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DESCARGA
CAPITULO XIV

Operaciones con Fracciones
Ejercicio 129

  1. De 1 x–4 restar 1 x–3
    1 x–4 – 1 x–3 = x–3–(x–4 ) (x–4 ) (x–3 ) = x –3– x +4 (x–4 ) (x–3 ) = 1 (x–4 ) (x–3 )
  2. De m–n m+n restar m+n m–n
    m–n m+n – m+n m–n = (m–n ) 2 – (m+n ) 2 (m+n ) (m–n ) = [(m–n ) +(m+n ) ] [(m–n ) –(m+n ) ] (m+n ) (m–n ) = [m– n +m+ n ] [ m –n– m –n ] m 2 – n 2 = 2m(–2n ) m 2 – n 2 = –4mn m 2 – n 2 = 4mn n 2 – m 2
  3. De 1–x 1+x restar 1+x 1–x
    1–x 1+x – 1+x 1–x = (1–x ) 2 – (1+x ) 2 (1–x ) (1+x ) = [(1–x ) +(1+x ) ] [(1–x ) –(1+x ) ] (1–x ) (1+x ) = [1– x +1+ x ] [ 1 –x– 1 –x ] 1– x 2 = 2(–2x ) 1– x 2 = –4x 1– x 2 = 4x x 2 –1
  4. De a+b a 2 +ab restar b–a ab+ b 2
    a+b a 2 +ab – b–a ab+ b 2 = a+b a(a+b ) – b–a b(a+b ) = b(a+b ) –a(b–a ) ab(a+b ) = ab + b 2 – ab + a 2 ab(a+b ) = a 2 + b 2 ab(a+b )
  5. De m+n m–n restar m 2 + n 2 m 2 – n 2
    m+n m–n – m 2 + n 2 m 2 – n 2 = m+n m–n – m 2 + n 2 (m+n ) (m–n ) = (m+n ) 2 –( m 2 + n 2 ) (m+n ) (m–n ) = m 2 +2mn+ n 2 – m 2 – n 2 (m+n ) (m–n ) = 2mn m 2 – n 2
  6. Restar 1 x– x 2 de 1 x+ x 2
    1 x+ x 2 – 1 x– x 2 = 1 x(1+x ) – 1 x(1–x ) = 1 x [ 1 1+x – 1 1–x ] = 1 x [ 1–x–(1+x ) (1+x ) (1–x ) ] = 1 x [ 1 –x– 1 –x (1+x ) (1–x ) ] = 1 x –2 x (1+x ) (1–x ) = –2 1– x 2 = 2 x 2 –1
  7. Restar x a 2 – x 2 de a+x (a–x ) 2
    a+x (a–x ) 2 – x a 2 – x 2 = a+x (a–x ) 2 – x (a–x ) (a+x ) = 1 a–x [ a+x (a–x ) – x a+x ] = 1 a–x [ (a+x ) 2 –x(a–x ) (a–x ) (a+x ) ] = 1 a–x [ a 2 +2ax+ x 2 –ax+ x 2 (a–x ) (a+x ) ] = 1 a–x [ a 2 +ax+2 x 2 (a–x ) (a+x ) ] = a 2 +ax+2 x 2 (a–x ) 2 (a+x )
  8. Restar 1 12a+6 de a+1 6a+3
    a+1 6a+3 – 1 12a+6 = a+1 3(2a+1 ) – 1 6(2a+1 ) = 1 3 [ a+1 2a+1 – 1 2(2a+1 ) ] = 1 3 [ 2(a+1 ) –1 2(2a+1 ) ] = 1 3 [ 2a+2–1 2(2a+1 ) ] = 2a+1 6(2a+1 )
  9. Restar a+3 a 2 +a–12 de a–4 a 2 –6a+9
    a–4 a 2 –6a+9 – a+3 a 2 +a–12 = a–4 (a–3 ) 2 – a+3 (a+4 ) (a–3 ) = 1 (a–3 ) [ a–4 a–3 – a+3 a+4 ] = 1 (a–3 ) [ (a–4 ) (a+4 ) –(a–3 ) (a+3 ) (a+4 ) (a–3 ) ] = 1 (a–3 ) [ a 2 –16–( a 2 –9 ) (a+4 ) (a–3 ) ] = 1 (a–3 ) [ a 2 –16– a 2 +9 (a+4 ) (a–3 ) ] = 1 (a–3 ) [ –7 (a+4 ) (a–3 ) ] = –7 (a+4 ) (a–3 ) 2
  10. Restar b a+3b de a 2 +4ab–3 b 2 a 2 –9 b 2
    a 2 +4ab–3 b 2 a 2 –9 b 2 – b a+3b = a 2 +4ab–3 b 2 (a+3b ) (a–3b ) – b a+3b = a 2 +4ab–3 b 2 –b(a–3b ) (a+3b ) (a–3b ) = a 2 +4ab– 3 b 2 –ab+ 3 b 2 (a+3b ) (a–3b ) = a 2 +3ab (a+3b ) (a–3b ) = a (a+3b ) (a+3b )(a–3b ) = a a–3b
    Simplificar:
  11. x x 2 –1 – x+1 (x–1 ) 2 = x (x–1 ) (x+1 ) – x+1 (x–1 ) 2 = 1 x–1 [ x x+1 – x+1 x–1 ] = 1 x–1 [ x(x–1 ) – (x+1 ) 2 (x–1 ) (x+1 ) ] = 1 x–1 [ x 2 –x–( x 2 +2x+1 ) (x–1 ) (x+1 ) ] = 1 x–1 [ x 2 –x– x 2 –2x–1 (x–1 ) (x+1 ) ] = 1 x–1 [ –3x–1 (x–1 ) (x+1 ) ] =– 3x+1 (x–1 ) 2 (x+1 )
  12. 1 a 3 – b 3 – 1 (a–b ) 3 = 1 (a–b ) ( a 2 +ab+ b 2 ) – 1 (a–b ) 3 = 1 a–b [ 1 a 2 +ab+ b 2 – 1 (a–b ) 2 ] = 1 a–b [ (a–b ) 2 –( a 2 +ab+ b 2 ) (a–b ) 2 ( a 2 +ab+ b 2 ) ] = 1 a–b [ a 2 –2ab+ b 2 – a 2 –ab– b 2 (a–b ) 2 ( a 2 +ab+ b 2 ) ] = 1 a–b [ –3ab (a–b ) 2 ( a 2 +ab+ b 2 ) ] = –3ab (a–b ) 3 ( a 2 +ab+ b 2 )
  13. x+3 6 x 2 +x–2 – 1 4 x 2 –4x+1 = x+3 6 x 2 –3x+4x–2 – 1 (2x–1 ) 2 = x+3 3x(2x–1 ) +2(2x–1 ) – 1 (2x–1 ) 2 = x+3 (3x+2 ) (2x–1 ) – 1 (2x–1 ) 2 = 1 2x–1 [ x+3 3x+2 – 1 2x–1 ] = 1 2x–1 [ (x+3 ) (2x–1 ) –(3x+2 ) (3x+2 ) (2x–1 ) ] = 1 2x–1 [ 2 x 2 –x+6x–3–3x–2 (3x+2 ) (2x–1 ) ] = 1 2x–1 [ 2 x 2 +2x–5 (3x+2 ) (2x–1 ) ] = 2 x 2 +2x–5 (3x+2 ) (2x–1 ) 2
  14. x–1 4x+4 – x+2 8x–8 = x–1 4(x+1 ) – x+2 8(x–1 ) = 1 4 [ x–1 x+1 – x+2 2(x–1 ) ] = 1 4 [ 2 (x–1 ) 2 –(x+1 ) (x+2 ) 2(x–1 ) (x+1 ) ] = 1 4 [ 2( x 2 –2x+1 ) –( x 2 +3x+2 ) 2(x–1 ) (x+1 ) ] = 1 4 [ 2 x 2 –4x+ 2 – x 2 –3x– 2 2(x–1 ) (x+1 ) ] = 1 4 [ x 2 –7x 2(x–1 ) (x+1 ) ] = x 2 –7x 8( x 2 –1 )
  15. x xy– y 2 – 1 y = x y(x–y ) – 1 y = 1 y [ x x–y –1 ] = 1 y [ x–(x–y ) x–y ] = 1 y [ x – x +y x–y ] = 1 y [ y x–y ] = 1 x–y
  16. b a 2 – b 2 – b a 2 +ab = b (a+b ) (a–b ) – b a(a+b ) = b a+b [ 1 a–b – 1 a ] = b a+b [ a–(a–b ) a(a–b ) ] = b a+b [ a – a +b a(a–b ) ] = b a+b [ b a(a–b ) ] = b 2 a( a 2 – b 2 )
  17. 2a–3 6a+9 – a–1 4 a 2 +12a+9 = 2a–3 3(2a+3 ) – a–1 (2a+3 ) 2 = 1 2a+3 [ 2a–3 3 – a–1 2a+3 ] = 1 2a+3 [ (2a+3 ) (2a–3 ) –3(a–1 ) 3(2a+3 ) ] = 1 2a+3 [ 4 a 2 –9–3a+3 3(2a+3 ) ] = 1 2a+3 [ 4 a 2 –3a–6 3(2a+3 ) ] = 4 a 2 –3a–6 3 (2a+3 ) 2
  18. x+1 x 2 +x+1 – x–1 x 2 –x+1 = (x+1 ) ( x 2 –x+1 ) –(x–1 ) ( x 2 +x+1 ) ( x 2 +x+1 ) ( x 2 –x+1 ) = x 3 +1–( x 3 –1 ) ( x 2 +x+1 ) ( x 2 –x+1 ) = x 3 +1– x 3 +1 ( x 2 +x+1 ) ( x 2 –x+1 ) = 2 ( x 2 +x+1 ) ( x 2 –x+1 )
  19. a–1 a 2 +a – 1 2a–2 – 1 2a+2 = a–1 a(a+1 ) – 1 2(a–1 ) – 1 2(a+1 ) = 2 (a–1 ) 2 –a(a+1 ) –a(a–1 ) 2a(a+1 ) (a–1 ) = 2( a 2 –2a+1 ) – a 2 – a – a 2 + a 2a(a+1 ) (a–1 ) = 2 a 2 –4a+2– 2 a 2 2a(a+1 ) (a–1 ) = 2–4a 2a(a+1 ) (a–1 ) = 2 (1–2a ) 2 a(a+1 ) (a–1 ) = 1–2a a( a 2 –1 )
  20. 1 4a+4 – 1 8a–8 – 1 12 a 2 +12 = 1 4(a+1 ) – 1 8(a–1 ) – 1 12( a 2 +1 ) = 1 4 [ 1 a+1 – 1 2(a–1 ) – 1 3( a 2 +1 ) ] = 1 4 [ 6(a–1 ) ( a 2 +1 ) –3(a+1 ) ( a 2 +1 ) –2(a+1 ) (a–1 ) 6(a+1 ) (a–1 ) ( a 2 +1 ) ] = 1 4 [ 6( a 3 +a– a 2 –1 ) –3( a 3 +a+ a 2 +1 ) –2( a 2 –1 ) 6(a+1 ) (a–1 ) ( a 2 +1 ) ] = 1 4 [ 6 a 3 +6a–6 a 2 –6–3 a 3 –3a–3 a 2 –3–2 a 2 +2 6(a+1 ) (a–1 ) ( a 2 +1 ) ] = 1 4 [ 3 a 3 –11 a 2 +3a–7 6(a+1 ) (a–1 ) ( a 2 +1 ) ] = 3 a 3 –11 a 2 +3a–7 24(a+1 ) (a–1 ) ( a 2 +1 )
  21. y x 2 –xy – 1 x – 1 x–y = y x(x–y ) – 1 x – 1 x–y = y–(x–y ) –x x(x–y ) = y–x+y–x x(x–y ) = –2x+2y x(x–y ) = –2 (x–y ) x (x–y ) =– 2 x
  22. a a 2 +ab – 1 a – 1 a+b = a a(a+b ) – 1 a – 1 a+b = a –(a+b ) – a a(a+b ) = – (a+b ) a (a+b ) =– 1 a
  23. 1 x 2 –xy – 1 x 2 +xy – 2y x 3 –x y 2 = 1 x(x–y ) – 1 x(x+y ) – 2y x( x 2 – y 2 ) = x+y–(x–y ) –2y x(x–y ) (x+y ) = x + y – x + y – 2y x(x–y ) (x+y ) =0
  24. x x 2 +x–2 – 3 x 2 +2x–3 – x x 2 +5x+6 = x (x+2 ) (x–1 ) – 3 (x+3 ) (x–1 ) – x (x+3 ) (x+2 ) = x(x+3 ) –3(x+2 ) –x(x–1 ) (x+2 ) (x–1 ) (x+3 ) = x 2 + 3x – 3x –6– x 2 +x (x+2 ) (x–1 ) (x+3 ) = x–6 (x+2 ) (x–1 ) (x+3 )
  25. 3 x 2 +x+1 – x+2 (x–1 ) 2 – 1–9x ( x 3 –1 ) (x–1 ) = 3 x 2 +x+1 – x+2 (x–1 ) 2 – 1–9x (x–1 ) ( x 2 +x+1 ) (x–1 ) = 3 x 2 +x+1 – x+2 (x–1 ) 2 – 1–9x (x–1 ) 2 ( x 2 +x+1 ) = 3 (x–1 ) 2 –(x+2 ) ( x 2 +x+1 ) –(1–9x ) (x–1 ) 2 ( x 2 +x+1 ) = 3( x 2 –2x+1 ) –( x 3 + x 2 +x+2 x 2 +2x+2 ) –1+9x (x–1 ) 2 ( x 2 +x+1 ) = 3 x 2 – 6x + 3 – x 3 – 3 x 2 – 3x – 2 – 1 + 9x (x–1 ) 2 ( x 2 +x+1 ) =– x 3 (x–1 ) ( x 3 –1 )
  26. a 2 + b 2 a 3 – b 3 – a+b 2 a 2 +2ab+2 b 2 – 1 2a–2b = a 2 + b 2 (a–b ) ( a 2 +ab+ b 2 ) – a+b 2( a 2 +ab+ b 2 ) – 1 2(a–b ) = 2( a 2 + b 2 ) –(a+b ) (a–b ) –( a 2 +ab+ b 2 ) 2(a–b ) ( a 2 +ab+ b 2 ) = 2 a 2 +2 b 2 –( a 2 – b 2 ) – a 2 –ab– b 2 2(a–b ) ( a 2 +ab+ b 2 ) = a 2 + b 2 – a 2 + b 2 –ab 2(a–b ) ( a 2 +ab+ b 2 ) = 2 b 2 –ab 2( a 3 – b 3 ) = b(2b–a ) 2( a 3 – b 3 )
  27. 3a 2 a 2 –2a–4 – a–1 4 a 2 +8a–32 – 10a–1 8 a 2 +40a+32 = 3a 2( a 2 –a–2 ) – a–1 4( a 2 +2a–8 ) – 10a–1 8( a 2 +5a+4 ) = 1 2 [ 3a (a–2 ) (a+1 ) – a–1 2(a+4 ) (a–2 ) – 10a–1 4(a+4 ) (a+1 ) ] = 1 2 [ 12a(a+4 ) –2(a–1 ) (a+1 ) –(10a–1 ) (a–2 ) 4(a–2 ) (a+1 ) (a+4 ) ] = 1 2 [ 12 a 2 +48a–2( a 2 –1 ) –(10 a 2 –20a–a+2 ) 4(a–2 ) (a+1 ) (a+4 ) ] = 1 2 [ 12 a 2 +48a– 2 a 2 + 2 – 10 a 2 +21a– 2 4(a–2 ) (a+1 ) (a+4 ) ] = 1 2 [ 69a 4(a–2 ) (a+1 ) (a+4 ) ] = 69a 8(a–2 ) (a+1 ) (a+4 )
  28. 1 4a–12x – a 2 +9 x 2 a 3 –27 x 3 – a 2( a 2 +3ax+9 x 2 ) = 1 4(a–3x ) – a 2 +9 x 2 (a–3x ) ( a 2 +3ax+9 x 2 ) – a 2( a 2 +3ax+9 x 2 ) = a 2 +3ax+9 x 2 –4( a 2 +9 x 2 ) –2a(a–3x ) 4(a–3x ) ( a 2 +3ax+9 x 2 ) = a 2 +3ax+9 x 2 –4 a 2 –36 x 2 –2 a 2 +6ax 4(a–3x ) ( a 2 +3ax+9 x 2 ) = –5 a 2 +9ax–27 x 2 4(a–3x ) ( a 2 +3ax+9 x 2 ) =– 5 a 2 –9ax+27 x 2 4( a 3 –27 x 3 )
  29. 2 a 2 –3 10a+10 – a+1 50 – 9 a 2 –14 50a+50 = 2 a 2 –3 10(a+1 ) – a+1 50 – 9 a 2 –14 50(a+1 ) = 1 10 [ 2 a 2 –3 a+1 – a+1 5 – 9 a 2 –14 5(a+1 ) ] = 1 10 [ 5(2 a 2 –3 ) – (a+1 ) 2 –(9 a 2 –14 ) 5(a+1 ) ] = 1 10 [ 10 a 2 –15–( a 2 +2a+1 ) –9 a 2 +14 5(a+1 ) ] = 1 10 [ a 2 –1– a 2 –2a–1 5(a+1 ) ] = 1 10 [ –2–2a 5(a+1 ) ] = 1 [ – 2 (1+a ) 5 (a+1 ) ] =– 1 25
Categories: Capítulo XIV