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Ejercicio 106

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DESCARGA
CAPITULO X

Descomposición Factorial
Ejercicio 106
MISCELANIA SOBRE LOS 10 CASOS DE DESCOMPOSICION EN FACTORES
Descomponer en factores:
  1. 5 a 2 +a=a( 5a+1 )
  2. m 2 +2mx+ x 2 = ( m+x ) 2
  3. a 2 +a–ab–b =a( a+1 ) –b( a+1 ) =( a+1 ) ( a–b )
  4. x 2 –36=( x–6 ) ( x+6 )
  5. 9 x 2 –6xy+ y 2 = ( 3x–y ) 2
  6. x 2 –3x–4 = x 2 +x–4x–4 . 4 1 | 4 =x( x+1 ) –4( x+1 ) =( x+1 ) ( x–4 )
  7. 6 x 2 –x–2 =6 x 2 +3x–4x–2 . 12 6 3 1 | 2 2 3 =3x( 2x+1 ) –2( 2x+1 ) =( 2x+1 ) ( 3x–2 )
  8. 1+ x 3 =( 1+x ) ( 1–x+ x 2 )
  9. 27 a 3 –1 =( 3a–1 ) [ ( 3a ) 2 +3a+1 ] =( 3a–1 ) ( 9 a 2 +3a+1 )
  10. x 5 + m 5 =( x+m ) ( x 4 – x 3 m+ x 2 m 2 –x m 3 + m 4 )
  11. a 3 –3 a 2 b+5a b 2 =a( a 2 –3ab+5 b 2 )
  12. 2xy–6y+xz–3z =2y( x–3 ) +z( x–3 ) =( x–3 ) ( 2y+z )
  13. 1–4b+4 b 2 = ( 1–2b ) 2
  14. 4 x 4 +3 x 2 y 2 + y 4 =4 x 4 +3 x 2 y 2 + y 4 + x 2 y 2 – x 2 y 2 =4 x 4 +4 x 2 y 2 + y 4 – x 2 y 2 = ( 2 x 2 + y 2 ) 2 – x 2 y 2 =[ ( 2 x 2 + y 2 ) +xy ] [ ( 2 x 2 + y 2 ) –xy ] =( 2 x 2 +xy+ y 2 ) ( 2 x 2 –xy+ y 2 )
  15. x 8 –6 x 4 y 4 + y 8 = x 8 –6 x 4 y 4 + y 8 +4 x 4 y 4 –4 x 4 y 4 = x 8 –2 x 4 y 4 + y 8 –4 x 4 y 4 = ( x 4 – y 4 ) 2 –4 x 4 y 4 =[ ( x 4 – y 4 ) –2 x 2 y 2 ] [ ( x 4 – y 4 ) +2 x 2 y 2 ] =( x 4 –2 x 2 y 2 – y 4 ) ( x 4 +2 x 2 y 2 – y 4 )
  16. a 2 –a–30 = a 2 –6a+5a–30 . 30 15 5 1 | 2 3 5 =a( a–6 ) +5( a–6 ) =( a–6 ) ( a+5 )
  17. 15 m 2 +11m–14 =15 m 2 –10m+21m–14 . 210 105 35 7 1 | 2 3 5 7 =5m( 3m–2 ) +7( 3m–2 ) =( 3m–2 ) ( 5m+7 )
  18. a 6 +1 =( a 2 +1 ) [ ( a 2 ) 2 – a 2 +1 ] =( a 2 +1 ) ( a 4 – a 2 +1 )
  19. 8 m 3 –27 y 6 =( 2m–3 y 2 ) [ ( 2m ) 2 +( 2m ) ( 3 y 2 ) + ( 3 y 2 ) 2 ] =( 2m–3 y 2 ) ( 4 m 2 +6m y 2 +9 y 4 )
  20. 16 a 2 –24ab+9 b 2 = ( 4a–3b ) 2
  21. 1+ a 7 =( 1+a ) ( 1–a+ a 2 – a 3 + a 4 – a 5 + a 6 )
  22. 8 a 3 –12 a 2 +6a–1 =( 8 a 3 –1 ) –( 12 a 2 –6a ) =( 2a–1 ) [ ( 2a ) 2 +2a+1 ] –6a( 2a–1 ) =( 2a–1 ) ( 4 a 2 +2a+1 ) –6a( 2a–1 ) =( 2a–1 ) [ ( 4 a 2 +2a+1 ) –6a ] =( 2a–1 ) [ 4 a 2 +2a+1–6a ] =( 2a–1 ) ( 4 a 2 –4a+1 ) =( 2a–1 ) ( 2a–1 ) 2 = ( 2a–1 ) 3
  23. 1– m 2 =( 1–m ) ( 1+m )
  24. x 4 +4 x 2 –21 = x 4 –3 x 2 +7 x 2 –21 . 21 7 1 | 3 7 = x 2 ( x 2 –3 ) +7( x 2 –3 ) =( x 2 –3 ) ( x 2 +7 )
  25. 125 a 6 +1 =( 5 a 2 +1 ) [ ( 5 a 2 ) 2 +5 a 2 +1 ] =( 5 a 2 +1 ) ( 25 a 4 +5 a 2 +1 )
  26. a 2 +2ab+ b 2 – m 2 = ( a+b ) 2 – m 2 =[ ( a+b ) –m ] [ ( a+b ) +m ] =( a+b–m ) ( a+b+m )
  27. 8 a 2 b+16 a 3 b–24 a 2 b 2 =8 a 2 b( 1+2a–3b )
  28. x 5 – x 4 +x–1 = x 4 ( x–1 ) +( x–1 ) =( x–1 ) ( x 4 +1 )
  29. 6 x 2 +19x–20 =6 x 2 +24x–5x–20 . 120 60 30 15 5 1 | 2 2 2 3 5 =6x( x+4 ) –5( x+4 ) =( x+4 ) ( 6x–5 )
  30. 25 x 4 –81 y 2 =( 5 x 2 +9y ) ( 5 x 2 –9y )
  31. 1– m 3 =( 1–m ) ( 1+m+ m 2 )
  32. x 2 – a 2 +2xy+ y 2 +2ab– b 2 =( x 2 +2xy+ y 2 ) –( a 2 –2ab+ b 2 ) = ( x+y ) 2 – ( a–b ) 2 =[ ( x+y ) +( a–b ) ] [ ( x+y ) –( a–b ) ] =( x+y+a–b ) ( x+y–a+b )
  33. 21 m 5 n–7 m 4 n 2 +7 m 3 n 3 –7 m 2 n=7 m 2 n( 3 m 3 – m 2 n+m n 2 –1 )
  34. a( x+1 ) –b( x+1 ) +c( x+1 ) =( x+1 ) ( a–b+c )
  35. 4+4( x–y ) + ( x–y ) 2 = [ 2+( x–y ) ] 2 = ( x–y+2 ) 2
  36. 1– a 2 b 4 =( 1+a b 2 ) ( 1–a b 2 )
  37. b 2 +12ab+36 a 2 = ( b+6a ) 2
  38. x 6 +4 x 3 –77 = x 6 –7 x 3 +11 x 3 –77 . 77 11 1 | 7 11 = x 3 ( x 3 –7 ) +11( x 3 –7 ) =( x 3 –7 ) ( x 3 +11 )
  39. 15 x 4 –17 x 2 –4 =15 x 4 –20 x 2 +3 x 2 –4 . 60 30 15 5 1 | 2 2 3 5 =5 x 2 ( 3 x 2 –4 ) +( 3 x 2 –4 ) =( 3 x 2 –4 ) ( 5 x 2 +1 )
  40. 1+ ( a–3b ) 3 =[ 1+( a–3b ) ] [ 1–( a–3b ) + ( a–3b ) 2 ] =( a–3b+1 ) ( 1–a+3b+ a 2 –6ab+9 b 2 )
  41. x 4 + x 2 +25 = x 4 + x 2 +25+9 x 2 –9 x 2 = x 4 +10 x 2 +25–9 x 2 = ( x 2 +5 ) 2 –9 x 2 =[ ( x 2 +5 ) –3x ] [ ( x 2 +5 ) +3x ] =( x 2 –3x+5 ) ( x 2 +3x+5 )
  42. a 8 –28 a 4 +36 = a 8 –28 a 4 +36+16 a 4 –16 a 4 = a 8 –12 a 4 +36–16 a 4 = ( a 4 –6 ) 2 –16 a 4 =[ ( a 4 –6 ) +4 a 2 ] [ ( a 4 –6 ) –4 a 2 ] =( a 4 +4 a 2 –6 ) ( a 4 –4 a 2 –6 )
  43. 343+8 a 3 =( 7+2a ) [ 7 2 –7( 2a ) + ( 2a ) 2 ] =( 7+2a ) ( 49–14a+4 a 2 )
  44. 12 a 2 bx–15 a 2 by=3 a 2 b( 4x–5y )
  45. x 2 +2xy–15 y 2 = x 2 –3xy+5xy–15 y 2 . 15 5 1 | 3 5 =x( x–3y ) +5y( x–3y ) =( x–3y ) ( x+5y )
  46. 6am–4an–2n+3m =6am+3m–4an–2n =3m( 2a+1 ) –2n( 2a+1 ) =( 2a+1 ) ( 3m–2n )
  47. 81 a 6 –4 b 2 c 8 =( 9 a 3 –2b c 4 ) ( 9 a 3 +2b c 4 )
  48. 16– ( 2a+b ) 2 =[ 4–( 2a+b ) ] [ 4+( 2a+b ) ] =( 4–2a+b ) ( 4+2a+b )
  49. 20–x– x 2 =20+4x–5x– x 2 . 20 10 5 1 | 2 2 5 =4( 5+x ) –5x( 5+x ) =( 5+x ) ( 4–5x )
  50. n 2 +n–42 = n 2 –6n+7n–42 . 42 21 7 1 | 2 3 7 =n( n–6 ) +7( n–6 ) =( n–6 ) ( n+7 )
  51. a 2 – d 2 + n 2 – c 2 –2an–2cd =( a 2 –2an+ n 2 ) –( d 2 +2cd+ c 2 ) = ( a–n ) 2 – ( d+c ) 2 =[ ( a–n ) +( d+c ) ] [ ( a–n ) –( d+c ) ] =( a–n+d+c ) ( a–n–d–c )
  52. 1+216 x 9 =( 1+6 x 3 ) [ 1–6 x 3 + ( 6 x 3 ) 2 ] =( 1+6 x 3 ) ( 1–6 x 3 +36 x 6 )
  53. x 3 –64=( x–4 ) ( x 2 +4x+16 )
  54. x 3 –64 x 4 = x 3 ( 1–64x )
  55. 18a x 5 y 3 –36 x 4 y 3 –54 x 2 y 8 =18 x 2 y 3 ( a x 3 –2 x 2 –3 y 5 )
  56. 49 a 2 b 2 –14ab+1= ( 7ab–1 ) 2
  57. ( x+1 ) 2 –81 =[ ( x+1 ) –9 ] [ ( x+1 ) +9 ] =( x+1–9 ) ( x+1+9 ) =( x–8 ) ( x+10 )
  58. a 2 – ( b+c ) 2 =[ a+( b+c ) ] [ a–( b+c ) ] =( a+b+c ) ( a–b–c )
  59. ( m+n ) 2 –6( m+n ) +9 = [ ( m+n ) –3 ] 2 = ( m+n–3 ) 2
  60. 7 x 2 +31x–20 =7 x 2 +35x–4x–20 . 140 70 35 1 | 2 2 35 =7x( x+5 ) –4( x+5 ) =( x+5 ) ( 7x–4 )
  61. 9 a 3 +63a–45 a 2 =9 a 3 –45 a 2 +63a =9a( a 2 –5a+7 )
  62. ax+a–x–1 =a( x+1 ) –( x+1 ) =( x+1 ) ( a–1 )
  63. 81 x 4 +25 y 2 –90 x 2 y =81 x 4 –90 x 2 y+25 y 2 = ( 9 x 2 –5y ) 2
  64. 1–27 b 2 + b 4 =1–27 b 2 + b 4 +25 b 2 –25 b 2 =1–2 b 2 + b 4 –25 b 2 = ( 1– b 2 ) 2 –25 b 2 =[ ( 1– b 2 ) +5b ] [ ( 1– b 2 ) –5b ] =( 1+5b– b 2 ) ( 1–5b– b 2 )
  65. m 4 + m 2 n 2 + n 4 = m 4 + m 2 n 2 + n 4 + m 2 n 2 – m 2 n 2 = m 4 +2 m 2 n 2 + n 4 – m 2 n 2 = ( m 2 + n 2 ) 2 – m 2 n 2 =[ ( m 2 + n 2 ) –mn ] [ ( m 2 + n 2 ) +mn ] =( m 2 –mn+ n 2 ) ( m 2 +mn+ n 2 )
  66. c 4 –4 d 4 =( c 2 –2 d 2 ) ( c 2 +2 d 2 )
  67. 15 x 4 –15 x 3 +20 x 2 =5 x 2 ( 3 x 2 –3x+4 )
  68. a 2 – x 2 –a–x =( a+x ) ( a–x ) –a–x =( a+x ) ( a–x ) –( a+x ) =( a+x ) [ ( a–x ) –1 ] =( a+x ) ( a–x–1 )
  69. x 4 –8 x 2 –240 = x 4 –20 x 2 +12 x 2 –240 . 240 120 60 30 15 5 1 | 2 2 2 2 3 5 = x 2 ( x 2 –20 ) +12( x 2 –20 ) =( x 2 –20 ) ( x 2 +12 )
  70. 6 m 4 +7 m 2 –20 =6 m 4 –8 m 2 +15 m 2 –20 . 120 60 30 15 1 | 2 2 2 15 =2 m 2 ( 3 m 2 –4 ) +5( 3 m 2 –4 ) =( 3 m 2 –4 ) ( 2 m 2 +5 )
  71. 9 n 2 +4 a 2 –12an =9 n 2 –12an+4 a 2 = ( 3n–2a ) 2
  72. 2 x 2 +2=2( x 2 +1 )
  73. 7a( x+y–1 ) –3b( x+y–1 ) =( x+y–1 ) ( 7a–3b )
  74. x 2 +3x–18 = x 2 +6x–3x–18 . 18 9 3 1 | 2 3 3 =x( x+6 ) –3( x+6 ) =( x+6 ) ( x–3 )
  75. ( a+m ) 2 – ( b+n ) 2 =[ ( a+m ) +( b+n ) ] [ ( a+m ) –( b+n ) ] =( a+b+m+n ) ( a–b+m–n )
  76. x 3 +6 x 2 y+12x y 2 +8 y 3 = ( x+2y ) 3
  77. 8 a 2 –22a–21 =8 a 2 +6a–28a–21 . 168 84 42 21 7 1 | 2 2 2 3 7 =2a( 4a+3 ) –7( 4a+3 ) =( 4a+3 ) ( 2a–7 )
  78. 1+18ab+81 a 2 b 2 = ( 1+9ab ) 2
  79. 4 a 6 –1=( 2 a 3 –1 ) ( 2 a 3 +1 )
  80. x 6 –4 x 3 –480 = x 6 –24 x 3 +20 x 3 –480 . 480 240 120 60 30 15 5 1 | 2 2 2 2 2 3 5 = x 3 ( x 3 –24 ) +20( x 3 –24 ) =( x 3 –24 ) ( x 3 +20 )
  81. ax–bx+b–a–by+ay =ax+ay–a–bx–by+b =a( x+y–1 ) –b( x+y–1 ) =( x+y–1 ) ( a–b )
  82. 6am–3m–2a+1 =3m( 2a–1 ) –( 2a–1 ) =( 2a–1 ) ( 3m–1 )
  83. 15+14x–8 x 2 =15+20x–6x–8 x 2 . 120 60 30 15 5 1 | 2 2 2 3 5 =5( 3+4x ) –2x( 3+4x ) =( 3+4x ) ( 5–2x )
  84. a 10 – a 8 + a 6 + a 4 = a 4 ( a 6 – a 4 + a 2 +1 )
  85. 2x( a–1 ) –a+1 =2x( a–1 ) –( a–1 ) =( a–1 ) ( 2x–1 )
  86. ( m+n ) ( m–n ) +3n( m–n ) =( m–n ) [ ( m+n ) +3n ] =( m–n ) ( m+n+3n ) =( m–n ) ( m+4n )
  87. a 2 – b 3 +2 b 3 x 2 –2 a 2 x 2 = a 2 – b 3 –2 a 2 x 2 +2 b 3 x 2 = a 2 – b 3 –2 x 2 ( a 2 – b 3 ) =( a 2 – b 3 ) –2 x 2 ( a 2 – b 3 ) =( a 2 – b 3 ) ( 1–2 x 2 )
  88. 2am–3b–c–cm–3bm+2a =2am–3bm–cm+2a–3b–c =m( 2a–3b–c ) +2a–3b–c =m( 2a–3b–c ) +( 2a–3b–c ) =( 2a–3b–c ) ( m+1 )
  89. x 2 – 2 3 x+ 1 9 = ( x– 1 3 ) 2
  90. 4 a 2n – b 4n =( 2 a n + b 2n ) ( 2 a n – b 2n )
  91. 81 x 2 – ( a+x ) 2 =[ 9x+( a+x ) ] [ 9x–( a+x ) ] =[ 9x+a+x ] [ 9x–a–x ] =( 10x+a ) ( 8x–a )
  92. a 2 +9–6a–16 x 2 = a 2 –6a+9–16 x 2 = ( a–3 ) 2 –16 x 2 =[ ( a–3 ) +4x ] [ ( a–3 ) –4x ] =( a+4x–3 ) ( a–4x–3 )
  93. 9 a 2 – x 2 –4+4x =9 a 2 – x 2 +4x–4 =9 a 2 –( x 2 –4x+4 ) =9 a 2 – ( x–2 ) 2 =[ 3a+( x–2 ) ] [ 3a–( x–2 ) ] =( 3a+x–2 ) ( 3a–x+2 )
  94. 9 x 2 – y 2 +3x–y =( 3x–y ) ( 3x+y ) +( 3x–y ) =( 3x–y ) [ ( 3x+y ) +1 ] =( 3x–y ) ( 3x+y+1 )
  95. x 2 –x–72 = x 2 –9x+8x–72 . 72 36 18 9 1 | 2 2 2 9 =x( x–9 ) +8( x–9 ) =( x–9 ) ( x+8 )
  96. 36 a 4 –120 a 2 b 2 +49 b 4 =36 a 4 –120 a 2 b 2 +49 b 4 +36 a 2 b 2 –36 a 2 b 2 =36 a 4 –84 a 2 b 2 +49 b 4 –36 a 2 b 2 = ( 6 a 2 –7 b 2 ) 2 –36 a 2 b 2 =[ ( 6 a 2 –7 b 2 ) +6ab ] [ ( 6 a 2 –7 b 2 ) –6ab ] =( 6 a 2 +6ab–7 b 2 ) ( 6 a 2 –6ab–7 b 2 )
  97. a 2 – m 2 –9 n 2 –6mn+4ab+4 b 2 = a 2 +4ab+4 b 2 –( m 2 +6mn+9 n 2 ) = ( a+2b ) 2 – ( m+3n ) 2 =[ ( a+2b ) +( m+3n ) ] [ ( a+2b ) –( m+3n ) ] =( a+2b+m+3n ) ( a+2b–m–3n )
  98. 1– 4 9 a 8 =( 1– 2 3 a 4 ) ( 1+ 2 3 a 4 )
  99. 81 a 8 +64 b 12 =81 a 8 +64 b 12 +144 a 4 b 6 –144 a 4 b 6 =81 a 8 +144 a 4 b 6 +64 b 12 –144 a 4 b 6 = ( 9 a 4 +8 b 6 ) 2 –144 a 4 b 6 =[ ( 9 a 4 +8 b 6 ) –12 a 2 b 3 ] [ ( 9 a 4 +8 b 6 ) +12 a 2 b 3 ] =( 9 a 4 –12 a 2 b 3 +8 b 6 ) ( 9 a 4 +12 a 2 b 3 +8 b 6 )
  100. 49 x 2 –77x+30 =49 x 2 –42x–35x+30 . 1470 735 245 49 7 1 | 2 3 5 7 7 =7x( 7x–6 ) –5( 7x–6 ) =( 7x–6 ) ( 7x–5 )
  101. x 2 –2abx–35 a 2 b 2 = x 2 –7abx+5abx–35 a 2 b 2 . 35 7 1 | 5 7 =x( x–7ab ) +5ab( x–7ab ) =( x–7ab ) ( x+5ab )
  102. 125 x 3 –225 x 2 +135x–27= ( 5x–3 ) 3
  103. ( a–2 ) 2 – ( a+3 ) 2 =[ ( a–2 ) +( a+3 ) ] [ ( a–2 ) –( a+3 ) ] =[ a–2+a+3 ] [ a –2– a –3 ] =–5( 2a+1 )
  104. 4 a 2 m+12 a 2 n–5bm–15bn =4 a 2 ( m+3n ) –5b( m+3n ) =( m+3n ) ( 4 a 2 –5b )
  105. 1+6 x 3 +9 x 6 = ( 1+3 x 3 ) 2
  106. a 4 +3 a 2 b–40 b 2 = a 4 +8 a 2 b–5 a 2 b–40 b 2 . 40 20 10 5 1 | 2 2 2 5 = a 2 ( a 2 +8b ) –5b( a 2 +8b ) =( a 2 +8b ) ( a 2 –5b )
  107. m 3 +8 a 3 x 3 =( m+2ax ) ( m 2 –2amx+4 a 2 x 2 )
  108. 1–9 x 2 +24xy–16 y 2 =1–( 9 x 2 –24xy+16 y 2 ) =1– ( 3x–4y ) 2 =[ 1–( 3x–4y ) ] [ 1+( 3x–4y ) ] =( 1–3x+4y ) ( 1+3x–4y )
  109. 1+11x+24 x 2 =1+3x+8x+24 x 2 . 24 12 6 3 1 | 2 2 2 3 =( 1+3x ) +8x( 1+3x ) =( 1+3x ) ( 1+8x )
  110. 9 x 2 y 3 –27 x 3 y 3 –9 x 5 y 3 =9 x 2 y 3 ( 1–3x– x 2 )
  111. ( a 2 + b 2 – c 2 ) 2 –9 x 2 y 2 =[ ( a 2 + b 2 – c 2 ) –3xy ] [ ( a 2 + b 2 – c 2 ) +3xy ] =( a 2 + b 2 – c 2 –3xy ) ( a 2 + b 2 – c 2 +3xy )
  112. 8 ( a+1 ) 3 –1 =[ 2( a+1 ) –1 ] { [ 2( a+1 ) ] 2 +2( a+1 ) +1 } =[ 2a+2–1 ] { 4 ( a+1 ) 2 +2a+2+1 } =( 2a+1 ) { 4( a 2 +2a+1 ) +2a+3 } =( 2a+1 ) { 4 a 2 +8a+4+2a+3 } =( 2a+1 ) ( 4 a 2 +10a+7 )
  113. 100 x 4 y 6 –121 m 4 =( 10 x 2 y 3 –11 m 2 ) ( 10 x 2 y 3 +11 m 2 )
  114. ( a 2 +1 ) 2 +5( a 2 +1 ) –24 = ( a 2 +1 ) 2 –3( a 2 +1 ) +8( a 2 +1 ) –24 . 24 12 6 3 1 | 2 2 2 3 =( a 2 +1 ) [ ( a 2 +1 ) –3 ] +8[ ( a 2 +1 ) –3 ] =[ ( a 2 +1 ) –3 ] [ ( a 2 +1 ) +8 ] =[ a 2 +1–3 ] [ a 2 +1+8 ] =( a 2 –2 ) ( a 2 +9 )
  115. 1+1000 x 6 =( 1+10 x 2 ) ( 1–10 x 2 + [ 10 x 2 ] 2 ) =( 1+10 x 2 ) ( 1–10 x 2 +100 x 4 )
  116. 49 a 2 – x 2 –9 y 2 +6xy =49 a 2 – x 2 +6xy–9 y 2 =49 a 2 –( x 2 –6xy+9 y 2 ) =49 a 2 – ( x–3y ) 2 =[ 7a–( x–3y ) ] [ 7a+( x–3y ) ] =( 7a–x+3y ) ( 7a+x–3y )
  117. x 4 – y 2 +4 x 2 +4–4yz–4 z 2 = x 4 +4 x 2 +4– y 2 –4yz–4 z 2 =( x 4 +4 x 2 +4 ) –( y 2 +4yz+4 z 2 ) = ( x 2 +2 ) 2 – ( y+2z ) 2 =[ ( x 2 +2 ) –( y+2z ) ] [ ( x 2 +2 ) +( y+2z ) ] =[ x 2 +2–y–2z ] [ x 2 +2+y+2z ] =( x 2 –y–2z+2 ) ( x 2 +y+2z+2 )
  118. a 3 –64 =( a–4 ) ( a 2 +4a+ 4 2 ) =( a–4 ) ( a 2 +4a+16 )
  119. a 5 + x 5 =( a+x ) ( a 4 – a 3 x+ a 2 x 2 –a x 3 + x 4 )
  120. a 6 –3 a 3 b–54 b 2 = a 6 +6 a 3 b–9 a 3 b–54 b 2 . 54 27 9 1 | 2 3 9 = a 3 ( a 3 +6b ) –9b( a 3 +6b ) =( a 3 +6b ) ( a 3 –9b )
  121. 165+4x– x 2 =165–11x+15x– x 2 . 165 55 11 1 | 3 5 11 =11( 15–x ) +x( 15–x ) =( 15–x ) ( 11+x )
  122. a 4 + a 2 +1 = a 4 + a 2 +1+ a 2 – a 2 = a 4 +2 a 2 +1– a 2 = ( a 2 +1 ) 2 – a 2 =[ ( a 2 +1 ) –a ] [ ( a 2 +1 ) +a ] =( a 2 –a+1 ) ( a 2 +a+1 )
  123. x 2 4 – y 6 81 =( x 2 – y 3 9 ) ( x 2 + y 3 9 )
  124. 16 x 2 + 8xy 5 + y 2 25 = ( 4x+ y 5 ) 2
  125. a 4 b 4 +4 a 2 b 2 –96 = a 4 b 4 –8 a 2 b 2 +12 a 2 b 2 –96 . 96 48 24 12 1 | 2 2 2 12 = a 2 b 2 ( a 2 b 2 –8 ) +12( a 2 b 2 –8 ) =( a 2 b 2 –8 ) ( a 2 b 2 +12 )
  126. 8 a 2 x+7y+21by–7ay–8 a 3 x+24 a 2 bx =8 a 2 x–8 a 3 x+24 a 2 bx+7y–7ay+21by =8 a 2 x( 1–a+3b ) +7y( 1–a+3b ) =( 1–a+3b ) ( 8 a 2 x+7y )
  127. x 4 +11 x 2 –390 = x 4 –15 x 2 +26 x 2 –390 . 390 195 65 13 1 | 2 3 5 13 = x 2 ( x 2 –15 ) +26( x 2 –15 ) =( x 2 –15 ) ( x 2 +26 )
  128. 7+33m–10 m 2 =7–2m+35m–10 m 2 . 70 35 1 | 2 35 =( 7–2m ) +5m( 7–2m ) =( 7–2m ) ( 1+5m )
  129. 4 ( a+b ) 2 –9 ( c+d ) 2 =[ 2( a+b ) –3( c+d ) ] [ 2( a+b ) +3( c+d ) ] =( 2a+2b–3c–3d ) ( 2a+2b+3c+3d )
  130. 729–125 x 3 y 12 =( 9–5x y 4 ) [ 9 2 +9( 5x y 4 ) + ( 5x y 4 ) 2 ] =( 9–5x y 4 ) ( 81+45x y 4 +25 x 2 y 8 )
  131. ( x+y ) 2 +x+y = ( x+y ) 2 +( x+y ) =( x+y ) [ ( x+y ) +1 ] =( x+y ) ( x+y+1 )
  132. 4–( a 2 + b 2 ) +2ab =4– a 2 – b 2 +2ab =4– a 2 +2ab– b 2 =4–( a 2 –2ab+ b 2 ) =4– ( a–b ) 2 =[ 2–( a–b ) ] [ 2+( a–b ) ] =( 2–a+b ) ( 2+a–b )
  133. x 3 – y 3 +x–y =( x 3 – y 3 ) +x–y =( x–y ) ( x 2 +xy+ y 2 ) +( x–y ) =( x–y ) [ ( x 2 +xy+ y 2 ) +1 ] =( x–y ) ( x 2 +xy+ y 2 +1 )
  134. a 2 – b 2 + a 3 – b 3 =( a 2 – b 2 ) +( a 3 – b 3 ) =( a–b ) ( a+b ) +( a–b ) ( a 2 +ab+ b 2 ) =( a–b ) [ ( a+b ) +( a 2 +ab+ b 2 ) ] =( a–b ) ( a+b+ a 2 +ab+ b 2 )
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