# 8 when is a polynomial factored completely a

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Chapter 6 / Exercise 76
Mathematical Excursions
Aufmann
Expert Verified
8. When is a polynomial factored completely? A polynomial is factored completely when it is expressed as a product of prime polynomials. Find the greatest common factor for each group of terms. See Example 1. 9. 48, 36 x 12 10. 42 a , 28 a 2 14 a 11. 9 wx , 21 wy , 15 xy 3 12. 70 x 2 , 84 x , 42 x 3 14 x 13. 24 x 2 y , 42 xy 2 , 66 xy 3 6 xy 14. 60 a 2 b 5 , 140 a 9 b 2 , 40 a 3 b 6 20 a 2 b 2 Factor out the greatest common factor in each expression. See Examples 2 and 3. 15. x 3 5 x x ( x 2 5) 16. 10 x 2 20 y 3 10( x 2 2 y 3 ) 17. 48 wx 36 wy 12 w (4 x 3 y ) 18. 42 wz 28 wa 14 w (3 z 2 a ) 19. 2 x 3 4 x 2 6 x 2 x ( x 2 2 x 3) 20. 6 x 3 12 x 2 18 x 6 x ( x 2 2 x 3) 21. 36 a 3 b 6 24 a 4 b 2 60 a 5 b 3 12 a 3 b 2 (3 b 4 2 a 5 a 2 b ) 22. 44 x 8 y 6 z 110 x 6 y 9 z 2 22 x 6 y 6 z (2 x 2 5 y 3 z ) 23. ( x 6) a ( x 6) b ( x 6)( a b ) 24. ( y 4)3 ( y 4) b ( y 4)(3 b ) 25. ( y 1) 2 y ( y 1) 2 z ( y 1) 2 ( y z ) 26. ( w 2) 2 w ( w 2) 2 3 ( w 2) 2 ( w 3) Factor out the greatest common factor, then factor out the opposite of the greatest common factor. See Example 4. 27. 2 x 2 y 2( x y ), 2( x y ) 28. 3 x 6 3( x 2), 3( x 2) 29. 6 x 2 3 x 3 x (2 x 1), 3 x ( 2 x 1) 30. 10 x 2 5 x 5 x (2 x 1), 5 x ( 2 x 1) 31. w 3 3 w 2 w 2 ( w 3), w 2 ( w 3) 32. 2 w 4 6 w 3 2 w 3 ( w 3), 2 w 3 ( w 3) 33. a 3 a 2 7 a a ( a 2 a 7), a ( a 2 a 7) 34. 2 a 4 4 a 3 6 a 2 2 a 2 ( a 2 2 a 3), 2 a 2 ( a 2 2 a 3) Factor each polynomial. See Example 5. 35. x 2 100 ( x 10)( x 10) 36. 81 y 2 (9 y )(9 y ) 37. 4 y 2 49 (2 y 7)(2 y 7) 38. 16 b 2 1 (4 b 1)(4 b 1) 5.6 Factoring Polynomials (5-47) 303 39. 9 x 2 25 a 2 (3 x 5 a )(3 x 5 a ) 40. 121 a 2 b 2 (11 a b )(11 a b ) 41. 144 w 2 z 2 1 (12 wz 1)(12 wz 1) 42. x 2 y 2 9 c 2 ( xy 3 c )( xy 3 c ) Factor each polynomial. See Example 6. 43. x 2 20 x 100 ( x 10) 2 44. y 2 10 y 25 ( y 5) 2 45. 4 m 2 4 m 1 (2 m 1) 2 46. 9 t 2 30 t 25 (3 t 5) 2 47. w 2 2 wt t 2 ( w t ) 2 48. 4 r 2 20 rt 25 t 2 (2 r 5 t ) 2 Factor. See Example 7. 49. a 3 1 ( a 1)( a 2 a 1) 50. w 3 1 ( w 1)( w 2 w 1) 51. w 3 27 ( w 3)( w 2 3 w 9) 52. x 3 64 ( x 4)( x 2 4 x 16) 53. 8 x 3 1 (2 x 1)(4 x 2 2 x 1) 54. 27 x 3 1 (3 x 1)(9 x 2 3 x 1) 55. a 3 8 ( a 2)( a 2 2 a 4) 56. m 3 8 ( m 2)( m 2 2 m 4) Factor each polynomial completely. See Example 8. 57. 2 x 2 8 2( x 2)( x 2) 58. 3 x 3 27 x 3 x ( x 3)( x 3) 59. x 3 10 x 2 25 x x ( x 5) 2 60. 5 a 4 m 45 a 2 m 5 a 2 m ( a 3)( a 3) 61. 4 x 2 4 x 1 (2 x 1) 2 62. ax 2 8 ax 16 a a ( x 4) 2 63. ( x 3) x ( x 3)7 ( x 3)( x 7) 64. ( x 2) x ( x 2)5 ( x 2)( x 5) 65. 6 y 2 3 y 3 y (2 y 1) 66. 4 y 2 y y (4 y 1) 67. 4 x 2 20 x 25 (2 x 5) 2 68. a 3 x 3 6 a 2 x 2 9 ax ax ( ax 3) 2 69. 2 m 4 2 mn 3 2 m ( m n )( m 2 mn n 2 ) 70. 5 x 3 y 2 y 5 y 2 (5 x 3 y 3 ) 71. (2 x 3) x (2 x 3)2 (2 x 3)( x 2) 72. (2 x 1) x (2 x 1)3 (2 x 1)( x 3) 73. 9 a 3 aw 2 a (3 a w )(3 a w ) 74. 2 bn 2 4 b 2 n 2 b 3 2 b ( n b ) 2 75. 5 a 2 30 a 45 5( a 3) 2 76. 2 x 2 50 2( x 5)( x 5) 77. 16 54 x 3 2(2 3 x )(4 6 x 9 x 2 ) 78. 27 x 2 y 64 x 2 y 4 x 2 y (3 4 y )(9 12 y 16 y 2 ) 79. 3 y 3 18 y 2 27 y 3 y ( y 3) 2 80. 2 m 2 n 8 mn 8 n 2 n ( m 2) 2 81. 7 a 2 b 2 7 7( ab 1)( ab 1) 82. 17 a 2 17 a 17 a ( a 1)
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Chapter 6 / Exercise 76
Mathematical Excursions
Aufmann
Expert Verified
Factor each polynomial completely. See Example 9. 83. x 10 9 ( x 5 3)( x 5 3) 84. y 8 4 ( y 4 2)( y 4 2) 85. z 12 6 z 6 9 ( z 6 3) 2 86. a 6 10 a 3 25 ( a 3 5) 2 87. 2 x 7 8 x 4 8 x 2 x ( x 3 2) 2 88. x 13 6 x 7 9 x x ( x 6 3) 2 89. 4 x 5 4 x 3 x x (2 x 2 1) 2 90. 18 x 6 24 x 3 8 2(3 x 3 2) 2 91. x 6 8 ( x 2 2)( x 4 2 x 2 4) 92. y 6 27 ( y 2 3)( y 4 3 y 2 9) 93. 2 x 9 16 2( x 3 2)( x 6 2 x 3 4) 94. x 13 x x ( x 4 1)( x 8 x 4 1) Factor each polynomial completely. The variables used as exponents represent positive integers. See Example 10.