( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.

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1 Chapter 6 Exercise Set A prime number is an integer greater than 1 that has exactly two factors, itself and To factor an expression means to write the expression as the product of factors. 5. The greatest common factor of two or more numbers is the greatest number that divides into all the numbers. 7. A factoring problem may be checked by multiplying the factors = 8 7 = = = = 9 10 = = = 4 49 = = = 2 2 5, 24 = 2 3 3, so the greatest common factor is 2 2 or = 2 5 7, 98 = 2 7 2, so the greatest common factor is 2 7 or = 2 4 5, 126 = so the greatest common factor is The greatest common factor is x. 23. The greatest common factor is 3x. 25. The greatest common factor is The greatest common factor is mn. 29. The greatest common factor is x 3 y The greatest common factor is The greatest common factor is x 2 y The greatest common factor is x. 37. The greatest common factor is x The greatest common factor is 2x The greatest common factor is 3w The greatest common factor is x The greatest common factor is x The greatest common factor is x The greatest common factor is 4. 4x - 8 = 4 x = 4( x - 2) 51. The greatest common factor is 5. 15x - 5 = 5 3x = 5( 3x -1) 53. The greatest common factor is 6. 6p + 12 = 6 p = 6 p The greatest common factor is 3x. 9x 2-12x = 3x 3x - 3x 4 = 3x( 3x - 4) 57. The greatest common factor is 2p. 26p 2-8p = 2p 13p - 2p 4 = 2p 13p The greatest common factor is 3x 2. 3x 5-12x 2 = 3x 2 x 3-3x 2 4 = 3x 2 x The greatest common factor is 12x 8. 36x x 8 = 12x 8 3x x 8 2 = 12x 8 3x The greatest common factor is 9y 3. 27y 15-9y 3 = 9y 3 3y 12-9y 3 1 = 9y 3 3y The greatest common factor is x. x + 3xy 2 = x 1+ x 3y 2 = x 1+ 3y 2 154

2 SSM: Elementary and Intermediate Algebra Chapter 6: Factoring 67. The greatest common factor is a 2. 7a 4 + 3a 2 = a 2 7a 2 + a 2 3 = a 2 7a The greatest common factor is 4xy. 16xy 2 z + 4x 3 y = 4xy 4yz + 4xy x 2 = 4xy 4yz + x The greatest common factor is 16mn 2. 48m 4 n 2-16mn 2 = 16mn 2 3m 3-16mn 2 1 = 16mn 2 3m The greatest common factor is 25x 2 yz. 25x 2 yz x 3 yz = 25x 2 yz z x 2 yz x = 25x 2 yz z 2 + x 75. The greatest common factor is y 2 z 3. 13y 5 z 3-11xy 2 z 5 = y 2 z 3 13 y 3 - y 2 z 3 11xz 2 = y 2 z 3 13y 3-11xz The greatest common factor is 4. 8c 2-4c - 32 = 4 2c 2-4 c = 4 2c 2 - c The greatest common factor is xy. 8x 2 y + 12xy 2 + 5xy = xy 8 x + xy 12 y + xy 5 = xy 8x + 12y The greatest common factor is x + 4. x( x + 4) + 3( x + 4) = ( x + 4) ( x + 3) 93. The greatest common factor is a b(a - 2) - 4(a - 2) = (a - 2)(3b - 4) 95. The greatest common factor is 2x x( 2x + 1) + 1( 2x + 1) = ( 2x + 1) ( 4 x + 1) 97. The greatest common factor is 2x x( 2x + 1) + 2x + 1 = 5x( 2x + 1) + 1( 2x + 1) = ( 2x + 1) ( 5x + 1) 99. The greatest common factor is 2z z(2z + 3) - 3(2z + 3) = (2z + 3)(4z - 3) = = 3 ( + 2 ) D 3-7D D = 7D 5D 2-7D D + 7D 2 = 7D 5D 2 - D The greatest common factor is 3. 9x x + 3 = 3 3x x = 3 3x 2 + 6x The greatest common factor is 4x. 4x 3-8x x = 4x x 2-4x 2x + 4x 3 = 4x x 2-2x The greatest common factor is 5. 35x 2-15y + 10 = 5 7x 2-5 3y = 5 7x 2-3y The greatest common factor is 3. 15p 2-6 p + 9 = 3 5p p = 3 5p 2-2p The greatest common factor is 3a. 9a 4-6a 3 + 3ab = 3a 3a 3-3a 2a 2 + 3a b = 3a 3a 3-2a 2 + b 155

3 Chapter 6: Factoring SSM: Elementary and Intermediate Algebra 105. The greatest common factor is 2( x - 3). 4x 2 ( x - 3) 3-6x( x - 3) 2 + 4( x - 3) = 2( x - 3) 2x 2 ( x - 3) 2-2( x - 3) 3x( x - 3) - 2( x - 3) 2 = 2( x - 3) 2x 2 ( x - 3) 2-3x( x - 3) + 2 [ ] 107. First factor x 1/3 from terms. x 7/3 + 5x 4 /3 + 2x 1/3 = x 1/3 x 2 + 5x + 2 3x 2 y 3 2 Ê ˆ 115. Á Ë 2x 5 y 2 = 3y 2 Ê ˆ Á Ë 2x 3 = (3y)2 (2x 3 ) 2 = 32 y (x 3 ) 2 = 9y2 4x x 2 + 2x + 3x + 6 = x x + x x ( x + 2) ( x + 3) = x x + 2 = x x - (x - 5) + 4(3 - x) = 2x - x x = x - 4x + 17 = -3x (x - 8) = x - 4(x + 2) 4 + 3x - 24 = x - 4x - 8 3x - 20 = -3x - 8 6x = 12 x = x - 5y = 20-5y = -4 x x + 20 y = V = 1 3 pr2 h y = 4 5 x - 4 = 1 3 p (4)2 (12) = 1 3 p (16)(12) = 64p in. 3 or in Let x = smaller number, then 2x 1 = the larger number. x + 2x - 1= 41 3x - 1= 41 3x = 42 x = 14 2x 1 = 2(14) 1 = 28 1 = 27 The numbers are 14 and 27. Exercise Set The first step in any factoring by grouping problem is to factor out a common factor, if one exists. 3. If you multiply ( x - 2) ( x + 4) by the FOIL method, you get the polynomial x 2 + 4x - 2x Answers will vary. 7. x 2 + 3x + 2x + 6 = x( x + 3) + 2( x + 3) = ( x + 3) ( x + 2) 9. x 2 + 5x + 4x + 20 = x( x + 5) + 4( x + 5) = ( x + 5) ( x + 4) 11. x 2 + 2x + 5x + 10 = x( x + 2) + 5( x + 2) = ( x + 2) ( x + 5) 13. x 2 + 3x - 5x - 15 = x( x + 3) - 5( x + 3) = ( x + 3) ( x - 5) 15. 4b 2-10b + 10b - 25 = 2b(2b - 5) + 5(2b - 5) = (2b - 5)(2b + 5) 17. 3x 2 + 9x + x + 3 = 3x( x + 3) + 1( x + 3) = ( x + 3) ( 3x + 1) 19. 6x 2 + 3x - 2x - 1 = 3x( 2x + 1) - 1( 2x + 1) = (2x + 1)(3x -1) 21. 8x x + x + 4 = 8x( x + 4) + 1( x + 4) = ( x + 4) ( 8x + 1) 156

4 SSM: Elementary and Intermediate Algebra Chapter 6: Factoring t 2-8t - 3t + 2 = 4t(3t - 2) - 1(3t - 2) = ( 3t - 2) ( 4t - 1) x 2-4x - 3x + 6 = 2x( x - 2) - 3( x - 2) = ( x - 2) ( 2x - 3) 29. x 2 + 2xy - 3xy - 6y 2 = x x + 2y - 3y( x + 2y) ( x - 3y) = x + 2y 31. 3x 2 + 2xy - 9xy - 6y 2 = x(3x + 2y) - 3y (3x + 2y) = (3x + 2y)(x - 3y) p p - 4 p - 10 = 3p 2 p + 5-2( 2p + 5) ( 3p - 2) = 2 p x 2-12xy - 25xy + 30y 2 = 2x 5x - 6y - 5y( 5x - 6y) ( 2x - 5y) = 5x - 6y 35. x 2 + bx + ax + ab = x( x + b) + a( x + b) = ( x + b) ( x + a) 37. xy + 5x - 3y - 15 = x y + 5-3( y + 5) ( x - 3) = y a 2 + 3a + ab + 3b = a( a + 3) + b( a + 3) = ( a + 3) ( a + b) + 5( y - 1) 41. xy - x + 5y - 5 = x y - 1 = ( x + 5) y x( 3 + 2y) y - 3x - 2xy = y = ( 4 - x) 3 + 2y 45. z 3 + 3z 2 + z + 3 = z 2 ( z + 3) + 1( z + 3) = z z x 3 + 4x 2-3x - 12 = x 2 ( x + 4) - 3( x + 4) = x 2-3 x x 2-12x + 8x - 48 = 2 x 2-2 6x x = 2 x 2-6x + 4 x - 24 [ + 4( x - 6) ] ( x + 4) = 2 x x - 6 = 2 x x 2 + 8x + 8x + 16 = 4 x x + 4 2x x 3 + 9x 2-2x 2-3x = 4 x 2 + 2x + 2x + 4 [ + 2( x + 2) ] ( x + 2) 2 = 4 x x + 2 = 4 x + 2 = 4 x + 2 = x 6x 2 + x 9x - x 2x - x 3 = x 6x 2 + 9x - 2x - 3 [ - 1( 2x + 3) ] ( 3x - 1) = x 3x 2x + 3 = x 2x

5 Chapter 6: Factoring SSM: Elementary and Intermediate Algebra 55. x 3 + 3x 2 y - 2x 2 y - 6xy 2 = x x 2 + x 3xy - x 2xy - x 6y 2 = x x 2 + 3xy - 2xy - 6y 2 [ - 2y( x + 3y) ] ( x - 2y) = x x x + 3y = x x + 3y 57. 5x + 3y + xy + 15 = xy + 5x + 3y ( y + 5) x + 3 = x y + 5 = y x + 5y + xy + 30 = 6x + xy + 5y + 30 = x( 6 + y) + 5 y + 6 = ( x + 5) y ax + by + ay + bx = ax + ay + bx + by = a( x + y) + b x + y = ( a + b) x + y 63. cd d + 3c = cd - 4d + 3c -12 = d( c - 4) + 3( c - 4) = ( d + 3) ( c - 4) 65. ac - bd - ad + bc = ac - ad + bc - bd = a( c - d) + b( c - d) = ( a + b) ( c - d) 67. Not any arrangement of the terms of a polynomial is factorable by grouping. xy + 2x + 5y +10 is factorable but xy x + 5y is not factorable in this arrangement = ( + 3) = ( + 3) ( - 5) 71. a. 3x x + 8 = 3x 2 + 6x + 4x + 8 b. 3x 2 + 6x + 4x + 8 = 3x( x + 2) + 4( x + 2) = ( x + 2) ( 3x + 4) 73. a. 2x 2-11x + 15 = 2x 2-6x - 5x + 15 b. 2 x 2-6x - 5x + 15 = 2x( x - 3) - 5( x - 3) = ( x - 3) ( 2x - 5) 75. a. 4x 2-17 x - 15 = 4x 2-20x + 3x - 15 b. 4x 2-20x + 3x -15 = 4x( x - 5) + 3( x - 5) = ( 4x + 3) ( x - 5) = ( + 3) + 2( + 3) (2x - 7) = 4(x + 5) x + 21= 4x x + 26 = 4x = 10x = x 5 6 = x = ( + 3)( + 2) 80. Let w = the number of pounds of chocolate wafers p = the number of pounds of peppermint candies w + p = w p = 4.75(50) or w + p = w p = Multiply the first equation by 2.5 and then add. 2.5[w + p = 50] gives -2.5w - 2.5p = w + 2.5p = w = w = 30 w + p = p = 50 p = 20 They should mix 30 pounds of chocolate wafers with 20 pounds of peppermint hard candies. 15x 3-6x 2-9x + 5 3x ) 82. x - 3 x 2-9 x 2 3x 3x - 9 3x x 2-9 x - 3 = x + 3 x + 3 = 15x3 3x - 6x 2 3x - 9x 3x + 5 3x = 5x 2-2x x 158

6 SSM: Elementary and Intermediate Algebra Chapter 6: Factoring Exercise Set Since 8000 is positive, both signs will be the same. Since 180 is positive, both signs will be positive. 3. Since 8000 is negative, one sign will be positive, the other will be negative. 5. Since 8000 is positive, both signs will be the same. Since 240 is negative, both signs will be negative. 7. The trinomial x 2 + 4xy 12y 2 is obtained by multiplying the factors using the FOIL method. 9. The trinomial 4a 2 4b 2 is obtained by multiplying all the factors and combing like terms. 11. The answer is not fully factored. A 2 can be factored from (2x 4). 13. To determine the factors when factoring a trinomial of the form x 2 + bx + c. First, find two numbers whose product is c, and whose sum is b. The factors are (x + first number) and (x + second number). 15. x 2-7x + 10 = ( x - 5) ( x - 2) 17. x 2 + 6x + 8 = ( x + 4) ( x + 2) 19. x 2 + 7x + 12 = ( x + 4) ( x + 3) 21. x 2 + 4x - 6 is prime. 23. y 2-13y + 12 = ( y -12)( y - 1) 25. a 2-2a - 8 = ( a - 4) ( x + 2) 27. r 2-2r - 15 = ( r - 5) ( r + 3) 29. b 2-11b + 18 = ( b - 9) ( b - 2) 31. x 2-8x - 15 is prime. 33. a a + 11 = ( a +1) ( a +11) 35. x 2-7x - 30 = ( x -10)( x + 3) 37. x 2 + 4x + 4 = ( x + 2) ( x + 2) = ( x + 2) p 2 + 6p + 9 = ( p + 3) ( p + 3) = ( p + 3) p 2-12p + 36 = (p - 6)( p - 6) = ( p - 6) w 2-18w + 45 = ( w - 15) ( w - 3) 45. x x - 39 = ( x + 13) ( x - 3) 47. x 2 - x - 20 = ( x - 5) ( x + 4) 49. y 2 + 9y +14 = ( y + 7) ( y + 2) 51. x x - 64 = ( x +16) ( x - 4) 53. s s - 24 is prime. 55. x 2-20x + 64 = ( x -16)( x - 4) 57. b 2-18b + 65 = ( b - 5) ( b -13) 59. x x = x 2 + 3x + 2 = (x + 2)(x + 1) 61. 7w 18 + w 2 = w 2 + 7w 18 = (w + 9)(w 2) 63. x 2-8xy + 15y 2 = ( x - 3y) ( x - 5y) 65. m 2-6mn + 9n 2 = ( m - 3n) ( m - 3n) = ( m - 3n) x 2 + 8xy + 15y 2 = ( x + 3y) ( x + 5y) 69. m 2 5mn 24n 2 = (m+ 3n)(m 8n) 71. 6x 2-30x + 24 = 6 x 2-5x + 4 = 6( x - 4) ( x -1) 73. 5x x + 15 = 5 x 2 + 4x + 3 = 5( x + 1) ( x + 3) 75. 2x 2-14x + 24 = 2 x 2-7x + 12 = 2( x - 4) ( x - 3) 77. b 3-7b b = b b 2-7b + 10 = b( b - 5) ( b - 2) 159

7 Chapter 6: Factoring SSM: Elementary and Intermediate Algebra 79. 3z 3-21z 2-54z = 3z z 2-7z - 18 = 3z ( z - 9) ( z + 2) 81. x 3 + 8x x = x( x 2 + 8x +16) = x(x + 4)( x + 4) = x( x + 4) a 2-24ab + 32b 2 = 4 a 2-6b + 8b 2 = 4( a - 4b) ( a - 2b) 85. r 2 s + 7rs s 3 = s r 2 + 7rs + 12s 2 = s( r + 3s) ( r + 4s) 87. x 4-4x 3-21x 2 = x 2 x 2-4x - 21 = x 2 ( x - 7) ( x + 3) 89. Sign of Coefficient of x-term Sign of Constant of Trinomial + both negative Signs of Constant Terms in the Binomial Factors one positive and one negative + one positive and one negative + + both positive 91. x 2 + 5x + 4 = ( x + 1) ( x + 4) 93. x x + 32 = ( x + 8) ( x + 4) 95. x x = ( x )( x + 0.2) 97. x x = Ê Á x + 1 ˆ Ë 5 Ê Á x + 1 ˆ Ë 5 = x Ê Á ˆ Ë x 2 + 5x = ( x + 20) ( x -15) ( 2x - 4) = 5x x -16 = 5x x - 5x -16 = 5x - 5x x -16 = 11 3x = x = 27 x = 9 160

8 SSM: Elementary and Intermediate Algebra Chapter 6: Factoring 102. Let x be the percent of acid in the mixture. Solution Strength Liters Amount 18% % Mixture x = 5x = x = x The mixture is a 19.6% acid solution. 5 5x ( 2x 2 + 5x - 6) ( x - 2) = 2x 2 ( x - 2) + 5x( x - 2) - 6( x - 2) = 2x 3-4 x 2 + 5x 2-10x - 6x + 12 = 2x 3 + x 2-16x + 12 ) 3x x - 4 3x 2-10 x x 2-12x 2x -10 2x x 2-10x - 10 x - 4 = 3x x x 2 + 5x - 6x - 10 = x( 3x + 5) - 2( 3x + 5) = ( 3x + 5) ( x - 2) Exercise Set Factoring trinomials is the reverse process of multiplying binomials. 3. When factoring` a trinomial of the form ax 2 + bx + c, the product of the constants in the binomial factors must equal the constant, c, of the trinomial. 5. 2x x + 5 = ( 2x + 1) ( x + 5) 7. 3x x + 8 = ( 3x + 2) ( x + 4) 9. 5x 2-9x - 2 = (5x +1)(x - 2) 11. 3r 2 +13r -10 = ( 3r - 2) ( r + 5) 13. 4z 2-12z + 9 = 2z - 3 ( 2z - 3) 2 = 2z y 2 - y - 4 = ( 5y + 4) ( y -1) 161

9 Chapter 6: Factoring SSM: Elementary and Intermediate Algebra 17. 5a 2-12a + 6 is prime z 2 + z -12 = ( 2z + 3) ( 3z - 4) 21. 3x 2 +11x + 4 is prime y 2-16 y + 3 = ( 5y -1)( y - 3) 25. 7x x + 6 = ( 7x + 1) ( x + 6) 27. 7x 2-8x + 1 = ( 7x - 1) ( x -1) 29. 5b 2-23b +12 = ( 5b - 3) ( b - 4) 31. 5z 2-6z - 8 = ( 5z + 4) ( z - 2) 33. 4y 2 + 5y - 6 = ( 4y - 3) ( y + 2) x 2-27x + 5 = ( 5x - 1) ( 2x - 5) d 2-7d -12 = ( 5d + 4) ( 2d - 3) 39. 6x 2-22x - 8 = 2 3x x = 2 3x 2-11x - 4 = 2( 3x + 1) ( x - 4) t t 2 = 7t 2 +10t + 3 = (7t + 3)(t +1) 43. 6x x +10 = 2 3x x = 2 3x 2 + 8x + 5 = 2( 3x + 5) ( x + 1) 45. 6x 3-5x 2-4x = x 6x 2 - x 5 x - x 4 = x 6x 2-5x - 4 = x( 2x +1) ( 3x - 4) x x 2 + 8x = 4x 3x 2 + 4x 7x + 4x 2 = 4x 3x 2 + 7x + 2 = 4x( 3x + 1) ( x + 2) 49. 4x 3-2x 2-12x = 2x 2x 2-2x x - 2x 6 = 2x 2x 2 - x - 6 ( x - 2) = 2x 2x z 2 + 6z - 6 = 6 6z z = 6 6z 2 + z -1 ( 2z +1) = 6 3z r 2-30r = 3r 2-30r + 72 = 3 r r = 3 r 2-10r + 24 = 3(r - 4)(r - 6) 55. 2x 2 + 5xy + 2y 2 = ( 2x + y) ( x + 2y) 57. 2x 2-7xy + 3y 2 = ( 2x - y) ( x - 3y) x xy - 8y 2 = 2 6x xy - 2 4y 2 = 2 6x 2 + 5xy - 4y 2 ( 3x + 4y) = 2 2x - y 61. 6x 2-9xy - 27y 2 = 3 2x 2-3 3xy - 3 9y 2 = 3 2x 2-3xy - 9y 2 = 3(x - 3y)(2x + 3y) 63. 6m 2 - mn - 2n 2 = (3m - 2n)(2m + n) 65. 8x 3 +10x 2 y + 3xy 2 = x 8x 2 + x 10xy + x 3y 2 = x(8x 2 +10xy + 3y 2 ) = x(4 x + 3y)(2x + y) 67. 4x 4 + 8x 3 y + 3x 2 y 2 = x 2 4x 2 + x 2 8xy + x 2 3y 2 = x 2 (4x 2 + 8xy + 3y 2 ) = x 2 (2x + y)(2x + 3y) 69. 3x 2-20x - 7. This polynomial was obtained by multiplying the factors x x This polynomial was obtained by multiplying the factors x 4 - x 3-3x 2. This polynomial was obtained by multiplying the factors. 75. a. The second factor can be found by dividing the trinomial by the binomial. 162

10 SSM: Elementary and Intermediate Algebra Chapter 6: Factoring ) 6x + 11 b. 3x x x x x 33x x The other factor is 6x x 2 + 9x - 20 = ( 6x - 5) ( 3x + 4) x 2-124x +160 = ( 5x - 8) ( 3x - 20) x x = 4 18 x x = 4 18x 2-45x - 50 = 4( 6x + 5) ( 3x - 10) 83. The other factor is 2x The product of the three first terms must equal 6x 3, and the product of the constants must equal 2250x x 2-4(y + 3) + 2y = -(-3) 2-4(-5 + 3) + 2(-5) 2 = -9-4(-2) + 2(25) = = ª His average speed was about miles per hour x 4 y 3-12xy x 5 y 6 = 12xy 2 3x 3 y - 12xy xy 2 2x 4 y 4 = 12xy 2 3x 3 y x 4 y x 2-15x + 54 = ( x - 9) ( x - 6) Exercise Set a. a 2 - b 2 = ( a + b) ( a - b) b. Answers will vary. 3. a. a 3 - b 3 = ( a - b) ( a 2 + ab + b 2 ) b. Answers will vary. 5. No, there is no special formula for factoring the sum of two squares. 7. x is prime. 9. 4a = 4( a 2 + 4) m n 2 = 4( 4m 2 + 9n 2 ) 13. y 2-25 = y = ( y + 5) ( y - 5) 15. z 2-81 = z = ( z + 9) ( z - 9) 17. x 2-49 = x = ( x + 7) ( x - 7) 19. x 2 - y 2 = ( x + y) ( x - y) 2-5z ( 3y - 5z) 21. 9y 2-25z 2 = 3y 2 = 3y + 5z [ 2 - ( 3b ) 2 ] a 2-36b 2 = 4 16a 2-9b 2 = 4 4a = 4( 4a + 3b) ( 4a - 3b) x 2-36 = ( 7x) = ( 7x + 6) ( 7x - 6) 27. z 4-81x 2 = z ( 9x) 2 ( z2-9x) = z 2 + 9x È 2 - ( 3y) 2 Î Í ( x 2-3y) 29. 9x 4-81y 2 = 9 x 4-9y 2 = 9 x 2 = 9 x 2 + 3y m 4-49n 2 = 6m ( 7n) 2 ( 6m2-7n) = 6m 2 + 7n 163

11 Chapter 6: Factoring SSM: Elementary and Intermediate Algebra x = 10 x 2-16 = 10 x = 10( x + 4) ( x - 4) È 2 - ( 5y 2 ) 2 Î Í ( 2x - 5y 2 ) x y 4 = 4 4x 2-25y 4 = 4 2x = 4 2x + 5y x 3 + y 3 = ( x + y) ( x 2 - xy + y 2 ) 39. a 3 - b 3 = ( a - b) ( a 2 + ab + b 2 ) 41. x = x x 3-27 = x a 3 +1 = a = ( x + 2) x 2-2x + 4 = ( x - 3) x 2 + 3x + 9 = ( a + 1) a 2 - a x 3-1 = ( 3x) = ( 3x -1) 9x 2 + 3x a = ( 3a) = ( 3a - 5) 9a 2 +15a y 3 = 3 3-2y y + 4y 2 = 3-2y m n 3 = ( 4m) 3 + ( 3n) a 3-27b 3 = (2a) 3 - (3b) 3 = ( 4m + 3n) 16m 2-12mn + 9n 2 = (2a - 3b)(4a 2 + 6ab + 9b 2 ) 57. 2x 2 + 8x + 8 = 2 x x + 4 = 2( x + 2) a 2 b - 25b = b(a 2-25) = b(a ) = b(a + 5)(a - 5) 61. 3c 2-18c + 27 = 3 c 2-6c + 9 = 3( c - 3) x 2-10x -15 = 5 x 2-2x - 3 = 5( x - 3) ( x + 1) + 3( y - 2) ( y - 2) 65. 3xy - 6x + 9y - 18 = 3 xy - 2x + 3y x 2-50 = 2(x 2-25) [ ] = 3 x y - 2 = 3 x + 3 = 2(x ) = 2(x + 5)( x - 5) = 3y( x ) 69. 3x 2 y - 27y = 3y x 2-9 = 3y( x + 3) ( x - 3) = 3y 2 ( x ) = 3y 2 ( x + 1) ( x 2 - x +1) 71. 3x 3 y 2 + 3y 2 = 3y 2 x x 3-16 = 2 x 3-8 = 2( x ) = 2( x - 2) ( x 2 + 2x + 4) x 2-50 = 2 9x 2-25 = 2 (3x) = 2( 3x + 5) ( 3x - 5) x x + 27 = 3 4 x x + 9 ( 2x + 3) = 3 2x + 3 = 3(2x + 3) 2 164

12 SSM: Elementary and Intermediate Algebra 79. 6x 2-4x + 24x - 16 = 2 3x 2-2x + 12x - 8 [ ] = 2 x( 3x - 2) + 4( 3x - 2) = 2( 3x - 2) ( x + 4) 81. 2rs 2-10rs - 48r = 2r s 2-5s - 24 = 2r( s + 3) ( s - 8) 83. 4x 2 + 5x - 6 = ( x + 2) ( 4x - 3) = 25( b ) b = 25 b 2-4 = 25( b + 2) ( b - 2) [ 2 ] 87. a 5 b 2-4a 3 b 4 = a 3 b 2 a 2-4b 2 = a 3 b 2 a 2-2b = a 3 b 2 ( a + 2b) ( a - 2b) 89. 3x 4-18x x 2 = 3x 2 x 2-6x x x = x( x ) 93. y 4-16 = y ( y 2-4) ( ( )( y2-2 2 ) ) y + 2 = y = y = y = 3x 2 ( x - 3) ( x - 3) = 3x 2 ( x - 3) 2 ( y - 2) a 2-15ab - 6b 2 = 3 12a 2-5ab - 2b 2 = 3( 3a - 2b) ( 4a + b) 97. 2ab - 3b + 4a - 6 = b( 2a - 3) + 2( 2a - 3) = ( 2a - 3) ( b + 2) y 4 = 9 1- y 4 È 2 Î Í ( 1- y 2 ) ( y 2 ) 1+ y = y 2 = 9 1+ y 2 = 9 1+ y 2 = 9 1+ y uæ + 2u + JÆ + 2J = u ( Æ+ 2) + J ( Æ+ 2) = ( u + J )( Æ+ 2) ( 1- y) u 2 Æ- 6uÆ - 20Æu + 30Æ = 2Æ 2u 2-3u -10u + 15 = 2Æ 2u 2-13u + 15 = 2Æ( 2u - 3) ( u - 5) 105. x 6-27y 9 = x 2 Chapter 6: Factoring 3 - ( 3y 3 ) 3 ( x4 + 3x 2 y 3 + 9y 6 ) = x 2-3y x x y 2 + 4y - 4 = ( x + 5) 2 - y 2-4y + 4 = ( x + 5) 2 - ( y - 2) 2 = [( x + 5) + ( y - 2) ] ( x + 5) - ( y - 2) = ( x + y + 3) ( x - y + 7) x - 2( x + 6) = 2x - 5 7x - 2x -12 = 2x - 5 5x -12 = 2x - 5 5x - 2x -12 = 2x - 2x - 5 3x -12 = -5 3x = x = 7 3x 3 = 7 3 [ ] x =

13 Chapter 6: Factoring SSM: Elementary and Intermediate Algebra 110. Substitute 36 for A, 6 for b, and 12 for d. A = 1 h(b + d) 2 36 = 1 2 h( ) 36 = 1 2 h(18) 36 = 9h 4 = h The height is 4 inches x + (5 2x) = 2 The sum of a number, and 5 decreased by twice the number is x 4 3 Ê y ˆ Á Ë 6xy 5 = 4 6 x 4 3 Ê x y ˆ Á Ë y 5 = 2 3 Ê 3 x3 1 ˆ Á Ë y 4 = 2x3 3 Ê ˆ Á Ë 3y 4 = 23 x y 4 3 = 8x9 27y x -2 x -3 = x -2-3 Exercise Set 6.6 = x -5 = 1 x 5 1. Answers will vary. 3. The standard form of a quadratic equation is ax 2 + bx + c = a. The zero-factor property may only be used when one side of the equation is equal to 0. b. ( x + 1) ( x - 2) = 4 x 2-2x + x - 2 = 4 x 2 - x - 6 = 0 ( x - 3) ( x + 2) = 0 x - 3 = 0 or x + 2 = 0 x = 3 x = x(x + 2) = 0 x = 0 or x + 2 = 0 x = 0 x = x( x - 8) = 0 x = 0 or x - 8 = 0 x = ( 2x + 5) ( x - 3) = 0 2x + 5 = 0 or x - 3 = 0 2x = -5 x = 3 x = x 2-9 = 0 ( x + 3) ( x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 x = x 2-12x = 0 x( x - 12) = 0 x = 0 or x - 12 = 0 x = x x = 0 9x( x + 3) = 0 x = 0 or x + 3 = x 2-8x + 16 = 0 ( x - 4) ( x - 4) = 0 x - 4 = 0 x = -3 x = x x = -20 x 2 +12x + 20 = 0 ( x + 10) ( x + 2) = 0 x +10 = 0 or x + 2 = 0 x = -10 x =

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