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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 )
Categories: Capítulo X