= 2544.625= 6545.9+16= 3 + 4 = 746.25-16= 5 - 4 = 147.9 + 16=25=548.25 – 16=9= 349.38 – 27=3–19‡-2.66850.481 – 16=465‡2.83951.3278=32738= 3252.38 · 64=38 ·364= 2·4 =853.(–2)2=4=254.(–1)2=1= 155.14(1 + 15)=164=164= 42= 256.19(3 + 33)=369=369= 63 = 257.a2b2=ab = ab58.a2b2=a2b2= ab59.(x+ 9)2= x + 9(x + 9 > 0becausexispositive)60.(x+ 9)2= x + 961.3x3(a3+b3)=3x33a3+b3= x3a3+b3(Notice:Notx(a + b).)62.4x4a4b4=4x44a44b4=xab63.4xy3x2y=4y2x=4y2x= 2yx4

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Solutions Section 0.264.4(x2+ y2)c2=4x2+ y2c2= 2x2+ y2c(Notice:Not2(x + y)/c.)65.3=31/266.8= 81/267.x3= x3/268.3x2= x2/369.3xy2= (xy2)1/370.x2y = (x2y)1/271.x2x= x2x1/2= x2-1/2= x3/272.xx=xx1/2= x1-1/2= x1/273.35x2=35x-274.25x-3=25x375.3x-1.22-13x2.1=32x-1.2- 13x-2.176.23x-1.2- x2.13= 23x1.2- 13x2.177.2x3- x0.12+43x1.1= 23x - 12x0.1+ 43x-1.178.4x23+ x3/26-23x2= 43x2+ 16x3/2- 23x-279.3x4-53x+43xx= 3x1/24-53x1/2+43xx1/2=34x1/2- 53x-1/2+ 43x-3/280.35x- 5x8+723x=35x1/2- 5x1/28+72x1/3= 35x-1/2- 58x1/2+ 72x-1/381.35x24-72x3= 3x2/54-72x3/2= 34x2/5- 72x-3/282.18xx-235x3=18x3/2-23x3/5= 18x-3/2-23x-3/583.1(x2+ 1)3-343(x2+ 1)=1(x2+ 1)3-34(x2+ 1)1/3=(x2+ 1)-3- 34(x2+ 1)-1/384.23(x2+ 1)-3- 33(x2+ 1)74=23(x2+ 1)3- 34(x2+ 1)7/385.22/3=32286.34/5=53487.x4/3=3x488.y7/4=4y789.(x1/2y1/3)1/5=5x3y90.x-1/3y3/2= y3/2x1/3=y33x5

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Solutions Section 0.291.- 32x-1/4= -32x1/4= -324x92.45 x3/2= 4x3593.0.2x-2/3+37x-1/2= 0.2x2/3+ 3x1/27= 0.23x2+ 3x794.3.1x-4/3- 117x-1/7= 3.1x4/3-117x1/7=3.13x4-1177x95.34(1 - x)5/2=34(1 –x)596.94(1 - x)-7/3= 9(1 - x)7/34=93(1 –x)7497.4-1/247/2= 4-1/2+7/2= 43= 6498.21/a22/a= 21/a-2/a= 2-1/a=121/a99.32/33-1/6= 32/3-1/6= 31/2=3100.21/32-122/32-1/3= 21/3-1+2/3-1/3= 2-1/3=121/3101.x3/2x5/2= x3/2-5/2= x-1= 1x102.y5/4y3/4= y5/4-3/4= y1/2=y103.x1/2y2x-1/2y= x1/2+1/2y2-1= xy104.x-1/2yx2y3/2= x-1/2-2y1-3/2= x-5/2y-1/2105.⎝⎛ ⎠⎞xy1/3⎝⎛ ⎠⎞yx2/3=⎝⎛ ⎠⎞yx-1/3⎝⎛ ⎠⎞yx2/3=⎝⎛ ⎠⎞yx1/3106.⎝⎛ ⎠⎞xy-1/3⎝⎛ ⎠⎞yx1/3=⎝⎛ ⎠⎞yx1/3⎝⎛ ⎠⎞yx1/3=⎝⎛ ⎠⎞yx2/3107.x2- 16 = 0, x2= 16, x = ±16= ±4108.x2- 1 = 0, x2= 1, x = ±1= ±1109.x2- 49= 0, x2= 49 , x = ±49= ± 23110.x2- 110 = 0, x2=110 , x = ±110 =±110‡ ±0.3162111.x2- (1 + 2x)2= 0, x2= (1 + 2x)2, x =±(1 + 2x);ifx = 1 + 2xthen-x = 1, x = -1; ifx =-(1 + 2x)then3x = -1, x = -1/3. So, x = -1or-1/3.112.x2- (2 - 3x)2= 0, x2= (2 - 3x)2, x =±(2 - 3x); ifx = 2 - 3xthen4x = 2, x = 1/2; ifx =-(2 - 3x)then-2x = -2, x = 1. So,x = 1or1/2.113.x5+ 32 = 0, x5= -32, x =5–32= -2114.x4- 81 = 0, x4= 81, x = ±481= ±3115.x1/2- 4 = 0, x1/2= 4, x = 42=16116.x1/3- 2 = 0, x1/3= 2, x = 23=8117.1 - 1x2= 0, 1 = 1x2, x2= 1, x = ±1=±1118.2x3- 6x4= 0, 2x3= 6x4, 2x4= 6x3, 2x = 6,x =3119.(x - 4)-1/3= 2, x - 4 = 2-3= 18 ,x = 4 + 18 = 338120.(x - 4)2/3+ 1 = 5, (x - 4)2/3= 4, x - 4 = ±43/2= ±8, x = 4 ± 8 = -4or126

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Solutions Section 0.3Section 0.31.x(4x + 6) = 4x2+ 6x2.(4y - 2)y = 4y2- 2y3.(2x - y)y = 2xy - y24.x(3x + y) = 3x2+ xy5.(x + 1)(x - 3) = x2+ x - 3x - 3 =x2- 2x - 36.(y + 3)(y + 4) = y2+ 3y + 4y + 12 =y2+ 7y + 127.(2y + 3)(y + 5) = 2y2+ 3y + 10y + 15 =2y2+ 13y + 158.(2x - 2)(3x - 4) = 6x2- 6x - 8x + 8 =6x2-14x + 89.(2x - 3)2= 4x2- 12x + 910.(3x + 1)2= 9x2+ 6x + 111.⎝⎛⎠⎞x + 1x2= x2+ 2 + 1x212.⎝⎛⎠⎞y - 1y2= y2- 2 + 1y213.(2x - 3)(2x + 3) = (2x)2- 32= 4x2- 914.(4 + 2x)(4 - 2x) = 42- (2x)2= 16 - 4x215.⎝⎛⎠⎞y - 1y⎝⎛⎠⎞y + 1y= y2-⎝⎛ ⎠⎞1y2= y2- 1y216.(x - x2)(x + x2) = x2- (x2)2= x2- x417.(x2+ x - 1)(2x + 4) = (x2+ x - 1)2x +(x2+ x - 1)4 = 2x3+ 2x2- 2x + 4x2+ 4x -4 = 2x3+ 6x2+ 2x - 418.(3x + 1)(2x2- x + 1) = 3x(2x2- x + 1) +1(2x2- x + 1) = 6x3- 3x2+ 3x + 2x2- x +1 = 6x3- x2+ 2x + 119.(x2- 2x + 1)2= (x2- 2x + 1)(x2- 2x + 1) =x2(x2- 2x + 1) - 2x(x2- 2x + 1) + (x2-2x + 1) = x4- 2x3+ x2- 2x3+ 4x2- 2x +x2- 2x + 1 = x4- 4x3+ 6x2- 4x + 120.(x + y - xy)2= (x + y - xy)(x + y - xy) = x(x+ y - xy) + y(x + y - xy) - xy(x + y - xy) = x2+xy - x2y + xy + y2- xy2- x2y - xy2+ x2y2= x2+y2- 2x2y - 2xy2+ 2xy + x2y221.(y3+ 2y2+ y)(y2+ 2y - 1) = y3(y2+ 2y - 1) +2y2(y2+ 2y - 1) + y(y2+ 2y - 1) = y5+ 2y4- y3+2y4+ 4y3- 2y2+ y3+ 2y2- y =y5+ 4y4+ 4y3- y22.(x3- 2x2+ 4)(3x2- x + 2) = x3(3x2- x + 2) -2x2(3x2- x + 2) + 4(3x2- x + 2) = 3x5- x4+ 2x3- 6x4+ 2x3- 4x2+ 12x2- 4x + 8 = 3x5- 7x4+4x3+ 8x2- 4x + 823.(x + 1)(x + 2) + (x + 1)(x + 3) =(x + 1)(x + 2 + x + 3) =(x + 1)(2x + 5)24.(x + 1)(x + 2)2+ (x + 1)2(x + 2) =(x + 1)(x + 2)(x + 2 + x + 1) =(x + 1)(x + 2)(2x + 3)25.(x2+ 1)5(x + 3)4+ (x2+ 1)6(x + 3)3=(x2+ 1)5(x + 3)3(x + 3 + x2+ 1) =(x2+ 1)5(x + 3)3(x2+ x + 4)26.10x(x2+ 1)4(x3+ 1)5+ 15x2(x2+ 1)5·(x3+ 1)4= 5x(x2+ 1)4(x3+ 1)4[2(x3+ 1) + 3x(x2+1)] = 5x(x2+ 1)4(x3+ 1)4(5x3+ 3x + 2)27.(x3+ 1)x+ 1- (x3+ 1)2x+ 1=(x3+ 1)x+ 1[1 - (x3+ 1)] = -x3(x3+ 1)x+ 128.(x2+ 1)x+ 1-(x+ 1)3=x+ 1·[x2+ 1 -(x+ 1)2] =x+ 1[x2+ 1 -(x + 1)] = (x2- x)x+ 1= x(x - 1)x+ 129.(x+ 1)3+(x+ 1)5=(x+ 1)3·[1 +(x+ 1)2] =(x+ 1)3(1 + x + 1) =7

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Solutions Section 0.3(x + 2)(x+ 1)330.(x2+ 1)3(x+ 1)4-3(x+ 1)7=3(x+ 1)4[x2+ 1 -3(x+ 1)3] =3(x+ 1)4[x2+ 1 - (x + 1)] =(x2- x)3(x+ 1)4= x(x - 1)3(x+ 1)431.(a)2x + 3x2= x(2 + 3x)(b)x(2 + 3x) = 0; x =0or2 + 3x = 0;x = 0or-2/332. (a)y2- 4y = y(y - 4)(b)y(y - 4) = 0; y = 0ory - 4 = 0;y = 0or433.(a)6x3- 2x2=2x2(3x - 1)(b)2x2(3x - 1) = 0; x = 0or3x - 1 = 0;x = 0or1/334.(a)3y3- 9y2= 3y2(y - 3)(b)3y2(y - 3) = 0; y= 0ory - 3 = 0;y = 0or335.(a)x2- 8x + 7 = (x - 1)(x - 7)(b)(x - 1)(x - 7) = 0; x - 1 = 0orx - 7 = 0;x =1or736.(a)y2+ 6y + 8 = (y + 2)(y + 4)(b)(y + 2)(y + 4) = 0; y + 2 = 0ory + 4 = 0;y =-2or-437.(a)x2+ x - 12 = (x - 3)(x + 4)(b)(x - 3)(x + 4) = 0; x - 3 = 0orx + 4 = 0;x =3or-438.(a)y2+ y - 6 =(y - 2)(y + 3)(b)(y - 2)(y + 3) = 0; y - 2 = 0ory + 3 = 0;y =2or-339.(a)2x2- 3x - 2 = (2x + 1)(x - 2)(b)(2x + 1)(x - 2) = 0; 2x + 1 = 0orx - 2 = 0;x= -1/2 or240.(a)3y2- 8y - 3 = (3y + 1)(y - 3)(b)(3y + 1)(y - 3) = 0; 3y + 1 = 0ory - 3 = 0;y= -1/3or341.(a)6x2+ 13x + 6 = (2x + 3)(3x + 2)(b)(2x + 3)(3x + 2) = 0; 2x + 3 = 0or3x + 2 = 0;x = -3/2or-2/342.(a)6y2+ 17y + 12 = (3y + 4)(2y + 3)(b)(3y + 4)(2y + 3) = 0; 3y + 4 = 0or2y + 3 = 0;y = -4/3or-3/243. (a)12x2+ x - 6 = (3x - 2)(4x + 3)(b)(3x - 2)(4x + 3) = 0; 3x - 2 = 0or4x + 3 = 0;x = 2/3or-3/444. (a)20y2+ 7y - 3 = (4y - 1)(5y + 3)(b)(4y - 1)(5y + 3) = 0; 4y - 1 = 0or5y + 3 = 0;y =1/4 or-3/545.(a)x2+ 4xy + 4y2= (x + 2y)2(b)(x + 2y)2= 0; x + 2y = 0;x = -2y46.(a)4y2- 4xy + x2= (2y - x)2(b)(2y - x)2= 0; 2y - x = 0;y = x/247. (a)x4- 5x2+ 4 =(x2- 1)(x2- 4) =(x - 1)(x + 1)(x - 2)(x + 2)(b)(x - 1)(x + 1)(x - 2)(x + 2) = 0; x - 1 = 0orx+ 1 = 0orx - 2 = 0orx + 2 = 0;x = ±1or±248.(a)y4+ 2y2- 3 = (y2- 1)(y2+ 3) =(y - 1)(y + 1)(y2+ 3)(b)(y - 1)(y + 1)(y2+ 3) = 0; y - 1 = 0ory + 1 = 0ory2+ 3 = 0;y = ±1(Notice thaty2+ 3 = 0has no real solutions.)8

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Solutions Section 0.4Section 0.41.x - 4x + 1·2x + 1x - 1= (x - 4)(2x + 1)(x + 1)(x - 1)= 2x2- 7x - 4x2- 12.2x - 3x - 2·x + 3x + 1= (2x - 3)(x + 3)(x - 2)(x + 1)=2x2+ 3x - 9x2- x - 23.x - 4x + 1 + 2x + 1x - 1=(x - 4)(x - 1) + (x + 1)(2x + 1)(x + 1)(x - 1)=3x2- 2x + 5x2- 14.2x - 3x - 2+ x + 3x + 1 =(2x - 3)(x + 1) + (x - 2)(x + 3)(x - 2)(x + 1)=3x2- 9x2- x - 25.x2x + 1 - x - 1x + 1 = x2- (x - 1)x + 1= x2- x + 1x + 16.x2- 1x - 2 -1x - 1 = (x2- 1)(x - 1) - (x - 2)(x - 2)(x - 1)=x3- x2- 2x + 3x2- 3x + 27.1⎝⎛⎠⎞xx - 1+ x - 1 = x - 1x+ x - 1 =x - 1 + x(x - 1)x= x2- 1x8.2⎝⎛⎠⎞x - 2x2-1x - 2 =2x2x - 2 -1x - 2= 2x2- 1x - 29.1x⎝⎛⎠⎞x - 3xy+ 1y= 1x⎝⎛⎠⎞x - 3 + xxy= 2x - 3x2y10.y2x⎝⎛⎠⎞2x - 3y+ xy= y2x⎝⎛⎠⎞2x - 3 + xy= y2(3x - 3)xy= y(3x - 3)x= 3xy - 3yx11.(x + 1)2(x + 2)3- (x + 1)3(x + 2)2(x + 2)6=(x + 1)2(x + 2)2[(x + 2) - (x + 1)](x + 2)6= (x + 1)2(x + 2)412.6x(x2+ 1)2(x3+ 2)3- 9x2(x2+ 1)3(x3+ 2)2(x3+ 2)6=3x(x2+ 1)2(x3+ 2)2[2(x3+ 2) - 3x(x2+ 1)](x3+ 2)6=3x(x2+ 1)2(-x3- 3x + 4)(x3+ 2)4=-3x(x2+ 1)2(x3+ 3x - 4)(x3+ 2)413.(x2- 1)x2+ 1-x4x2+ 1x2+ 1= (x2- 1)(x2+ 1) - x4(x2+ 1)x2+ 1=-1(x2+ 1)314.xx3– 1-3x4x3– 1x3- 1= x(x3- 1) - 3x4(x3- 1)x3– 1= -2x4- x(x3– 1)3= -x(2x3+ 1)(x3– 1)315.1(x + y)2- 1x2y= x2- (x + y)2yx2(x + y)2= x2- x2- 2xy - y2yx2(x + y)2= -y(2x + y)yx2(x + y)2= -(2x + y)x2(x + y)216.1(x + y)3- 1x3y= x3- (x + y)3yx3(x + y)3=x3- x3- 3x2y - 3xy2- y3yx3(x + y)3= -y(3x2+ 3xy + y2)yx3(x + y)3=-(3x2+ 3xy + y2)x3(x + y)39

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Solutions Section 0.5Section 0.51.x + 1 = 0, x = -12.x - 3 = 1, x = 43.-x + 5 = 0, x =54.2x + 4 = 1, 2x = -3, x = -3/25.4x - 5 = 8, 4x = 13, x = 13/46.34x + 1 = 0, 34 x = -1, x =-4/37.7x + 55 = 98, 7x = 43, x = 43/78.3x + 1 = x, 2x = -1, x = -1/29.x + 1 = 2x + 2, -x = 1, x = -110.x + 1 = 3x + 1, -2x = 0, x = 011.ax + b = c, ax = c - b, x = (c - b)/a12.x - 1 = cx + d, (1 - c)x = d + 1, x = d + 11 - c13.2x2+ 7x - 4 = 0, (2x - 1)(x + 4) = 0, x = -4,1214.x2+ x + 1 = 0,∆= -3 < 0, so this equation hasno real solutions15.x2- x + 1 = 0,∆= -3 < 0, so this equation hasno real solutions16.2x2- 4x + 3 = 0,∆= -8 < 0, so this equationhas no real solutions17.2x2- 5 = 0, x2= 52 , x = ±5218.3x2- 1 = 0, x2= 13 , x = ±1319.-x2- 2x - 1 = 0, -(x + 1)2= 0, x = -120.2x2- x - 3 = 0, (2x - 3)(x + 1) = 0, x = 3/2, -121.12x2- x -32= 0, x2- 2x - 3 = 0,(x + 1)(x - 3) = 0, x = -1, 322.-12x2-12x + 1 = 0, x2+ x - 2 = 0,(x + 2)(x - 1) = 0, x = -2, 123.x2- x = 1, x2- x - 1 = 0, x = 1 ±52by thequadratic formula24.16x2= -24x - 9, 16x2+ 24x + 9 = 0,(4x + 3)2= 0, x = - 3/425.x = 2 - 1x , x2= 2x - 1, x2- 2x + 1 = 0, (x -1)2= 0, x = 126.x + 4 =1x - 2 , (x + 4)(x - 2) = 1, x2+2x - 8 = 1, x2+ 2x - 9 = 0, x = -2 ±402= -1±10 by the quadratic formula27.x4- 10x2+ 9 = 0, (x2- 1)(x2- 9) = 0,x2= 1orx2= 0, x = ±1, ±328.x4- 2x2+ 1 = 0, (x2- 1)2= 0, x = ±129.x4+ x2- 1 = 0, x2= -1 ±52by the quadraticformula,x = ±-1 ±5230.x3+ 2x2+ x = 0, x(x2+ 2x + 1) = 0,x(x + 1)2= 0, x = 0, -131.x3+ 6x2+ 11x + 6 = 0,(x + 1)(x + 2)(x + 3) = 0, x = -1, -2, -332.x3- 6x2+ 12x - 8 = 0, (x - 2)3= 0, x = 233.x3+ 4x2+ 4x + 3 = 0,10

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Solutions Section 0.5(x + 3)(x2+ x + 1) = 0, x = -3(Forx2+ x +1 = 0,∆= -3 < 0, so there are no real solutions tothis quadratic equation.)34.y3+ 64 = 0, y3= -64, y =3–64= -435.x3- 1 = 0, x3= 1, x =31= 136.x3- 27 = 0, x3= 27, x =327= 337.y3+ 3y2+ 3y + 2 = 0,(y + 2)(y2+ y + 1) = 0, y = -2(Fory2+ y +1 = 0,∆= -3 < 0, so there are no real solutions tothis quadratic equation.)38.y3- 2y2- 2y - 3 = 0,(y - 3)(y2+ y + 1) = 0, y = 3(Fory2+ y +1 = 0,∆= -3 < 0, so there are no real solutions tothis quadratic equation.)39.x3- x2- 5x + 5 = 0, (x - 1)(x2- 5) = 0,x = 1, ±540.x3- x2- 3x + 3 = 0, (x - 1)(x2- 3) = 0,x = 1, ±341.2x6- x4- 2x2+ 1 = 0, (2x2- 1)(x4- 1) = 0,[or(2x2- 1)(x2- 1)(x2+ 1) = 0; in any case, thinkof the cubic you get by substitutingyforx2], x = ±1, ±1242.3x6- x4- 12x2+ 4 = 0,(3x2- 1)(x4- 4) = 0, [or(3x2- 1)(x2- 2)(x2+ 2)= 0], x =±2 , ±1343.(x2+ 3x + 2)(x2- 5x + 6) = 0,(x + 2)(x + 1)(x - 2)(x - 3) = 0, x = -2, -1, 2, 344.(x2- 4x + 4)2(x2+ 6x + 5)3= 0,(x - 2)4(x + 1)3(x + 5)3= 0, x = -5, -1, 211

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Solutions Section 0.6Section 0.61.x4- 3x3= 0, x3(x - 3) = 0, x = 0, 32.x6- 9x4= 0, x4(x2- 9) = 0, x = 0, ±33.x4- 4x2= -4, x4- 4x2+ 4 = 0, (x2- 2)2= 0, x= ±24.x4- x2= 6, x4- x2- 6 = 0,(x2- 3)(x2+ 2) = 0, x = ±35.(x + 1)(x + 2) + (x + 1)(x + 3) = 0,(x + 1)(x + 2 + x + 3) = 0, (x + 1)(2x + 5) = 0, x=-1, -5/26.(x + 1)(x + 2)2+ (x + 1)2(x + 2) = 0,(x + 1)(x + 2)(x + 2 + x + 1) = 0,(x + 1)(x + 2)(2x + 3) = 0, x = -1, -2, -3/27.(x2+ 1)5(x + 3)4+ (x2+ 1)6(x + 3)3= 0,(x2+ 1)5(x + 3)3(x + 3 + x2+ 1) = 0,(x2+ 1)5(x + 3)3(x2+ x + 4) = 0, x = -3(Neitherx2+ 1 = 0norx2+ x + 4 = 0has a real solution.)8.10x(x2+ 1)4(x3+ 1)5- 10x2(x2+ 1)5(x3+ 1)4=0, 10x(x2+ 1)4(x3+ 1)4[x3+ 1 - x(x2+ 1)] = 0,10x(x2+ 1)4(x3+ 1)4(1 - x) = 0, x = -1, 0, 19.(x3+ 1)x+ 1- (x3+ 1)2x+ 1= 0,(x3+ 1)x+ 1[1 - (x3+ 1)] = 0,-x3(x3+ 1)x+ 1= 0, x = 0, -110.(x2+ 1)x+ 1-(x+ 1)3= 0,x+ 1[x2+ 1 - (x + 1)] = 0, (x2- x)x+ 1= 0,x(x - 1)x+ 1= 0, x = -1, 0, 111.(x+ 1)3+(x+ 1)5= 0,(x+ 1)3(1 +x + 1) = 0, (x + 2)(x+ 1)3= 0, x = -1 (x = -2isnot a solution because(x+ 1)3is not defined forx =-2.)12.(x2+ 1)3(x+ 1)4-3(x+ 1)7= 0,3(x+ 1)4[x2+ 1 - (x + 1)] = 0,(x2- x)3(x+ 1)4= 0, x(x - 1)3(x+ 1)4= 0,x = -1, 0, 113.(x + 1)2(2x + 3) - (x + 1)(2x + 3)2= 0,(x + 1)(2x + 3)(x + 1 - 2x - 3) = 0,(x + 1)(2x + 3)(-x - 2) = 0, x = -2, -3/2, -114.(x2- 1)2(x + 2)3- (x2- 1)3(x + 2)2= 0,(x2- 1)2(x + 2)2(x + 2 - x2+ 1) = 0,-(x2- 1)2(x + 2)2(x2- x - 3) = 0, x = -2, -1, 1,(1 ±13)/215.(x + 1)2(x + 2)3- (x + 1)3(x + 2)2(x + 2)6= 0,(x + 1)2(x + 2)2[(x + 2) - (x + 1)](x + 2)6= 0, (x + 1)2(x + 2)4=0, (x + 1)2= 0, x = -116.6x(x2+ 1)2(x2+ 2)4- 8x(x2+ 1)3(x2+ 2)3(x2+ 2)8= 0,2x(x2+ 1)2(x2+ 2)3[3(x2+ 2) - 4(x2+ 1)](x2+ 2)8= 0,-2x(x2+ 1)2(x2- 2)(x2+ 2)5= 0,-2x(x2+ 1)2(x2- 2) = 0, x =0, ±217.2(x2- 1)x2+ 1-x4x2+ 1x2+ 1= 0,2(x2- 1)(x2+ 1) - x4(x2+ 1)x2+ 1= 0,x4- 2(x2+ 1)x2+ 1= 0, x4- 2 = 0, x = ±4218.4xx3– 1-3x4x3– 1x3- 1= 0,4x(x3- 1) - 3x4(x3- 1)x3– 1= 0,x4- 4x(x3- 1)x3- 1 = 0,x4- 4x = 0, x(x3- 4) = 0, x = 0, ±3419.x - 1x = 0, x2- 1 = 0, x = ±112

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Solutions Section 0.620.1 - 4x2= 0, x2- 4 = 0, x = ±221.1x - 9x3= 0, x2- 9 = 0, x = ±322.1x2-1x + 1 = 0, x + 1 - x2= 0,x2- x - 1 = 0, x = (1 ±5)/223.x - 4x + 1-xx - 1 = 0, (x - 4)(x - 1) - x(x + 1)(x + 1)(x - 1)=0,-6x + 4(x + 1)(x - 1)= 0, -6x + 4 = 0, x = 2/324.2x - 3x - 1- 2x + 3x + 1= 0,(2x - 3)(x + 1) - (2x + 3)(x - 1)(x - 1)(x + 1)= 0,-2x(x - 1)(x + 1) = 0, -2x = 0, x = 025.x + 4x + 1 + x + 43x= 0, 3x(x + 4) + (x + 1)(x + 4)3x(x + 1)=0, (x + 4)(3x + x + 1)3x(x + 1)= 0, (x + 4)(4x + 1)3x(x + 1)= 0, (x+ 4)(4x + 1) = 0, x =-4, -1/426.2x - 3x- 2x - 3x + 1= 0,(2x - 3)(x + 1) - x(2x - 3)x(x + 1)= 0,(2x - 3)(x + 1 - x)x(x + 1)= 0,2x - 3x(x + 1) = 0,2x - 3 = 0, x = 3/213

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