Ejercicio 119

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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 y3x y 2 = xy 3 xy (xy ) = 1 3(xy )
  3. 2ax+4bx 3ay+6by = 2x (a+2b ) 3y (a+2b ) = 2x 3y
  4. x 2 2x3 x3 = (x3 )(x+1 ) x3 =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 (ab ) = b 3 c 8(ab )
  6. x 2 4 5ax+10a = (x2 )(x+2 ) 5a (x+2 ) = x2 5a
  7. 3 x 2 4x15 x 2 5x+6 = 3 x 2 9x+5x15 (x3 ) (x2 ) = 3x(x3 ) +5(x3 ) (x3 ) (x2 ) = (x3 )(3x+5 ) (x3 )(x2 ) = 3x+5 x2
  8. 15 a 2 bn45 a 2 bm 10 a 2 b 2 n30 a 2 b 2 m = a 2 b (n3m ) a 2 b 2 (n3m ) = 3 2b
  9. x 2 y 2 x 2 +2xy+ y 2 = (x+y )(xy ) (x+y ) 2 = xy x+y
  10. 3 x 2 y+15xy x 2 25 = 3xy (x+5 ) (x5 )(x+5 ) = 3xy x5
  11. a 2 4ab+4 b 2 a 3 8 b 3 = (a2b ) 2 (a2b )( a 2 +2ab+4 b 2 ) = a2b a 2 +2ab+4 b 2
  12. x 3 +4 x 2 21x x 3 9x = x ( x 2 +4x21 ) x ( x 2 9 ) = x 2 +4x21 x 2 9 = (x+7 )(x3 ) (x3 )(x+3 ) = x+7 x+3
  13. 6 x 2 +5x6 15 x 2 7x2 = 6 x 2 +9x4x6 15 x 2 10x+3x2 = 3x(2x+3 ) 2(2x+3 ) 5x(3x2 ) +(3x2 ) = (3x2 )(2x+3 ) (3x2 )(5x+1 ) = 2x+3 5x+1
  14. a 3 +1 a 4 a 3 +a1 = (a+1 ) ( a 2 a+1 ) a 3 (a1 ) +(a1 ) = (a+1 ) ( a 2 a+1 ) (a1 )( a 3 +1 ) = 1 a1
  15. 2ax+ay4bx2by ax4a2bx+8b = a(2x+y ) 2b(2x+y ) a(x4 ) 2b(x4 ) = (a2b )(2x+y ) (a2b )(x4 ) = 2x+y x4
  16. a 2 ab6 b 2 a 3 x6 a 2 bx+9a b 2 x = (a3b ) (a+2b ) ax( a 2 6ab+9 b 2 ) = (a3b )(a+2b ) ax (a3b ) 2 = a+2b ax(a3b )
  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. (mn ) 2 m 2 n 2 = (mn ) 2 (m+n )(mn ) = mn m+n
  20. (ax ) 3 a 3 x 3 = (ax ) (ax )( a 2 +ax+ x 2 ) = (ax ) 2 a 2 +ax+ x 2
  21. a 2 a20 a 2 7a+10 = (a5 )(a+4 ) (a5 )(a2 ) = a+4 a2
  22. (1 a 2 ) 2 a 2 +2a+1 = [(1a ) (1+a ) ] 2 (a+1 ) 2 = (1a ) 2 (1+a ) 2 (1+a ) 2 = (1a ) 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 )(xy ) (xy )( 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 5n6 = n( n 2 1 ) (n6 ) (n+1 ) = n(n1 )(n+1 ) (n6 )(n+1 ) = n(n1 ) n6
  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 (bc ) 2 (a+b ) 2 c 2 = [a(bc ) ] [a+(bc ) ] [(a+b ) +c ] [(a+b ) c ] = (ab+c )(a+bc ) (a+b+c )(a+bc ) = ab+c a+b+c
  29. (a+b ) 2 (cd ) 2 (a+c ) 2 (bd ) 2 = [(a+b ) +(cd ) ] [(a+b ) (cd ) ] [(a+c ) +(bd ) ] [(a+c ) (bd ) ] = (a+b+cd )(a+bc+d ) (a+c+bd )(a+cb+d ) = a+bc+d ab+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 (a2b ) x.3 a (a2b ) = 4a 3x
  33. x 3 6 x 2 x 2 12x+36 = x 2 (x6 ) (x6 ) 2 = x 2 x6
  34. (x4y ) 2 x 5 64 x 2 y 3 = (x4y ) 2 x 2 ( x 3 64 y 3 ) = (x4y ) 2 x 2 (x4y )( x 2 +4xy+16 y 2 ) = x4y 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 ) (m3 )( m 2 +3m+9 ) = mn m3
  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 8a20 = 4 a 4 16 a 2 + a 2 4 (a10 ) (a+2 ) = 4 a 2 ( a 2 4 ) +( a 2 4 ) (a10 ) (a+2 ) = ( a 2 4 ) (4 a 2 +1 ) (a10 ) (a+2 ) = (a+2 )(a2 ) (4 a 2 +1 ) (a10 )(a+2 ) = (a2 ) (4 a 2 +1 ) a10
  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 +an30a = a 2 ( n 2 36 ) a ( n 2 +n30 ) = a (n+6 )(n6 ) (n+6 )(n5 ) = a(n6 ) n5
  43. 3 m 2 +5mn8 n 2 m 3 n 3 = 3 m 2 3mn+8mn8 n 2 (mn ) ( m 2 +mn+ n 2 ) = 3m(mn ) +8n(mn ) (mn ) ( m 2 +mn+ n 2 ) = (mn )(3m+8n ) (mn )( m 2 +mn+ n 2 ) = 3m+8n m 2 +mn+ n 2
  44. 15 a 3 b18 a 2 b 20 a 2 b 2 24a b 2 = 3 a 2 b (5a6 ) 4 a b 2 (5a6 ) = 3a 4b
  45. 9 x 2 24x+16 9 x 4 16 x 2 = (3x4 ) 2 x 2 (9 x 2 16 ) = (3x4 ) 2 x 2 (3x+4 )(3x4 ) = 3x4 x 2 (3x+4 )
  46. 16 a 2 x25x 12 a 3 7 a 2 10a = x(16 a 2 25 ) a(12 a 2 7a10 ) = x(4x5 ) (4x+5 ) a(12 a 2 +8a15a10 ) = x(4x5 ) (4x+5 ) a[4a(3a+2 ) 5(3a+2 ) ] = x (4x5 )(4x+5 ) a (4x5 )(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 ) = (2xy )(4 x 2 +2xy+ y 2 ) x (2xy ) 2 = 4 x 2 +2xy+ y 2 x(2xy )
  48. 3an4a6bn+8b 6 n 2 5n4 = a(3n4 ) 2b(3n4 ) 6 n 2 +3n8n4 = (3n4 ) (a2b ) 3n(2n+1 ) 4(2n+1 ) = (3n4 )(a2b ) (3n4 )(2n+1 ) = a2b 2n+1
  49. x 4 49 x 2 x 3 +2 x 2 63x = x 2 ( x 2 49 ) x ( x 2 +2x63 ) = x(x+7 )(x7 ) (x+9 )(x7 ) = x(x+7 ) x+9
  50. x 4 +x x 3 yy x 3 x x 2 y+y = x( x 3 +1 ) y( x 3 +1 ) x( x 2 1 ) y( x 2 1 ) = (xy )( x 3 +1 ) (xy )( x 2 1 ) = (x+1 )( x 2 x+1 ) (x+1 )(x1 ) = x 2 x+1 x1
  51. 2 x 3 +6 x 2 x3 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 m4am+ a 3 n4an a 4 4 a 3 12 a 2 = am( a 2 4 ) +an( a 2 4 ) a 2 ( a 2 4a12 ) = ( a 2 4 ) (am+an ) a 2 (a6 ) (a+2 ) = a (a2 )(a+2 )(m+n ) a 2 (a6 )(a+2 ) = (a2 ) (m+n ) a(a6 )
  53. 4 a 2 (x3 ) 2 (2a+x ) 2 9 = [2a(x3 ) ] [2a+(x3 ) ] [(2a+x ) 3 ] [(2a+x ) +3 ] = (2ax+3 )(2a+x3 ) (2a+x3 )(2a+x+3 ) = 2ax+3 2a+x+3
  54. mam+nan 13a+3 a 2 a 3 = m(1a ) +n(1a ) (1 a 3 ) 3a(1a ) = (1a ) (m+n ) (1a ) (1+a+ a 2 ) 3a(1a ) = (1a )(m+n ) (1a )[(1+a+ a 2 ) 3a ] = m+n 12a+ a 2 = m+n (1a ) 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 (1a ) (1a ) ( a 2 +1 ) a( a 2 +1 ) = (1a )( a 2 1 ) ( a 2 +1 )(1a ) = 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 +3xy2xy 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 )(3xy ) = 4 x 2 2xy+ y 2 +6xy 3xy = 4 x 2 +4xy+ y 2 3xy = (2x+y ) 2 3xy
  58. 8 n 3 125 2520n+4 n 2 = (2n5 ) (4 n 2 +10n+25 ) 4 n 2 20n+25 = (2n5 )(4 n 2 +10n+25 ) (2n5 ) 2 = 4 n 2 +10n+25 2n5
  59. 6x x 2 15+2x x 2 = ( x 2 +x6 ) ( x 2 2x15 ) = x 2 +x6 x 2 2x15 = (x+3 )(x2 ) (x5 )(x+3 ) = x2 x5
  60. 3+2x8 x 2 4+5x6 x 2 = (8 x 2 2x3 ) (6 x 2 5x4 ) = 8 x 2 2x3 6 x 2 5x4 = 8 x 2 +4x6x3 6 x 2 +3x8x4 = 4x(2x+1 ) 3(2x+1 ) 3x(2x+1 ) 4(2x+1 ) = (2x+1 )(4x3 ) (2x+1 )(3x4 ) = 4x3 3x4
  61. m 2 n 2 +3mn10 44mn+ m 2 n 2 = (mn+5 ) (mn2 ) m 2 n 2 4mn+4 = (mn+5 )(mn2 ) (mn2 ) 2 = mn+5 mn2
  62. x 3 + x 2 y4 b 2 x4 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 ) (x2b ) 2 = (x+y ) (x+2b )(x2b ) (x2b ) 2 = (x+y ) (x+2b ) x2b
  63. x 6 + x 3 2 x 4 x 3 yx+y = ( x 3 +2 ) ( x 3 1 ) x 3 (xy ) (xy ) = ( x 3 +2 )( x 3 1 ) (xy )( x 3 1 ) = x 3 +2 xy
  64. ( x 2 x2 ) ( x 2 9 ) ( x 2 2x3 ) ( x 2 +x6 ) = (x2 ) (x+1 ) (x3 ) (x+3 ) (x3 ) (x+1 ) (x+3 ) (x2 ) =1
  65. ( a 2 4a+4 ) (4 a 2 4a+1 ) ( a 2 +a6 ) (2 a 2 5a+2 ) = (a2 ) 2 (2a1 ) 2 (a+3 )(a2 )(2 a 2 4aa+2 ) = (a2 ) (2a1 ) 2 (a+3 ) [2a(a2 ) (a2 ) ] = (a2 ) (2a1 ) 2 (a+3 )(a2 ) (2a1 ) = 2a1 a+3
  66. ( x 3 3x ) ( x 3 1 ) ( x 4 + x 3 + x 2 ) ( x 2 1 ) = x ( x 2 3 )(x1 ) ( x 2 +x+1 ) x 2 ( x 2 +x+1 )(x+1 )(x1 ) = x 2 3 x(x+1 )
  67. (4 n 2 +4n3 ) ( n 2 +7n30 ) (2 n 2 7n+3 ) (4 n 2 +12n+9 ) = (4 n 2 2n+6n3 ) (n+10 ) (n3 ) (2 n 2 n6n+3 ) (2n+3 ) 2 = [2n(2n1 ) +3(2n1 ) ] (n+10 ) (n3 ) [n(2n1 ) 3(2n1 ) ] (2n+3 ) 2 = (2n1 ) (2n+3 )(n+10 )(n3 ) (2n1 ) (n3 ) (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 8x12 = x 3 +2 x 2 + x 2 4 x 3 4x+ x 2 4x12 = ( x 3 +2 x 2 ) +( x 2 4 ) ( x 3 4x ) +( x 2 4x12 ) = x 2 (x+2 ) +(x+2 ) (x2 ) x( x 2 4 ) +(x6 ) (x+2 ) = (x+2 ) [ x 2 +(x2 ) ] x(x+2 ) (x2 ) +(x6 ) (x+2 ) = (x+2 )( x 2 +x2 ) (x+2 )[x(x2 ) +(x6 ) ] = (x+2 ) (x1 ) x 2 2x+x6 = (x+2 ) (x1 ) x 2 x6 = (x+2 )(x1 ) (x3 )(x+2 ) = x1 x3
  70. x 3 x 2 8x+12 x 4 2 x 3 7 x 2 +20x12 = ( x 3 4x ) ( x 2 +4x12 ) ( x 4 2 x 3 ) (7 x 2 20x+12 ) = x( x 2 4 ) (x+6 ) (x2 ) x 3 (x2 ) (7 x 2 14x6x+12 ) = x(x2 ) (x+2 ) (x+6 ) (x2 ) x 3 (x2 ) [7x(x2 ) 6(x2 ) ] = (x2 ) [x(x+2 ) (x+6 ) ] x 3 (x2 ) (x2 ) (7x6 ) = (x2 )( x 2 +2xx6 ) (x2 )[ x 3 (7x6 ) ] = x 2 +x6 x 3 7x+6 = (x+3 ) (x2 ) x 3 x6x+6 = (x+3 ) (x2 ) x( x 2 1 ) 6(x1 ) = (x+3 ) (x2 ) x(x1 ) (x+1 ) 6(x1 ) = (x+3 ) (x2 ) (x1 ) [x(x+1 ) 6 ] = (x+3 ) (x2 ) (x1 )( x 2 +x6 ) = 1 x1
  71. x 4 7 x 2 2x+8 x 4 2 x 3 9 x 2 +10x+24 = x 4 x7 x 2 x+8 x 4 9 x 2 2 x 3 +10x+24 = ( x 4 x ) (7 x 2 +x8 ) ( x 4 9 x 2 ) (2 x 3 10x24 ) = x( x 3 1 ) (7 x 2 7x+8x8 ) x 2 ( x 2 9 ) 2( x 3 5x12 ) aplicoelmétododelasraices = x(x1 ) ( x 2 +x+1 ) [7x(x1 ) +8(x1 ) ] x 2 (x3 ) (x+3 ) 2(x3 ) ( x 2 +3x+4 ) = x(x1 ) ( x 2 +x+1 ) (x1 ) (7x+8 ) (x3 ) [ x 2 (x+3 ) 2( x 2 +3x+4 ) ] = (x1 ) [x( x 2 +x+1 ) (7x+8 ) ] (x3 ) [ x 3 +3 x 2 2 x 2 6x8 ] = (x1 ) [ x 3 + x 2 +x7x8 ] (x3 ) [ x 3 + x 2 6x8 ] = (x1 ) ( x 3 + x 2 6x8 ) (x3 ) (x+2 ) ( x 2 x4 ) = (x1 ) [( x 3 4x ) +( x 2 2x8 ) ] (x3 ) (x+2 ) ( x 2 x4 ) = (x1 ) [x( x 2 4 ) +(x4 ) (x+2 ) ] (x3 ) (x+2 ) ( x 2 x4 ) = (x1 ) [x(x+2 ) (x2 ) +(x4 ) (x+2 ) ] (x3 ) (x+2 ) ( x 2 x4 ) = (x1 )(x+2 )[x(x2 ) +(x4 ) ] (x3 )(x+2 )( x 2 x4 ) = (x1 ) ( x 2 2x+x4 ) (x3 ) ( x 2 x4 ) = (x1 )( x 2 x4 ) (x3 )( x 2 x4 ) = x1 x3
  72. a 5 a 3 a 2 +1 a 5 2 a 4 6 a 3 +8 a 2 +5a6 = ( a 5 a 3 ) ( a 2 1 ) ( a 5 +5a6 ) (2 a 4 +6 a 3 8 a 2 ) = a 3 ( a 2 1 ) ( a 2 1 ) (a1 ) ( a 4 + a 3 + a 2 +a+6 ) 2 a 2 ( a 2 +3a4 ) = ( a 2 1 ) ( a 3 1 ) (a1 ) ( a 4 + a 3 + a 2 +a+6 ) 2 a 2 (a+4 ) (a1 ) = (a1 )(a+1 ) (a1 ) ( a 2 +a+1 ) (a1 )[( a 4 + a 3 + a 2 +a+6 ) 2 a 2 (a+4 ) ] = (a+1 ) (a1 ) ( a 2 +a+1 ) a 4 + a 3 + a 2 +a+62 a 3 8 a 2 = (a+1 ) (a1 ) ( a 2 +a+1 ) a 4 a 3 7 a 2 +a+6 = (a+1 ) (a1 ) ( a 2 +a+1 ) ( a 4 a 3 ) (7 a 2 a6 ) = (a+1 ) (a1 ) ( a 2 +a+1 ) a 3 (a1 ) (7 a 2 7a+6a6 ) = (a+1 ) (a1 ) ( a 2 +a+1 ) a 3 (a1 ) [7a(a1 ) +6(a1 ) ] = (a+1 ) (a1 ) ( a 2 +a+1 ) a 3 (a1 ) (a1 ) (7a+6 ) = (a+1 )(a1 )( a 2 +a+1 ) (a1 )[ a 3 (7a+6 ) ] = (a+1 ) ( a 2 +a+1 ) a 3 7a6 = (a+1 ) ( a 2 +a+1 ) a 3 a6a6 = (a+1 ) ( a 2 +a+1 ) a( a 2 1 ) 6(a+1 ) = (a+1 ) ( a 2 +a+1 ) a(a+1 ) (a1 ) 6(a+1 ) = (a+1 )( a 2 +a+1 ) (a+1 )[a(a1 ) 6 ] = a 2 +a+1 a 3 a6 = a 2 +a+1 (a3 ) (a+2 )