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Review Chapter 6 1. a. X4*X5=(x*x*x*x)(x*x*x*x*x) =x*x*x*x*x*x*x*x*x =x9 b. (If=(y*y)(y*y)(y*y)(y*y) =y*y*y*y*y*y*y*y = yB c. (x *y)5 = (x * y)(x * y)(x * y)(x * y)(x *y) =x*y*x*y*x*y*x*y*x*y = (x *x* x* x* x)(y* y* y* y* y) =x5/ 2. a. (2xi)6 =26x6vt =64x6i8 b. 4y*7/ =4*7* y* l =28i+5 =28l 3. Volume = side3 = (2.1 * 104 inches Y = 2.13 *(104)3 inches3 = 2.13 * 1012 inches3 = 9.261 * 1012 inches3 98 Chapter 6: Exponents and Factoring . s 4. a. X4X-7 = X4+(7) =X"3 8. a. 2 xy 4 --x S-2y 1-4 =X3y"3 =( 3)C3) b. X -4X7 =X -4+7 =-X3 l =X3 Y J4x2Y y3 b. (4x2J =43(X2)3 l 64x6 -7 5. distance = rate*time ;; ( C. --,-- x 3-7y 2-lf _( XY xy 3 2J 1.1341103feet/second* 10-6seconds =(X"4/f =1.13*1034110-6 feet '" x -4'4i'4 s 1.13* lOJ+n) feet = X"16 y4 = 1.13* 10 ) fool = .0013 loot x7 6. a. -r;;;x x 7-3 =C6 4 ""x )( 4 ) =L XI6 c. (;J = ;: 7. a. 3. Quotient law b. 5. Power-of-a-power law c. 2. Quotient law and defmition of negative integer exponents d. 1. Productlaw e. 4. Negative exponent in the denominator _27l - 8XI2 @ Houghton Mifflin Company. All rights reserved. -- - .-- 99 Chapter Review 9. 5... 13. The GCF of5, 15, and 10 is 5. The powers of x are ~, ~, xS. The smallest power is~. Zx'y The powers of y are y1,l, 4~1 wx' G 4xJ(5x + 2x4y) = 20x4 + 8x7Y 10. a. 5x(6x- 1) = 30x2- 5x =~z b. 2xz(4x+ 7y-3) + 14xyz- 6xz c. -(x2-8x+2) =-~+8x-2 11. a. The powers of x are Xl and xS. The smallest power is x. The powers of y are l and yl. The smallest power is y. Taking the smallest powers of x and y gives a GCF of x*y = xy. b. The GCF of 18,3, and 9 is 3. The powers of x are ~, x3,x4. The smallest power is~' y and z are not factors of all three expressions, so they don't appear in the GCF. Taking the;smallest powers gives a OCF of 3"'~ = 3x1. c. Using the factor tree we have 60 = 2~*3*5and 105 = 3*5*7. 2 is not a fRctorof IOSand 7 is not Iifactor of 60, so 2 and 7 do not appear in the GCF. The powers oD arc 31 and 31. The smallest power is 3. The powers of5 are 51 and 51. The smallest poweris 5. x andy are not factors of both expressions, so they don't appear in the GCF. Taking the smallest powers gives a GCF of 3.5 = 15. 60 - 5~# The smallest power 3xy3 + ~y) 14. a. -2x(4x- 5) b. -3(3x+ 4y) 15. a. Thetwotermshavea GCFofz. Factoring gives: z(4x - 3) b. We have two terms 4x(y + 7) and -3(y + 7). Common to both terms is the binomial (y + 7). Factoring (y + 7) from both terms gives (y + 7)(4x -3). c. (a) and (b) are basically the same. Let z replace y + 7 in part (b). 16. a. ~ x2 + 5x .. 0 x(x + 5) = 0 x;;;Oorx+5=O x""Oorx"'-S ~-8x=0 ~x-~=O x=Omx-8=0 x=Omx=8 17. a. ... 2 ...~ A 30 ,~ 2 21'.15 (x+4)(x+ 31'.5 ;. isy. z is not a factor of all three expressions, so it does not appear in the GCF. Taking the smallest powers gives a GCF of 5*~*y = 5~y. Factoring the GCF from all three terms, we have 2) = x2 + 6x+ 8 b. 5... 12. The GCFof 8; and 12yis 4y. ... US sx: .. -6x ; "I :~ I~i! 8y2 + 12y = 4y(2y + 3) @ Houghton MifflinCompany. All rights reserved. '"'12.tt .' -15:.- 18 (x-3)(4x2 + 5x- 6) =4xJ -7x2 -21x+ 18 100 Chapter 6: Exponents and Factoring 18. a. By the diagonal test for grouping, any placement of tenns in the area diagram that gives equal diagonal products will work. One arrangement has xy and "18along one diagonal and 2x and "9y along the other diagonal. x -9 19. a. Factorable; the product of the fIrst term and the last term 4x3*"5= "20x3,equals the product of the middle two terms IO~*-2x = -20~. Thereis no needto rearrange these terms because the product of the fIrst and last term equals the product of the middle two terms. 4x3 + 10x2- 2x- 5 =(4x3 + 10x2) + C2x2 2x =2x2(2x+5)+ -1(2x+5) 2x - 9y + xy - 18 = (y + 2)(x - 9) To factor symbolically, we want the product of the fIrst and last terms to equal the product of the middle two terms. Several groupings will accomplish this. 2x-9y+xy-18 =xy-9y+2x-18 = (xy-9y)+ (2x-18) = y(x-9)+ 2(x -9) = (x - 9)(y + 2) or (y + 2)(x - 9) b. :J,r -1)'2 "';'tV 3.l I - 4y + 3z -12yz b. Not factorable; we can't pair terms to make the product of one pair equal the product of the other pair. c. Factorable; the product of the fIrst term and the second term 3x*8y = 24xy, equals the product of the third term and the last term 6xy*4 = 24xy. 3x + 8y + 6xy + 4 Arranging the terms so that the fIrst and last term have the same product as the middle two terms. we have 6xy + 3x + 8y + 4 = (6xy + 3x) + (Sy + 4) = 3x(2y + I) + 4(2y + 1) (2y + 1)(3x + 4) 20. a. (x + y)2 = x2+ 2xy + l + 1 =C 4y + 1)(3z + I) Factorin b. (X+2y)2 =x2 +2*x*2y+(2y)2 =x2 +4xy+4y2 c. (5x-3y)2 =(5x+ -3y)2 = (5X)2+2*5x* -3Y+C3y)2 =-12yz-4y+3z+1 + (3z + 1) = -4y(3z+I)+1(3z+1) =(3z + I)C4y+ 1) or C4y+ = (2x + 5)(2x2 -1) · ) i' g symbolically gives -4y+3z-12yz+l =C12yz - 4y) 5) =25x2 -30xy+9y2 1)(3z+ 1) 21. a. The diagonal product is the product of the fIrst and last term: ~*24 = 24x2. b. Ix and 24x, 2x and 12x, 3x and 8x, 4x and 6x c. 3x and 8x sum to Ilx. d. x2 + llx + 24 = = ~ + 3x + 8x + 24 e. x 8 r xI 3~ 24 + llx+ 24 = (x+ 3)(x+ 8) @ Houghton Mifflin Company. All rights reserved. -- --- -= - Chapter Review 23. a. Diagonal product = ~*5 = 5~ The diagonal product is positive and the middle term is negative, so both factors must be negative. 'I * '5 = 5 'Ix + -5x = '6x No There are no factor pairs of 5~ that add to equal the middle term, '2x, so ~ - 2x + 5 does not factor. This is called a prime polynomial. 22. a. Split the middle term by fmding a factor pair of the diagonal product, -14~, that adds to equal the middle term, -5x. The diagonal product is negative so we need one positive factor and one negative factor. The middle term is negative so the negative factor must have a larger absolute value than the positive factor. 1*-14 = '14 Ix + '14x = '13x No 2x + -7x = -5x Yes 2*-7 = '14 x2-5x-14= ~+2x-7x-14 = x(x + 2) - 7(x + 2) = (x + 2)(x - 7) b. Diagonal product = ~*6 = 1~ The diagonal product is positive and the middle term is positive, so the both factors must be positive. 1 * 12 b. Diagonal product = 5x2*-2 = -1O~. Look for a factor pair of 1O~ that adds to the middle term, 9x. The diagonal product is negative, so we needonepositivefactorandonenegative factor. The middle term is positive so the positive factor must have a larger absolute value than the negative factor. -1*10 = -10 -2*5= '10 5~ + 9x 'Ix + lOx = 9x Yes -2x + 5x = 3x No - 2 = 5x2 - Ix + lOx - 2 "'x(5x-l)+2(5x-l) =(5x - 1)(x + 2) d. Arranging the terms in descending order 7~ + 26x - 8. The diagonal product = 7x2*-8= -56x2. Look for a factor pair of -56~ that adds to 26x. The diagonal product is negative, so one factor is positive and the other is negative. The middle term is positive, so the positive factor must have a larger absolute value than the negative factor. -1*56 = -56 -Ix + 56x = 55x No -2*28 = -56 '2x + 28x = 26x Yes -4*14 = -56 '4x + 14x = lOx No -7*8 = -56 -7x + 8x = Ix No 7x2 + 26x - 8 = 7x2 - . = 12 2x + 28x - 8 =x(7x - 2) + 4(7x - 2) =(7x - 2)(x + 4) @ Houghton MifflinCompany. All rights reserved. Ix + 12x = 13x no 2 * 6 = 12 2x + 6x = 8x no 3 * 4 = 12 3x + 4x = 7x no There are no factor pairs of 1~ that add to equal the middle term, x, so ~ + x + 6 is a prime polynomial. 24. a. Diagonal product = ~*21i = 21x2i Look for factor pairs of 21x2i that add to lOX)'. The diagonal product and middle term are positive so both factors arc:positive. J*21.. 21 lxy + 21xy= 2~' No 3*7'" 21 c. Diagonal product - 3~*' 4 = l2x2. Look for a factor pair of -12x2that adds to 11x. The diagonal product is negative so we need a negative and positive factor. The middle term is negative so the negative factor must have a larger absolute value than the positive factor. 1*-12=-12 lx+-l2x='llx Yes 2*-6='12 2x+-6x=Ax No 3*A = '12 3x + Ax = -Ix No 3x2- llx - 4 = 3~ + Ix - 12x - 4 =x(3x+ 1)-4(3x+ 1) =(3x+ 1)(x-4) 101 . 3xy+ 7,ty= lO.\;yYes ~+ lOxy+2ly ...~ + 3xy + txy + 21l = x(x + 3y) 7y(x + 3y) = (x + 3y)(x + 7y) b. Diagonalproduct= 2x4*-lS'" -30x4 Look for factor pairs of'30x4 that add to "x'-. The diagonal product is negative, so one factor is positive and the other negative. The middle term is negative so the negative factor must have a larger absolute value than the positive factor. 1*-30 = -30 l~ + -30~ = -29x2 No 2*-15 = -30 2x2 + -15~ = -13~ No 3*"10= -30 3x2+ -10~ = -7~ No 5*-6= '30 5~ + -6~ = -lx2 Yes 2x4-~ -15 = 2x4+ 5~ - 6x2- 15 =x2(~ + 5)- 3(~ + 5) = (~ + 5)(~ - 3) 25. a. ~ is the squareofx and25 is the squareof 5. x2- 25 = (x - 5)(x+ 5) b. 9 is the squareof 3 andi is the squareofy. 9 - i = (3 - y)(3 + y) c. 4~ is the squareof 2x and81i is the square of9y. 4~ - 81i = (2x- 9y)(2x+ 9y) 102 Chapter 6: Exponents and Factoring - 36x + 16 =2(9x2 - 18x + 8) =2(9x2 - 6x - 12x + 8) =2(3x(3x - 2) - 4(3x - 2)) =2(3x - 2)(3x - 4) 29. a. 6 - 48z3 =6(1- b. 8Z3) =6(1- 26. a. 182 - 2z)(1 + 2z + 4Z2) = 2(125x3+ i) b. 250x3 + 21 =2(5x + y)(2Sx2 - Sxy+ l) c. 64-/ =(8-l)(8 +i) =(2- y)(4+2y+l)(2+ y)(4-2y+ l) x/- xy + 2/- 2y =y(xy - x + 2y - 2) = y(x(y - 1) + 2(y - 1)) = y(y - l)(x + 2) Chapter 6 Test c. 32y - 300y = 3y(x2 - 100) = 3y(x - 10)(x + 10) 1. (xy3Z-4f=x2(l)2(z-4f 27. a. Attempt to factor the left side of the equation. The diagonal product is 40x2. Look for a factor pair of 40x2 that adds to 13x. Both the diagonal product and the middle term are positive, so the both factors are positive. Of the four factor pairs of 40x2,only 5x and 8x add to 13x. x2 + 13x + 40 x2 + 5x + 8x + 40 =0 x(x + 5) + 8(x + 5) =0 =X2y6z-g x2/ =7 2.~ 6 -3 20x-2 3x3y-3 =0 =10 3x3 = 10y3 (x+5)(x+8)=0 x+5=Oorx+8=0 x=~50rx=~8 b. " x2 -+-2x '" 3 x2+2x-3"'O "Ix and 3x is the factor pair 0f -3x2that adds to 2x. x2 - Ix + 3x - 3 = 0 x(x-1)+3(x-1)=0 (x-1)(x+3)=0 x-1=00rx3=0 x = lor x = -3 3. :" -, l3:)-) - =(3X4~lyHf -3 =(3xJl) -3 = (3x3 * 1) -3 =(3i) =3 -3(x 3 -3 ) c.3x2-19x+6=0 "Ix and "18x is the factor pair of 182 that adds to "19x. 3x2 -Ix -18x + 6 x(3x -1) - 6(3x -1) (3x -l)(x - 6) 3x-1=00rx-6=0 3x = 1 or x = 6 x=- =0 =0 =0 1 orx=6 3 28. a. x3+ 64 = (x + 4)(2 - 4x + 16)Sumof two cubes a = x, b = 4 b. 1 - / = (1 - y)(l + Y + l) Difference of two cubes a = 1, b = y c. 8x3+ 1000/ = (2x+ 10y)(4x2- 20xy + 100l) Sum of two cubes a = 2x, b = 10y 4. signals = 2.0 * 107 signals/second * 8.64 * 104 seconds =2.0*8.64*107 =17.28*107+4 *104 signals signals = 17.28 * 1011signals = 1.728 * 1012 signals 5. a. 3a*(a + 4) = 3a2 + 12a b. -6xy2 * CX2 - 2xy + Sl) =6x3l + 12x21- 30xl d. / + 27i = (y + 31)(y2 - 3yl + 9i) Sumof two cubes a = y, b = 31 @ Houghton Mifflin Company. All rights reserved. Chapter Test 6. -6ab2c3 -3bc2 _12a3b4c3 = -3bc2(2abc+1 + 4a3b3c) 7. 2x2-10x=0 2x(x- 5)= 0 2x=0 or x-5=0 x=O or x=5 8. a. (2x + 5)(x-7) = 2x(x -7) + 5(x - 7) =2x2-14x + 5x-35 =2X2-9x-35 b. (x-4)(3x2 -x+6) =x(3x2 -x+ 6) - 4(3x2 - x + 6) =3x3 - x2 + 6x -12x2 + 4x - 24 =3x' _13x2 + 10x- 24 9. a. 2x+ 4xy + 3... 6y = 2x(1+ 2)') + 3(1+ 7y) . (1+ 2y)(2x + 3) b. 6x' - 9x - 2X2 or3 = 6x3 - 2X2 - 9x + 3 '" 2x2(3x -1) - 3(3x -I) '" (3x - 1)(2x2 - 3) 10. (2x-7y)2 ",(2x+ -7y)2 a (2X)2 +2*2x.C7)')+C7y)2 '" 4x2 - 28xy + 49y2 11. a. 4x2-23x-6=4x2 +1x-24x-6 =x(4x+I)-6(4x+1) = (4x+ b. x2+ 2xy -351 1)(x-6) =x2 -5xy+ =x(x-5y)+ 7xy- 35y2 7y(x- 5)') =(x-5y)(x+7y) 12. 64x2-l =(8x - y)(8x + y) 13. a. 2x2y+6xy2-80l =2y(x2 +3xy-401) =2y(x2- 5xy+ 8xy- 401) =2y(x(x -5y) + 8y(x-5y)) =2y(x- 5y)(x+ 8y) b. 12-3x2 =3(4-X2) =3(2-x)(2+x) 6x2-17 x + 5 =0 14. 6x2 - 2x -15x + 5 = 0 2x(3x -1) - 5(3x -1) =0 (3x-1)(2x-5)=0 3x -1 = 0 or 2x - 5 = 0 3x = 1or 2x = 5 , 1 5 x=- orx=3 2 @ Houghton MifflinCompany. All rights reserved. 103