Ejercicio 120

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CAPITULO XIII

Simplificación de fracciones
Simplificación de fracciones cuyos términos sean polinomios. Caso en que hay que cambiar el signo a uno o más factores
Ejercicio 120
Simplificar o reducir a su más simple expresión:
  1. 44x 6x6 = 4x4 6x6 = (x1 ) (x1 ) = 2 3
  2. a 2 b 2 b 2 a 2 = a 2 b 2 a 2 b 2 =1
  3. m 2 n 2 (nm ) 2 = n 2 m 2 (nm ) 2 = (nm )(n+m ) (nm ) 2 = n+m nm
  4. x 2 x12 16 x 2 = x 2 x12 x 2 16 = (x4 )(x+3 ) (x4 )(x+4 ) = x+3 x+4
  5. 3y6x 2mxmy2nx+ny = 3(y2x ) 2mx2nxmy+ny = 3(y2x ) 2x(mn ) y(mn ) = 3(y2x ) (mn ) (2xy ) = 3 (y2x ) (nm )(y2x ) = 3 nm
  6. 2 x 2 9x5 10+3x x 2 = 2 x 2 +x10x5 x 2 3x10 = x(2x+1 ) 5(2x+1 ) (x5 ) (x+2 ) = (2x+1 )(x5 ) (x5 )(x+2 ) = 2x+1 x+2
  7. 8 a 3 a 2 +2a8 = a 3 8 a 2 +2a8 = (a2 )( a 2 +2a+4 ) (a+4 )(a2 ) = a 2 +2a+4 a+4
  8. a 2 +a2 nanm+am = (a+2 ) (a1 ) nm(anam ) = (a+2 ) (a1 ) nma(nm ) = (a+2 ) (a1 ) (1a ) (nm ) = (a+2 )(a1 ) (a1 )(mn ) = a+2 mn
  9. 4 x 2 4xy+ y 2 5y10x = (2xy ) 2 5(y2x ) = (2xy ) 2 5 (2xy ) = 2xy 5
  10. 3mxnx3my+ny n y 2 n x 2 3m y 2 +3m x 2 = 3mx3my(nxny ) (n y 2 n x 2 ) (3m y 2 3m x 2 ) = 3m(xy ) n(xy ) n( y 2 x 2 ) 3m( y 2 x 2 ) = (3mn ) (xy ) ( y 2 x 2 ) (n3m ) = (3mn )(xy ) ( x 2 y 2 )(3mn ) = xy (xy )(x+y ) = 1 x+y
  11. 96x+ x 2 x 2 7x+12 = (3x ) 2 (x4 ) (x3 ) = (3x ) 2 (4x )(3x ) = 3x 4x
  12. a 2 b 2 b 3 a 3 = (ab ) (a+b ) (ba ) ( b 2 +ab+ a 2 ) = (ba )(a+b ) (ba )( b 2 +ab+ a 2 ) = a+b b 2 +ab+ a 2
  13. 3ax3bx6a+6b 2b2abx+ax = 3x(ab ) 6(ab ) 2(ba ) x(ba ) = (3x6 ) (ab ) (ba ) (2x ) = 3 (x2 ) (ab ) (ab ) (x2 ) =3
  14. a 2 x 2 x 2 ax3x+3a = (ax ) (a+x ) x(xa ) 3(xa ) = (ax ) (a+x ) (xa ) (x3 ) = (ax )(a+x ) (ax )(3x ) = a+x 3x
  15. 3bx6x 8 b 3 = 3x(b2 ) (2b ) (4+2b+ b 2 ) = 3x (b2 ) (b2 )(4+2b+ b 2 ) = 3x 4+2b+ b 2
  16. (1a ) 3 a1 = (1a ) 1a = (1a ) 2
  17. 2 x 3 2 x 2 y2x y 2 3 y 3 +3x y 2 3 x 2 y = 2x( x 2 xy y 2 ) 3y( y 2 +xy x 2 ) = 2x ( x 2 xy y 2 ) 3y ( x 2 xy y 2 ) = 2x 3y
  18. (ab ) 3 (ba ) 2 = (ab ) 3 (ab ) 2 =ab
  19. 2 x 2 22x+60 753 x 2 = 2( x 2 11x+30 ) 3(25 x 2 ) = 2(x6 ) (x5 ) 3(5x ) (5+x ) = 2(6x )(5x ) 3 (5x )(5+x ) = 2(6x ) 3(5+x )
  20. 6a n 2 3 b 2 n 2 b 4 4a b 2 +4 a 2 = 3 n 2 (2a b 2 ) ( b 2 2a ) 2 = 3 n 2 ( b 2 2a ) ( b 2 2a ) 2 = 3 n 2 2a b 2
  21. (xy ) 2 z 2 (y+z ) 2 x 2 = [(xy ) z ] [(xy ) +z ] [(y+z ) x ] [(y+z ) +x ] = (xyz ) (xy+z ) (y+zx ) (y+z+x ) = (y+zx )(yxz ) (y+zx )(y+z+x ) = yxz y+z+x
  22. 3 a 2 3ab bdadbc+ac = 3a(ab ) d(ba ) c(ba ) = 3a(ab ) (dc ) (ba ) = 3a (ab ) (cd )(ab ) = 3a cd
  23. (x5 ) 3 125 x 3 = (x5 ) 3 x 3 125 = (x5 ) (x5 )( x 2 +5x+25 ) = (x5 ) 2 x 2 +5x+25
  24. 13x66 x 2 6 x 2 13x+6 = 6 x 2 13x+6 6 x 2 13x+6 =1
  25. 2 x 3 2x y 2 + x 2 y 2 2x y 2 + y 2 2 x 3 x 2 = 2x( x 2 y 2 ) +( x 2 y 2 ) y 2 (2x+1 ) x 2 (2x+1 ) = (2x+1 )( x 2 y 2 ) ( y 2 x 2 )(2x+1 ) = x 2 y 2 x 2 y 2 =1
  26. 30 x 2 y45x y 2 20 x 3 8 x 3 +27 y 3 = 5x(6xy9 y 2 4 x 2 ) (2x+3y ) (4 x 2 6xy+9 y 2 ) = 5x (4 x 2 6xy+9 y 2 ) (2x+3y )(4 x 2 6xy+9 y 2 ) = 5x 2x+3y
  27. n+1 n 3 n 2 n 3 n2 n 2 +2 = n+1 n 2 (n+1 ) n( n 2 1 ) 2( n 2 1 ) = (1 n 2 ) (n+1 ) ( n 2 1 ) (n2 ) = ( n 2 1 )(n+1 ) ( n 2 1 )(2n ) = n+1 2n
  28. (x2 ) 2 ( x 2 +x12 ) (2x ) (3x ) 2 = (2x ) 2 (x+4 ) (x3 ) (2x ) (3x ) 2 = (x2 ) (x+4 )(3x ) (3x ) 2 = (x2 ) (x+4 ) 3x
  29. 5 x 3 15 x 2 y 90 x 3 y 2 10 x 5 = 5 x 2 (x3y ) x 3 (9 y 2 x 2 ) = x3y 2x(3yx ) (3y+x ) = x3y 2x (x3y )(3y+x ) = 1 2x(3y+x )
  30. ( x 2 1 ) ( x 2 8x+16 ) ( x 2 4x ) (1 x 2 ) = ( x 2 1 )( x 2 8x+16 ) (4x x 2 )( x 2 1 ) = (x4 ) 2 x(4x ) = (x4 ) 2 x (x4 ) = x4 x = 4x x