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Ejercicio 107

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DESCARGA
CAPITULO X

Descomposición Factorial
Ejercicio 107
Descomponer en tres factores:
  1. 3a x 2 –3a =3a( x 2 –1 ) =3a( x–1 ) ( x+1 )
  2. 3 x 2 –3x–6 =3( x 2 –x–2 ) =3( x 2 +x–2x–2 ) =3[ x( x+1 ) –2( x+1 ) ] =3( x+1 ) ( x–2 )
  3. 2 a 2 x–4abx+2 b 2 x =2x( a 2 –2ab+ b 2 ) =2x ( a–b ) 2
  4. 2 a 3 –2 =2( a 3 –1 ) =2( a–1 ) ( a 2 +a+1 )
  5. a 3 –3 a 2 –28a =a( a 2 –3a–28 ) . 28 14 7 1 | 2 2 7 =a( a 2 –4a+7a–28 ) =a[ a( a–4 ) +7( a–4 ) ] =a( a–4 ) ( a+7 )
  6. x 3 –4x+ x 2 –4 =x( x 2 –4 ) +( x 2 –4 ) =( x 2 –4 ) ( x+1 ) =( x–2 ) ( x+2 ) ( x+1 )
  7. 3a x 3 +3a y 3 =3a( x 3 + y 3 ) =3a( x+y ) ( x 2 –xy+ y 2 )
  8. 4a b 2 –4abn+a n 2 =a( 4 b 2 –4bn+ n 2 ) =a ( 2b–n ) 2
  9. x 4 –3 x 2 –4 = x 4 + x 2 –4 x 2 –4 . 4 1 | 4 = x 2 ( x 2 +1 ) –4( x 2 +1 ) =( x 2 +1 ) ( x 2 –4 ) =( x 2 +1 ) ( x–2 ) ( x+2 )
  10. a 3 – a 2 –a+1 = a 2 ( a–1 ) –( a–1 ) =( a–1 ) ( a 2 –1 ) =( a–1 ) ( a–1 ) ( a+1 ) = ( a–1 ) 2 ( a+1 )
  11. 2a x 2 –4ax+2a =2a( x 2 –2x+1 ) =2a ( x–1 ) 2
  12. x 3 –x+ x 2 y–y =x( x 2 –1 ) +y( x 2 –1 ) =( x 2 –1 ) ( x+y ) =( x+1 ) ( x–1 ) ( x+y )
  13. 2 a 3 +6 a 2 –8a =2a( a 2 +3a–4 ) . 4 1 | 4 =2a( a 2 –a+4a–4 ) =2a[ a( a–1 ) +4( a–1 ) ] =2a( a–1 ) ( a+4 )
  14. 16 x 3 –48 x 2 y+36x y 2 =4x( 4 x 2 –12xy+9 y 2 ) =4x ( 2x–3y ) 2
  15. 3 x 3 – x 2 y–3x y 2 + y 3 =3 x 3 –3x y 2 – x 2 y+ y 3 =3x( x 2 – y 2 ) –y( x 2 – y 2 ) =( x 2 – y 2 ) ( 3x–y ) =( x+y ) ( x–y ) ( 3x–y )
  16. 5 a 4 +5a =5a( a 3 +1 ) =5a( a+1 ) ( a 2 –a+1 )
  17. 6a x 2 –ax–2a =a( 6 x 2 –x–2 ) . 12 6 3 1 | 2 2 3 =a( 6 x 2 +3x–4x–2 ) =a[ 3x( 2x+1 ) –2( 2x+1 ) ] =a( 2x+1 ) ( 3x–2 )
  18. n 4 –81 =( n 2 +9 ) ( n 2 –9 ) =( n 2 +9 ) ( n+3 ) ( n–3 )
  19. 8a x 2 –2a =2a( 4 x 2 –1 ) =2a( 2x–1 ) ( 2x+1 )
  20. a x 3 +10a x 2 +25ax =ax( x 2 +10x+25 ) =ax ( x+5 ) 2
  21. x 3 –6 x 2 –7x =x( x 2 –6x–7 ) . 7 1 | 7 =x( x 2 +x–7x–7 ) =x[ x( x+1 ) –7( x+1 ) ] =x( x+1 ) ( x–7 )
  22. m 3 +3 m 2 –16m–48 = m 2 ( m+3 ) –16( m+3 ) =( m+3 ) ( m 2 –16 ) =( m+3 ) ( m–4 ) ( m+4 )
  23. x 3 –6 x 2 y+12x y 2 –8 y 3 = ( x–2y ) 3 =( x–2y ) ( x–2y ) ( x–2y )
  24. ( a+b ) ( a 2 – b 2 ) –( a 2 – b 2 ) =( a 2 – b 2 ) [ ( a+b ) –1 ] =( a 2 – b 2 ) ( a+b–1 ) =( a–b ) ( a+b ) ( a+b–1 )
  25. 32 a 5 x–48 a 3 bx+18a b 2 x =2ax( 16 a 4 –24 a 2 b+9 b 2 ) =2ax ( 4 a 2 –3b ) 2
  26. x 4 – x 3 + x 2 –x = x 3 ( x–1 ) +x( x–1 ) =( x–1 ) ( x 3 +x ) =x( x–1 ) ( x 2 +1 )
  27. 4 x 2 +32x–36 =4( x 2 +8x–9 ) . 9 1 | 9 =4( x 2 –x+9x–9 ) =4[ x( x–1 ) +9( x–1 ) ] =4( x–1 ) ( x+9 )
  28. a 4 – ( a+2 ) 2 =[ a 2 –( a+2 ) ] [ a 2 +( a+2 ) ] =( a 2 –a–2 ) ( a 2 +a+2 )
  29. x 6 –25 x 3 –54 = x 6 –27 x 3 +2 x 3 –54 . 54 27 1 | 2 27 = x 3 ( x 3 –27 ) +2( x 3 –27 ) =( x 3 –27 ) ( x 3 +2 ) =( x–3 ) ( x 2 +3x+9 ) ( x 3 +2 )
  30. a 6 +a =a( a 5 +1 ) =a( a+1 ) ( a 4 – a 3 + a 2 –a+1 )
  31. a 3 b+2 a 2 bx+ab x 2 –ab y 2 =ab( a 2 +2ax+ x 2 – y 2 ) =ab[ ( a+x ) 2 – y 2 ] =ab[ ( a+x ) +y ] [ ( a+x ) –y ] =ab( a+x+y ) ( a+x–y )
  32. 3ab m 2 –3ab =3ab( m 2 –1 ) =3ab( m–1 ) ( m+1 )
  33. 81 x 4 y+3x y 4 =3xy( 27 x 3 + y 3 ) =3xy( 3x+y ) ( 9 x 2 –3xy+ y 2 )
  34. a 4 – a 3 +a–1 = a 3 ( a–1 ) +( a–1 ) =( a–1 ) ( a 3 +1 ) =( a–1 ) ( a+1 ) ( a 2 –a+1 )
  35. x–3 x 2 –18 x 3 =x( 1–3x–18 x 2 ) . 18 9 3 1 | 2 3 3 =x( 1–6x+3x–18 x 2 ) =x[ ( 1–6x ) +3x( 1–6x ) ] =x( 1–6x ) ( 1+3x )
  36. 6ax–2bx+6ab–2 b 2 =6ax+6ab–2bx–2 b 2 =6a( x+b ) –2b( x+b ) =( x+b ) ( 6a–2b ) =2( x+b ) ( 3a–b )
  37. a m 3 –7a m 2 +12am =am( m 2 –7m+12 ) . 12 6 3 1 | 2 2 3 =am( m 2 –4m–3m+12 ) =am[ m( m–4 ) –3( m–4 ) ] =am( m–4 ) ( m–3 )
  38. 4 a 2 x 3 –4 a 2 =4 a 2 ( x 3 –1 ) =4 a 2 ( x–1 ) ( x 2 +x+1 )
  39. 28 x 3 y–7x y 3 =7xy( 4 x 2 – y 2 ) =7xy( 2x–y ) ( 2x+y )
  40. 3ab x 2 –3abx–18ab =3ab( x 2 –x–6 ) . 6 3 1 | 2 3 =3ab( x 2 –3x+2x–6 ) =3ab[ x( x–3 ) +2( x–3 ) ] =3ab( x–3 ) ( x+2 )
  41. x 4 –8 x 2 –128 = x 4 –16 x 2 +8 x 2 –128 . 128 64 32 16 1 | 2 2 2 16 = x 2 ( x 2 –16 ) +8( x 2 –16 ) =( x 2 –16 ) ( x 2 +8 ) =( x+4 ) ( x–4 ) ( x 2 +8 )
  42. 18 x 2 y+60x y 2 +50 y 3 =2y( 9 x 2 +30xy+25 y 2 ) =2y ( 3x+5y ) 2
  43. ( x 2 –2xy ) ( a+1 ) + y 2 ( a+1 ) =( a+1 ) [ ( x 2 –2xy ) + y 2 ] =( a+1 ) [ x 2 –2xy+ y 2 ] =( a+1 ) ( x–y ) 2
  44. x 3 +2 x 2 y–3x y 2 =x( x 2 +2xy–3 y 2 ) . 3 1 | 3 =x( x 2 –xy+3xy–3 y 2 ) =x[ x( x–y ) +3y( x–y ) ] =x( x–y ) ( x+3y )
  45. a 2 x–4 b 2 x+2 a 2 y–8 b 2 y = a 2 x+2 a 2 y–4 b 2 x–8 b 2 y = a 2 ( x+2y ) –4 b 2 ( x+2y ) = a 2 ( x+2y ) ( a 2 –4 b 2 ) = a 2 ( x+2y ) ( a–2b ) ( a+2b )
  46. 45 a 2 x 4 –20 a 2 =5 a 2 ( 9 x 4 –4 ) =5 a 2 ( 3 x 2 –2 ) ( 3 x 2 +2 )
  47. a 4 – ( a–12 ) 2 =[ a 2 –( a–12 ) ] [ a 2 +( a–12 ) ] =( a 2 –a+12 ) ( a 2 +a–12 )
  48. b x 2 –b– x 2 +1 =b( x 2 –1 ) –( x 2 –1 ) =( x 2 –1 ) ( b–1 ) =( x–1 ) ( x+1 ) ( b–1 )
  49. 2 x 4 +6 x 3 –56 x 2 =2 x 2 ( x 2 +3x–28 ) . 28 14 7 1 | 2 2 7 =2 x 2 ( x 2 –4x+7x–28 ) =2 x 2 [ x( x–4 ) +7( x–4 ) ] =2 x 2 ( x–4 ) ( x+7 )
  50. 30 a 2 –55a–50 =5( 6 a 2 –11a–10 ) . 60 30 15 1 | 2 2 15 =5( 6 a 2 –15a+4a–10 ) =5[ 3a( 2a–5 ) +2( 2a–5 ) ] =5( 2a–5 ) ( 3a+2 )
  51. 9 ( x–y ) 3 –( x–y ) =( x–y ) [ 9 ( x–y ) 2 –1 ] =( x–y ) [ 3( x–y ) –1 ] [ 3( x–y ) +1 ] =( x–y ) ( 3x–3y–1 ) ( 3x–3y+1 )
  52. 6 a 2 x–9 a 3 –a x 2 =–a x 2 +6 a 2 x–9 a 3 =–a( x 2 –6ax+9 a 2 ) =–a ( x–3a ) 2 =–a( x–3a ) ( x–3a ) =a( 3a–x ) ( x–3a )
  53. 64a–125 a 4 =a( 64–125 a 3 ) =a( 4–5a ) ( 16+20a+25 a 2 )
  54. 70 x 4 +26 x 3 –24 x 2 =2 x 2 ( 35 x 2 +13x–12 ) . 420 210 105 35 7 1 | 2 2 3 5 7 =2 x 2 ( 35 x 2 –15x+28x–12 ) =2 x 2 [ 5x( 7x–3 ) +4( 7x–3 ) ] =2 x 2 ( 7x–3 ) ( 5x+4 )
  55. a 7 +6 a 5 –55 a 3 = a 3 ( a 4 +6 a 2 –55 ) . 55 11 1 | 5 11 = a 3 ( a 4 +11 a 2 –5 a 2 –55 ) = a 3 [ a 2 ( a 2 +11 ) –5( a 2 +11 ) ] = a 3 ( a 2 +11 ) ( a 2 –5 )
  56. 16 a 5 b–56 a 3 b 3 +49a b 5 =ab( 16 a 4 –56 a 2 b 2 +49 b 4 ) =ab ( 4 a 2 –7 b 2 ) 2
  57. 7 x 6 +32 a 2 x 4 –15 a 4 x 2 = x 2 ( 7 x 4 +32 a 2 x 2 –15 a 4 ) . 105 35 1 | 3 35 = x 2 ( 7 x 4 –3 a 2 x 2 +35 a 2 x 2 –15 a 4 ) = x 2 [ x 2 ( 7 x 2 –3 a 2 ) +5 a 2 ( 7 x 2 –3 a 2 ) ] = x 2 ( 7 x 2 –3 a 2 ) ( x 2 +5 a 2 )
  58. x 2m+2 – x 2 y 2n = x 2 ( x 2m – y 2n ) = x 2 ( x m – y n ) ( x m + y n )
  59. 2 x 4 +5 x 3 –54x–135 =2 x 4 –54x+5 x 3 –135 =2x( x 3 –27 ) +5( x 3 –27 ) =( x 3 –27 ) ( 2x+5 ) =( x–3 ) ( x 2 +3x+9 ) ( 2x+5 )
  60. a x 3 +a x 2 y+ax y 2 –2a x 2 –2axy–2a y 2 =ax( x 2 +xy+ y 2 ) –2a( x 2 +xy+ y 2 ) =( x 2 +xy+ y 2 ) ( ax–2a ) =a( x 2 +xy+ y 2 ) ( x–2 )
  61. ( x+y ) 4 –1 =[ ( x+y ) 2 –1 ] [ ( x+y ) 2 +1 ] =[ ( x+y ) –1 ] [ ( x+y ) +1 ] [ x 2 +2xy+ y 2 +1 ] =( x+y–1 ) ( x+y+1 ) ( x 2 +2xy+ y 2 +1 )
  62. 3 a 5 +3 a 3 +3a =3a( a 4 + a 2 +1 ) =3a( a 4 + a 2 +1+ a 2 – a 2 ) =3a( a 4 +2 a 2 +1– a 2 ) =3a{ ( a 2 +1 ) 2 – a 2 } =3a{ [ ( a 2 +1 ) –a ] [ ( a 2 +1 ) +a ] } =3a( a 2 –a+1 ) ( a 2 +a+1 )
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