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Ejercicio 119

Gestor Baldor
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DESCARGA
CAPITULO XIII

Fracciones algebraicas
Simplificación de fracciones cuyos términos sean polinomios
Ejercicio 119
Simplificar o reducir a su más simple expresión:
  1. 3ab 2 a 2 x+2 a 3 = 3 a b 2 a 2 ( x+a ) = 3b 2a( x+a )
  2. xy 3 x 2 y–3x y 2 = xy 3 xy ( x–y ) = 1 3( x–y )
  3. 2ax+4bx 3ay+6by = 2x ( a+2b ) 3y ( a+2b ) = 2x 3y
  4. x 2 –2x–3 x–3 = ( x–3 ) ( x+1 ) x–3 =x+1
  5. 10 a 2 b 3 c 80( a 3 – a 2 b ) = 1 0 a 2 b 3 c 8 0 a 2 ( a–b ) = b 3 c 8( a–b )
  6. x 2 –4 5ax+10a = ( x–2 ) ( x+2 ) 5a ( x+2 ) = x–2 5a
  7. 3 x 2 –4x–15 x 2 –5x+6 = 3 x 2 –9x+5x–15 ( x–3 ) ( x–2 ) = 3x( x–3 ) +5( x–3 ) ( x–3 ) ( x–2 ) = ( x–3 ) ( 3x+5 ) ( x–3 ) ( x–2 ) = 3x+5 x–2
  8. 15 a 2 bn–45 a 2 bm 10 a 2 b 2 n–30 a 2 b 2 m = a 2 b ( n–3m ) a 2 b 2 ( n–3m ) = 3 2b
  9. x 2 – y 2 x 2 +2xy+ y 2 = ( x+y ) ( x–y ) ( x+y ) 2 = x–y x+y
  10. 3 x 2 y+15xy x 2 –25 = 3xy ( x+5 ) ( x–5 ) ( x+5 ) = 3xy x–5
  11. a 2 –4ab+4 b 2 a 3 –8 b 3 = ( a–2b ) 2 ( a–2b ) ( a 2 +2ab+4 b 2 ) = a–2b a 2 +2ab+4 b 2
  12. x 3 +4 x 2 –21x x 3 –9x = x ( x 2 +4x–21 ) x ( x 2 –9 ) = x 2 +4x–21 x 2 –9 = ( x+7 ) ( x–3 ) ( x–3 ) ( x+3 ) = x+7 x+3
  13. 6 x 2 +5x–6 15 x 2 –7x–2 = 6 x 2 +9x–4x–6 15 x 2 –10x+3x–2 = 3x( 2x+3 ) –2( 2x+3 ) 5x( 3x–2 ) +( 3x–2 ) = ( 3x–2 ) ( 2x+3 ) ( 3x–2 ) ( 5x+1 ) = 2x+3 5x+1
  14. a 3 +1 a 4 – a 3 +a–1 = ( a+1 ) ( a 2 –a+1 ) a 3 ( a–1 ) +( a–1 ) = ( a+1 ) ( a 2 –a+1 ) ( a–1 ) ( a 3 +1 ) = 1 a–1
  15. 2ax+ay–4bx–2by ax–4a–2bx+8b = a( 2x+y ) –2b( 2x+y ) a( x–4 ) –2b( x–4 ) = ( a–2b ) ( 2x+y ) ( a–2b ) ( x–4 ) = 2x+y x–4
  16. a 2 –ab–6 b 2 a 3 x–6 a 2 bx+9a b 2 x = ( a–3b ) ( a+2b ) ax( a 2 –6ab+9 b 2 ) = ( a–3b ) ( a+2b ) ax ( a–3b ) 2 = a+2b ax( a–3b )
  17. m 2 + n 2 m 4 – n 4 = m 2 + n 2 ( m 2 – n 2 ) ( m 2 + n 2 ) = 1 m 2 – n 2
  18. x 3 + y 3 ( x+y ) 3 = ( x+y ) ( x 2 –xy+ y 2 ) ( x+y ) = x 2 –xy+ y 2 ( x+y ) 2
  19. ( m–n ) 2 m 2 – n 2 = ( m–n ) 2 ( m+n ) ( m–n ) = m–n m+n
  20. ( a–x ) 3 a 3 – x 3 = ( a–x ) ( a–x ) ( a 2 +ax+ x 2 ) = ( a–x ) 2 a 2 +ax+ x 2
  21. a 2 –a–20 a 2 –7a+10 = ( a–5 ) ( a+4 ) ( a–5 ) ( a–2 ) = a+4 a–2
  22. ( 1– a 2 ) 2 a 2 +2a+1 = [ ( 1–a ) ( 1+a ) ] 2 ( a+1 ) 2 = ( 1–a ) 2 ( 1+a ) 2 ( 1+a ) 2 = ( 1–a ) 2
  23. a 4 b 2 – a 2 b 4 a 4 – b 4 = a 2 b 2 ( a 2 – b 2 ) ( a 2 + b 2 ) ( a 2 – b 2 ) = a 2 b 2 a 2 + b 2
  24. x 2 – y 2 x 3 – y 3 = ( x+y ) ( x–y ) ( x–y ) ( x 2 +xy+ y 2 ) = x+y x 2 +xy+ y 2
  25. 24 a 3 b+8 a 2 b 2 36 a 4 +24 a 3 b+4 a 2 b 2 = a 2 b( 3a+b ) 4 a 2 ( 9 a 2 +6ab+ b 2 ) = 2b ( 3a+b ) ( 3a+b ) 2 = 2b 3a+b
  26. n 3 –n n 2 –5n–6 = n( n 2 –1 ) ( n–6 ) ( n+1 ) = n( n–1 ) ( n+1 ) ( n–6 ) ( n+1 ) = n( n–1 ) n–6
  27. 8 n 3 +1 8 n 3 –4 n 2 +2n = ( 2n+1 ) ( 4 n 2 –2n+1 ) 2n ( 4 n 2 –2n+1 ) = 2n+1 2n
  28. a 2 – ( b–c ) 2 ( a+b ) 2 – c 2 = [ a–( b–c ) ] [ a+( b–c ) ] [ ( a+b ) +c ] [ ( a+b ) –c ] = ( a–b+c ) ( a+b–c ) ( a+b+c ) ( a+b–c ) = a–b+c a+b+c
  29. ( a+b ) 2 – ( c–d ) 2 ( a+c ) 2 – ( b–d ) 2 = [ ( a+b ) +( c–d ) ] [ ( a+b ) –( c–d ) ] [ ( a+c ) +( b–d ) ] [ ( a+c ) –( b–d ) ] = ( a+b+c–d ) ( a+b–c+d ) ( a+c+b–d ) ( a+c–b+d ) = a+b–c+d a–b+c+d
  30. 3 x 3 +9 x 2 x 2 +6x+9 = 3 x 2 ( x+3 ) ( x+3 ) 2 = 3 x 2 x+3
  31. 10 a 2 ( a 3 + b 3 ) 6 a 4 –6 a 3 b+6 a 2 b 2 = a 2 ( a+b ) ( a 2 –ab+ b 2 ) a 2 ( a 2 –ab+ b 2 ) = 5( a+b ) 3
  32. a( 4 a 2 –8ab ) x( 3 a 2 –6ab ) = a.4 a ( a–2b ) x.3 a ( a–2b ) = 4a 3x
  33. x 3 –6 x 2 x 2 –12x+36 = x 2 ( x–6 ) ( x–6 ) 2 = x 2 x–6
  34. ( x–4y ) 2 x 5 –64 x 2 y 3 = ( x–4y ) 2 x 2 ( x 3 –64 y 3 ) = ( x–4y ) 2 x 2 ( x–4y ) ( x 2 +4xy+16 y 2 ) = x–4y x 2 ( x 2 +4xy+16 y 2 )
  35. x 3 –3x y 2 x 4 –6 x 2 y 2 +9 y 4 = x ( x 2 –3 y 2 ) ( x 2 –3 y 2 ) 2 = x x 2 –3 y 2
  36. m 3 n+3 m 2 n+9mn m 3 –27 = mn ( m 2 +3m+9 ) ( m–3 ) ( m 2 +3m+9 ) = mn m–3
  37. x 4 –8 x 2 +15 x 4 –9 = ( x 2 –5 ) ( x 2 –3 ) ( x 2 –3 ) ( x 2 +3 ) = x 2 –5 x 2 +3
  38. a 4 +6 a 2 –7 a 4 +8 a 2 –9 = ( a 2 +7 ) ( a 2 –1 ) ( a 2 +9 ) ( a 2 –1 ) = a 2 +7 a 2 +9
  39. 3 x 2 +19x+20 6 x 2 +17x+12 = 3 x 2 +15x+4x+20 6 x 2 +9x+8x+12 = 3x( x+5 ) +4( x+5 ) 3x( 2x+3 ) +4( 2x+3 ) = ( x+5 ) ( 3x+4 ) ( 3x+4 ) ( 2x+3 ) = x+5 2x+3
  40. 4 a 4 –15 a 2 –4 a 2 –8a–20 = 4 a 4 –16 a 2 + a 2 –4 ( a–10 ) ( a+2 ) = 4 a 2 ( a 2 –4 ) +( a 2 –4 ) ( a–10 ) ( a+2 ) = ( a 2 –4 ) ( 4 a 2 +1 ) ( a–10 ) ( a+2 ) = ( a+2 ) ( a–2 ) ( 4 a 2 +1 ) ( a–10 ) ( a+2 ) = ( a–2 ) ( 4 a 2 +1 ) a–10
  41. 125a+ a 4 2 a 3 +20 a 2 +50a = a ( 125+ a 3 ) 2 a ( a 2 +10a+25 ) = ( a+5 ) ( a 2 –5a+25 ) 2 ( a+5 ) 2 = a 2 –5a+25 2( a+5 )
  42. a 2 n 2 –36 a 2 a n 2 +an–30a = a 2 ( n 2 –36 ) a ( n 2 +n–30 ) = a ( n+6 ) ( n–6 ) ( n+6 ) ( n–5 ) = a( n–6 ) n–5
  43. 3 m 2 +5mn–8 n 2 m 3 – n 3 = 3 m 2 –3mn+8mn–8 n 2 ( m–n ) ( m 2 +mn+ n 2 ) = 3m( m–n ) +8n( m–n ) ( m–n ) ( m 2 +mn+ n 2 ) = ( m–n ) ( 3m+8n ) ( m–n ) ( m 2 +mn+ n 2 ) = 3m+8n m 2 +mn+ n 2
  44. 15 a 3 b–18 a 2 b 20 a 2 b 2 –24a b 2 = 3 a 2 b ( 5a–6 ) 4 a b 2 ( 5a–6 ) = 3a 4b
  45. 9 x 2 –24x+16 9 x 4 –16 x 2 = ( 3x–4 ) 2 x 2 ( 9 x 2 –16 ) = ( 3x–4 ) 2 x 2 ( 3x+4 ) ( 3x–4 ) = 3x–4 x 2 ( 3x+4 )
  46. 16 a 2 x–25x 12 a 3 –7 a 2 –10a = x( 16 a 2 –25 ) a( 12 a 2 –7a–10 ) = x( 4x–5 ) ( 4x+5 ) a( 12 a 2 +8a–15a–10 ) = x( 4x–5 ) ( 4x+5 ) a[ 4a( 3a+2 ) –5( 3a+2 ) ] = x ( 4x–5 ) ( 4x+5 ) a ( 4x–5 ) ( 3a+2 ) = x( 4x+5 ) a( 3a+2 )
  47. 8 x 4 –x y 3 4 x 4 –4 x 3 y+ x 2 y 2 = x ( 8 x 3 – y 3 ) x 2 ( 4 x 2 –4xy+ y 2 ) = ( 2x–y ) ( 4 x 2 +2xy+ y 2 ) x ( 2x–y ) 2 = 4 x 2 +2xy+ y 2 x( 2x–y )
  48. 3an–4a–6bn+8b 6 n 2 –5n–4 = a( 3n–4 ) –2b( 3n–4 ) 6 n 2 +3n–8n–4 = ( 3n–4 ) ( a–2b ) 3n( 2n+1 ) –4( 2n+1 ) = ( 3n–4 ) ( a–2b ) ( 3n–4 ) ( 2n+1 ) = a–2b 2n+1
  49. x 4 –49 x 2 x 3 +2 x 2 –63x = x 2 ( x 2 –49 ) x ( x 2 +2x–63 ) = x( x+7 ) ( x–7 ) ( x+9 ) ( x–7 ) = x( x+7 ) x+9
  50. x 4 +x– x 3 y–y x 3 –x– x 2 y+y = x( x 3 +1 ) –y( x 3 +1 ) x( x 2 –1 ) –y( x 2 –1 ) = ( x–y ) ( x 3 +1 ) ( x–y ) ( x 2 –1 ) = ( x+1 ) ( x 2 –x+1 ) ( x+1 ) ( x–1 ) = x 2 –x+1 x–1
  51. 2 x 3 +6 x 2 –x–3 x 3 +3 x 2 +x+3 = 2 x 2 ( x+3 ) –( x+3 ) x 2 ( x+3 ) +( x+3 ) = ( x+3 ) ( 2 x 2 –1 ) ( x+3 ) ( x 2 +1 ) = 2 x 2 –1 x 2 +1
  52. a 3 m–4am+ a 3 n–4an a 4 –4 a 3 –12 a 2 = am( a 2 –4 ) +an( a 2 –4 ) a 2 ( a 2 –4a–12 ) = ( a 2 –4 ) ( am+an ) a 2 ( a–6 ) ( a+2 ) = a ( a–2 ) ( a+2 ) ( m+n ) a 2 ( a–6 ) ( a+2 ) = ( a–2 ) ( m+n ) a( a–6 )
  53. 4 a 2 – ( x–3 ) 2 ( 2a+x ) 2 –9 = [ 2a–( x–3 ) ] [ 2a+( x–3 ) ] [ ( 2a+x ) –3 ] [ ( 2a+x ) +3 ] = ( 2a–x+3 ) ( 2a+x–3 ) ( 2a+x–3 ) ( 2a+x+3 ) = 2a–x+3 2a+x+3
  54. m–am+n–an 1–3a+3 a 2 – a 3 = m( 1–a ) +n( 1–a ) ( 1– a 3 ) –3a( 1–a ) = ( 1–a ) ( m+n ) ( 1–a ) ( 1+a+ a 2 ) –3a( 1–a ) = ( 1–a ) ( m+n ) ( 1–a ) [ ( 1+a+ a 2 ) –3a ] = m+n 1–2a+ a 2 = m+n ( 1–a ) 2
  55. 6 x 2 +3 42 x 5 –9 x 3 –15x = 3 ( 2 x 2 +1 ) 3 x( 14 x 4 –3 x 2 –5 ) = 2 x 2 +1 x( 14 x 4 +7 x 2 –10 x 2 –5 ) = 2 x 2 +1 x[ 7 x 2 ( 2 x 2 +1 ) –5( 2 x 2 +1 ) ] = 2 x 2 +1 x ( 2 x 2 +1 ) ( 7 x 2 –5 ) = 1 x( 7 x 2 –5 )
  56. a 2 – a 3 –1+a a 2 +1– a 3 –a = a 2 ( 1–a ) –( 1–a ) ( a 2 +1 ) –a( a 2 +1 ) = ( 1–a ) ( a 2 –1 ) ( a 2 +1 ) ( 1–a ) = a 2 –1 a 2 +1
  57. 8 x 3 +12 x 2 y+6x y 2 + y 3 6 x 2 +xy– y 2 = ( 8 x 3 + y 3 ) +6xy( 2x+y ) 6 x 2 +3xy–2xy– y 2 = ( 2x+y ) ( 4 x 2 –2xy+ y 2 ) +6xy( 2x+y ) 3x( 2x+y ) –y( 2x+y ) = ( 2x+y ) [ ( 4 x 2 –2xy+ y 2 ) +6xy ] ( 2x+y ) ( 3x–y ) = 4 x 2 –2xy+ y 2 +6xy 3x–y = 4 x 2 +4xy+ y 2 3x–y = ( 2x+y ) 2 3x–y
  58. 8 n 3 –125 25–20n+4 n 2 = ( 2n–5 ) ( 4 n 2 +10n+25 ) 4 n 2 –20n+25 = ( 2n–5 ) ( 4 n 2 +10n+25 ) ( 2n–5 ) 2 = 4 n 2 +10n+25 2n–5
  59. 6–x– x 2 15+2x– x 2 = –( x 2 +x–6 ) –( x 2 –2x–15 ) = x 2 +x–6 x 2 –2x–15 = ( x+3 ) ( x–2 ) ( x–5 ) ( x+3 ) = x–2 x–5
  60. 3+2x–8 x 2 4+5x–6 x 2 = –( 8 x 2 –2x–3 ) –( 6 x 2 –5x–4 ) = 8 x 2 –2x–3 6 x 2 –5x–4 = 8 x 2 +4x–6x–3 6 x 2 +3x–8x–4 = 4x( 2x+1 ) –3( 2x+1 ) 3x( 2x+1 ) –4( 2x+1 ) = ( 2x+1 ) ( 4x–3 ) ( 2x+1 ) ( 3x–4 ) = 4x–3 3x–4
  61. m 2 n 2 +3mn–10 4–4mn+ m 2 n 2 = ( mn+5 ) ( mn–2 ) m 2 n 2 –4mn+4 = ( mn+5 ) ( mn–2 ) ( mn–2 ) 2 = mn+5 mn–2
  62. x 3 + x 2 y–4 b 2 x–4 b 2 y 4 b 2 –4bx+ x 2 = x 2 ( x+y ) –4 b 2 ( x+y ) x 2 –4bx+4 b 2 = ( x+y ) ( x 2 –4 b 2 ) ( x–2b ) 2 = ( x+y ) ( x+2b ) ( x–2b ) ( x–2b ) 2 = ( x+y ) ( x+2b ) x–2b
  63. x 6 + x 3 –2 x 4 – x 3 y–x+y = ( x 3 +2 ) ( x 3 –1 ) x 3 ( x–y ) –( x–y ) = ( x 3 +2 ) ( x 3 –1 ) ( x–y ) ( x 3 –1 ) = x 3 +2 x–y
  64. ( x 2 –x–2 ) ( x 2 –9 ) ( x 2 –2x–3 ) ( x 2 +x–6 ) = ( x–2 ) ( x+1 ) ( x–3 ) ( x+3 ) ( x–3 ) ( x+1 ) ( x+3 ) ( x–2 ) =1
  65. ( a 2 –4a+4 ) ( 4 a 2 –4a+1 ) ( a 2 +a–6 ) ( 2 a 2 –5a+2 ) = ( a–2 ) 2 ( 2a–1 ) 2 ( a+3 ) ( a–2 ) ( 2 a 2 –4a–a+2 ) = ( a–2 ) ( 2a–1 ) 2 ( a+3 ) [ 2a( a–2 ) –( a–2 ) ] = ( a–2 ) ( 2a–1 ) 2 ( a+3 ) ( a–2 ) ( 2a–1 ) = 2a–1 a+3
  66. ( x 3 –3x ) ( x 3 –1 ) ( x 4 + x 3 + x 2 ) ( x 2 –1 ) = x ( x 2 –3 ) ( x–1 ) ( x 2 +x+1 ) x 2 ( x 2 +x+1 ) ( x+1 ) ( x–1 ) = x 2 –3 x( x+1 )
  67. ( 4 n 2 +4n–3 ) ( n 2 +7n–30 ) ( 2 n 2 –7n+3 ) ( 4 n 2 +12n+9 ) = ( 4 n 2 –2n+6n–3 ) ( n+10 ) ( n–3 ) ( 2 n 2 –n–6n+3 ) ( 2n+3 ) 2 = [ 2n( 2n–1 ) +3( 2n–1 ) ] ( n+10 ) ( n–3 ) [ n( 2n–1 ) –3( 2n–1 ) ] ( 2n+3 ) 2 = ( 2n–1 ) ( 2n+3 ) ( n+10 ) ( n–3 ) ( 2n–1 ) ( n–3 ) ( 2n+3 ) 2 = n+10 2n+3
  68. ( x 6 – y 6 ) ( x+y ) ( x 3 – y 3 ) ( x 3 + x 2 y+x y 2 + y 3 ) = ( x 3 – y 3 ) ( x 3 + y 3 ) ( x+y ) ( x 3 – y 3 ) [ x 2 ( x+y ) + y 2 ( x+y ) ] = ( x 3 + y 3 ) ( x+y ) ( x 2 + y 2 ) ( x+y ) = x 3 + y 3 x 2 + y 2
  69. x 3 +3 x 2 –4 x 3 + x 2 –8x–12 = x 3 +2 x 2 + x 2 –4 x 3 –4x+ x 2 –4x–12 = ( x 3 +2 x 2 ) +( x 2 –4 ) ( x 3 –4x ) +( x 2 –4x–12 ) = x 2 ( x+2 ) +( x+2 ) ( x–2 ) x( x 2 –4 ) +( x–6 ) ( x+2 ) = ( x+2 ) [ x 2 +( x–2 ) ] x( x+2 ) ( x–2 ) +( x–6 ) ( x+2 ) = ( x+2 ) ( x 2 +x–2 ) ( x+2 ) [ x( x–2 ) +( x–6 ) ] = ( x+2 ) ( x–1 ) x 2 –2x+x–6 = ( x+2 ) ( x–1 ) x 2 –x–6 = ( x+2 ) ( x–1 ) ( x–3 ) ( x+2 ) = x–1 x–3
  70. x 3 – x 2 –8x+12 x 4 –2 x 3 –7 x 2 +20x–12 = ( x 3 –4x ) –( x 2 +4x–12 ) ( x 4 –2 x 3 ) –( 7 x 2 –20x+12 ) = x( x 2 –4 ) –( x+6 ) ( x–2 ) x 3 ( x–2 ) –( 7 x 2 –14x–6x+12 ) = x( x–2 ) ( x+2 ) –( x+6 ) ( x–2 ) x 3 ( x–2 ) –[ 7x( x–2 ) –6( x–2 ) ] = ( x–2 ) [ x( x+2 ) –( x+6 ) ] x 3 ( x–2 ) –( x–2 ) ( 7x–6 ) = ( x–2 ) ( x 2 +2x–x–6 ) ( x–2 ) [ x 3 –( 7x–6 ) ] = x 2 +x–6 x 3 –7x+6 = ( x+3 ) ( x–2 ) x 3 –x–6x+6 = ( x+3 ) ( x–2 ) x( x 2 –1 ) –6( x–1 ) = ( x+3 ) ( x–2 ) x( x–1 ) ( x+1 ) –6( x–1 ) = ( x+3 ) ( x–2 ) ( x–1 ) [ x( x+1 ) –6 ] = ( x+3 ) ( x–2 ) ( x–1 ) ( x 2 +x–6 ) = 1 x–1
  71. x 4 –7 x 2 –2x+8 x 4 –2 x 3 –9 x 2 +10x+24 = x 4 –x–7 x 2 –x+8 x 4 –9 x 2 –2 x 3 +10x+24 = ( x 4 –x ) –( 7 x 2 +x–8 ) ( x 4 –9 x 2 ) –( 2 x 3 –10x–24 ) = x( x 3 –1 ) –( 7 x 2 –7x+8x–8 ) x 2 ( x 2 –9 ) –2( x 3 –5x–12 ) ⇒ aplicoelmétododelasraices = x( x–1 ) ( x 2 +x+1 ) –[ 7x( x–1 ) +8( x–1 ) ] x 2 ( x–3 ) ( x+3 ) –2( x–3 ) ( x 2 +3x+4 ) = x( x–1 ) ( x 2 +x+1 ) –( x–1 ) ( 7x+8 ) ( x–3 ) [ x 2 ( x+3 ) –2( x 2 +3x+4 ) ] = ( x–1 ) [ x( x 2 +x+1 ) –( 7x+8 ) ] ( x–3 ) [ x 3 +3 x 2 –2 x 2 –6x–8 ] = ( x–1 ) [ x 3 + x 2 +x–7x–8 ] ( x–3 ) [ x 3 + x 2 –6x–8 ] = ( x–1 ) ( x 3 + x 2 –6x–8 ) ( x–3 ) ( x+2 ) ( x 2 –x–4 ) = ( x–1 ) [ ( x 3 –4x ) +( x 2 –2x–8 ) ] ( x–3 ) ( x+2 ) ( x 2 –x–4 ) = ( x–1 ) [ x( x 2 –4 ) +( x–4 ) ( x+2 ) ] ( x–3 ) ( x+2 ) ( x 2 –x–4 ) = ( x–1 ) [ x( x+2 ) ( x–2 ) +( x–4 ) ( x+2 ) ] ( x–3 ) ( x+2 ) ( x 2 –x–4 ) = ( x–1 ) ( x+2 ) [ x( x–2 ) +( x–4 ) ] ( x–3 ) ( x+2 ) ( x 2 –x–4 ) = ( x–1 ) ( x 2 –2x+x–4 ) ( x–3 ) ( x 2 –x–4 ) = ( x–1 ) ( x 2 –x–4 ) ( x–3 ) ( x 2 –x–4 ) = x–1 x–3
  72. a 5 – a 3 – a 2 +1 a 5 –2 a 4 –6 a 3 +8 a 2 +5a–6 = ( a 5 – a 3 ) –( a 2 –1 ) ( a 5 +5a–6 ) –( 2 a 4 +6 a 3 –8 a 2 ) = a 3 ( a 2 –1 ) –( a 2 –1 ) ( a–1 ) ( a 4 + a 3 + a 2 +a+6 ) –2 a 2 ( a 2 +3a–4 ) = ( a 2 –1 ) ( a 3 –1 ) ( a–1 ) ( a 4 + a 3 + a 2 +a+6 ) –2 a 2 ( a+4 ) ( a–1 ) = ( a–1 ) ( a+1 ) ( a–1 ) ( a 2 +a+1 ) ( a–1 ) [ ( a 4 + a 3 + a 2 +a+6 ) –2 a 2 ( a+4 ) ] = ( a+1 ) ( a–1 ) ( a 2 +a+1 ) a 4 + a 3 + a 2 +a+6–2 a 3 –8 a 2 = ( a+1 ) ( a–1 ) ( a 2 +a+1 ) a 4 – a 3 –7 a 2 +a+6 = ( a+1 ) ( a–1 ) ( a 2 +a+1 ) ( a 4 – a 3 ) –( 7 a 2 –a–6 ) = ( a+1 ) ( a–1 ) ( a 2 +a+1 ) a 3 ( a–1 ) –( 7 a 2 –7a+6a–6 ) = ( a+1 ) ( a–1 ) ( a 2 +a+1 ) a 3 ( a–1 ) –[ 7a( a–1 ) +6( a–1 ) ] = ( a+1 ) ( a–1 ) ( a 2 +a+1 ) a 3 ( a–1 ) –( a–1 ) ( 7a+6 ) = ( a+1 ) ( a–1 ) ( a 2 +a+1 ) ( a–1 ) [ a 3 –( 7a+6 ) ] = ( a+1 ) ( a 2 +a+1 ) a 3 –7a–6 = ( a+1 ) ( a 2 +a+1 ) a 3 –a–6a–6 = ( a+1 ) ( a 2 +a+1 ) a( a 2 –1 ) –6( a+1 ) = ( a+1 ) ( a 2 +a+1 ) a( a+1 ) ( a–1 ) –6( a+1 ) = ( a+1 ) ( a 2 +a+1 ) ( a+1 ) [ a( a–1 ) –6 ] = a 2 +a+1 a 3 –a–6 = a 2 +a+1 ( a–3 ) ( a+2 )
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