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CAPITULO X

Descomposición Factorial
Ejercicio 107
Descomponer en tres factores:
  1. 3a x 2 3a =3a( x 2 1 ) =3a(x1 ) (x+1 )
  2. 3 x 2 3x6 =3( x 2 x2 ) =3( x 2 +x2x2 ) =3[x(x+1 ) 2(x+1 ) ] =3(x+1 ) (x2 )
  3. 2 a 2 x4abx+2 b 2 x =2x( a 2 2ab+ b 2 ) =2x (ab ) 2
  4. 2 a 3 2 =2( a 3 1 ) =2(a1 ) ( a 2 +a+1 )
  5. a 3 3 a 2 28a =a( a 2 3a28 ) . 28 14 7 1 | 2 2 7 =a( a 2 4a+7a28 ) =a[a(a4 ) +7(a4 ) ] =a(a4 ) (a+7 )
  6. x 3 4x+ x 2 4 =x( x 2 4 ) +( x 2 4 ) =( x 2 4 ) (x+1 ) =(x2 ) (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 (2bn ) 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 ) (x2 ) (x+2 )
  10. a 3 a 2 a+1 = a 2 (a1 ) (a1 ) =(a1 ) ( a 2 1 ) =(a1 ) (a1 ) (a+1 ) = (a1 ) 2 (a+1 )
  11. 2a x 2 4ax+2a =2a( x 2 2x+1 ) =2a (x1 ) 2
  12. x 3 x+ x 2 yy =x( x 2 1 ) +y( x 2 1 ) =( x 2 1 ) (x+y ) =(x+1 ) (x1 ) (x+y )
  13. 2 a 3 +6 a 2 8a =2a( a 2 +3a4 ) . 4 1 | 4 =2a( a 2 a+4a4 ) =2a[a(a1 ) +4(a1 ) ] =2a(a1 ) (a+4 )
  14. 16 x 3 48 x 2 y+36x y 2 =4x(4 x 2 12xy+9 y 2 ) =4x (2x3y ) 2
  15. 3 x 3 x 2 y3x 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 ) (3xy ) =(x+y ) (xy ) (3xy )
  16. 5 a 4 +5a =5a( a 3 +1 ) =5a(a+1 ) ( a 2 a+1 )
  17. 6a x 2 ax2a =a(6 x 2 x2 ) . 12 6 3 1 | 2 2 3 =a(6 x 2 +3x4x2 ) =a[3x(2x+1 ) 2(2x+1 ) ] =a(2x+1 ) (3x2 )
  18. n 4 81 =( n 2 +9 ) ( n 2 9 ) =( n 2 +9 ) (n+3 ) (n3 )
  19. 8a x 2 2a =2a(4 x 2 1 ) =2a(2x1 ) (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 6x7 ) . 7 1 | 7 =x( x 2 +x7x7 ) =x[x(x+1 ) 7(x+1 ) ] =x(x+1 ) (x7 )
  22. m 3 +3 m 2 16m48 = m 2 (m+3 ) 16(m+3 ) =(m+3 ) ( m 2 16 ) =(m+3 ) (m4 ) (m+4 )
  23. x 3 6 x 2 y+12x y 2 8 y 3 = (x2y ) 3 =(x2y ) (x2y ) (x2y )
  24. (a+b ) ( a 2 b 2 ) ( a 2 b 2 ) =( a 2 b 2 ) [(a+b ) 1 ] =( a 2 b 2 ) (a+b1 ) =(ab ) (a+b ) (a+b1 )
  25. 32 a 5 x48 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 (x1 ) +x(x1 ) =(x1 ) ( x 3 +x ) =x(x1 ) ( x 2 +1 )
  27. 4 x 2 +32x36 =4( x 2 +8x9 ) . 9 1 | 9 =4( x 2 x+9x9 ) =4[x(x1 ) +9(x1 ) ] =4(x1 ) (x+9 )
  28. a 4 (a+2 ) 2 =[ a 2 (a+2 ) ] [ a 2 +(a+2 ) ] =( a 2 a2 ) ( 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 ) =(x3 ) ( 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+xy )
  32. 3ab m 2 3ab =3ab( m 2 1 ) =3ab(m1 ) (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 +a1 = a 3 (a1 ) +(a1 ) =(a1 ) ( a 3 +1 ) =(a1 ) (a+1 ) ( a 2 a+1 )
  35. x3 x 2 18 x 3 =x(13x18 x 2 ) . 18 9 3 1 | 2 3 3 =x(16x+3x18 x 2 ) =x[(16x ) +3x(16x ) ] =x(16x ) (1+3x )
  36. 6ax2bx+6ab2 b 2 =6ax+6ab2bx2 b 2 =6a(x+b ) 2b(x+b ) =(x+b ) (6a2b ) =2(x+b ) (3ab )
  37. a m 3 7a m 2 +12am =am( m 2 7m+12 ) . 12 6 3 1 | 2 2 3 =am( m 2 4m3m+12 ) =am[m(m4 ) 3(m4 ) ] =am(m4 ) (m3 )
  38. 4 a 2 x 3 4 a 2 =4 a 2 ( x 3 1 ) =4 a 2 (x1 ) ( x 2 +x+1 )
  39. 28 x 3 y7x y 3 =7xy(4 x 2 y 2 ) =7xy(2xy ) (2x+y )
  40. 3ab x 2 3abx18ab =3ab( x 2 x6 ) . 6 3 1 | 2 3 =3ab( x 2 3x+2x6 ) =3ab[x(x3 ) +2(x3 ) ] =3ab(x3 ) (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 ) (x4 ) ( 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 ) (xy ) 2
  44. x 3 +2 x 2 y3x y 2 =x( x 2 +2xy3 y 2 ) . 3 1 | 3 =x( x 2 xy+3xy3 y 2 ) =x[x(xy ) +3y(xy ) ] =x(xy ) (x+3y )
  45. a 2 x4 b 2 x+2 a 2 y8 b 2 y = a 2 x+2 a 2 y4 b 2 x8 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 ) (a2b ) (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 (a12 ) 2 =[ a 2 (a12 ) ] [ a 2 +(a12 ) ] =( a 2 a+12 ) ( a 2 +a12 )
  48. b x 2 b x 2 +1 =b( x 2 1 ) ( x 2 1 ) =( x 2 1 ) (b1 ) =(x1 ) (x+1 ) (b1 )
  49. 2 x 4 +6 x 3 56 x 2 =2 x 2 ( x 2 +3x28 ) . 28 14 7 1 | 2 2 7 =2 x 2 ( x 2 4x+7x28 ) =2 x 2 [x(x4 ) +7(x4 ) ] =2 x 2 (x4 ) (x+7 )
  50. 30 a 2 55a50 =5(6 a 2 11a10 ) . 60 30 15 1 | 2 2 15 =5(6 a 2 15a+4a10 ) =5[3a(2a5 ) +2(2a5 ) ] =5(2a5 ) (3a+2 )
  51. 9 (xy ) 3 (xy ) =(xy ) [9 (xy ) 2 1 ] =(xy ) [3(xy ) 1 ] [3(xy ) +1 ] =(xy ) (3x3y1 ) (3x3y+1 )
  52. 6 a 2 x9 a 3 a x 2 =a x 2 +6 a 2 x9 a 3 =a( x 2 6ax+9 a 2 ) =a (x3a ) 2 =a(x3a ) (x3a ) =a(3ax ) (x3a )
  53. 64a125 a 4 =a(64125 a 3 ) =a(45a ) (16+20a+25 a 2 )
  54. 70 x 4 +26 x 3 24 x 2 =2 x 2 (35 x 2 +13x12 ) . 420 210 105 35 7 1 | 2 2 3 5 7 =2 x 2 (35 x 2 15x+28x12 ) =2 x 2 [5x(7x3 ) +4(7x3 ) ] =2 x 2 (7x3 ) (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 b56 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 54x135 =2 x 4 54x+5 x 3 135 =2x( x 3 27 ) +5( x 3 27 ) =( x 3 27 ) (2x+5 ) =(x3 ) ( x 2 +3x+9 ) (2x+5 )
  60. a x 3 +a x 2 y+ax y 2 2a x 2 2axy2a y 2 =ax( x 2 +xy+ y 2 ) 2a( x 2 +xy+ y 2 ) =( x 2 +xy+ y 2 ) (ax2a ) =a( x 2 +xy+ y 2 ) (x2 )
  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+y1 ) (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 )