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Ejercicio 130

Gestor Baldor
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CAPITULO XIV

Operaciones con Fracciones
Ejercicio 130
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  1. 2 x–3 + 3 x+2 – 4x–7 x 2 –x–6 = 2 x–3 + 3 x+2 – 4x–7 ( x–3 ) ( x+2 ) = 2( x+2 ) +3( x–3 ) –( 4x–7 ) ( x–3 ) ( x+2 ) = 2x+4+3x–9–4x+7 ( x–3 ) ( x+2 ) = x+2 ( x–3 ) ( x+2 ) = 1 x–3
  2. a 3a+6 – 1 6a+12 + a+12 12a+24 = a 3( a+2 ) – 1 6( a+2 ) + a+12 12( a+2 ) = 1 3( a+2 ) [ a– 1 2 + a+12 4 ] = 1 3( a+2 ) [ 4a–2+a+12 4 ] = 1 3( a+2 ) [ 5a+10 4 ] = 1 3 ( a+2 ) [ 5 ( a+2 ) 4 ] = 5 12
  3. x x 2 +1 + 1 3x – 1 x 2 = 3 x 3 +x( x 2 +1 ) –3( x 2 +1 ) 3 x 2 ( x 2 +1 ) = 3 x 3 + x 3 +x–3 x 2 –3 3 x 2 ( x 2 +1 ) = 4 x 3 –3 x 2 +x–3 3 x 2 ( x 2 +1 )
  4. a+3 a 2 –1 + a–1 2a+2 + a–4 4a–4 = a+3 ( a–1 ) ( a+1 ) + a–1 2( a+1 ) + a–4 4( a–1 ) = 4( a+3 ) +2 ( a–1 ) 2 +( a+1 ) ( a–4 ) 4( a–1 ) ( a+1 ) = 4a+12+2( a 2 –2a+1 ) + a 2 –3a–4 4( a–1 ) ( a+1 ) = a+8+2 a 2 –4a+2+ a 2 4( a 2 –1 ) = 3 a 2 –3a+10 4( a 2 –1 )
  5. a–b a 2 +ab + a+b ab – a ab+ b 2 = a–b a( a+b ) + a+b ab – a b( a+b ) = b( a–b ) + ( a+b ) 2 – a 2 ab( a+b ) = ab– b 2 + a 2 +2ab+ b 2 – a 2 ab( a+b ) = 3 ab ab ( a+b ) = 3 a+b
  6. x–y x+y – x+y x–y + 4 x 2 x 2 – y 2 = x–y x+y – x+y x–y + 4 x 2 ( x+y ) ( x–y ) = ( x–y ) 2 – ( x+y ) 2 +4 x 2 ( x+y ) ( x–y ) = [ ( x–y ) +( x+y ) ] [ ( x–y ) –( x+y ) ] +4 x 2 ( x+y ) ( x–y ) = [ x– y +x+ y ] [ x –y– x –y ] +4 x 2 ( x+y ) ( x–y ) = 2x( –2y ) +4 x 2 ( x+y ) ( x–y ) = 4 x 2 –4xy ( x+y ) ( x–y ) = 4x ( x–y ) ( x+y ) ( x–y ) = 4x x+y
  7. x a 2 –ax + 1 a + 1 x = x a( a–x ) + 1 a + 1 x = x 2 +x( a–x ) +a( a–x ) ax( a–x ) = x 2 + ax – x 2 + a 2 – ax ax( a–x ) = a 2 a x( a–x ) = a x( a–x )
  8. x+1 x 2 –x–20 – x+4 x 2 –4x–5 + x+5 x 2 +5x+4 = x+1 ( x–5 ) ( x+4 ) – x+4 ( x–5 ) ( x+1 ) + x+5 ( x+4 ) ( x+1 ) = ( x+1 ) 2 – ( x+4 ) 2 +( x–5 ) ( x+5 ) ( x+1 ) ( x–5 ) ( x+4 ) = [ ( x+1 ) +( x+4 ) ] [ ( x+1 ) –( x+4 ) ] + x 2 –25 ( x+1 ) ( x–5 ) ( x+4 ) = [ x+1+x+4 ] [ x +1– x –4 ] + x 2 –25 ( x+1 ) ( x–5 ) ( x+4 ) = –3( 2x+5 ) + x 2 –25 ( x+1 ) ( x–5 ) ( x+4 ) = –6x–15+ x 2 –25 ( x+1 ) ( x–5 ) ( x+4 ) = x 2 –6x–40 ( x+1 ) ( x–5 ) ( x+4 ) = ( x–10 ) ( x+4 ) ( x+1 ) ( x–5 ) ( x+4 ) = x–10 ( x+1 ) ( x–5 )
  9. 2x+1 12x+8 – x 2 6 x 2 +x–2 + 2x 16x–8 = 2x+1 4( 3x+2 ) – x 2 6 x 2 –3x+4x–2 + 2x 8( 2x–1 ) = 2x+1 4( 3x+2 ) – x 2 3x( 2x–1 ) +2( 2x–1 ) + 2x 8( 2x–1 ) = 2x+1 4( 3x+2 ) – x 2 ( 3x+2 ) ( 2x–1 ) + 2x 8( 2x–1 ) = 2( 2x–1 ) ( 2x+1 ) –8 x 2 +2x( 3x+2 ) 8( 3x+2 ) ( 2x–1 ) = 2( 4 x 2 –1 ) –8 x 2 +6 x 2 +4x 8( 3x+2 ) ( 2x–1 ) = 8 x 2 –2–2 x 2 +4x 8( 3x+2 ) ( 2x–1 ) = 6 x 2 +4x–2 8( 3x+2 ) ( 2x–1 ) = 2 ( 3 x 2 +2x–1 ) ( 3x+2 ) ( 2x–1 ) = 3 x 2 +2x–1 4( 3x+2 ) ( 2x–1 )
  10. 1 ax – 1 a 2 +ax + 1 a+x = 1 ax – 1 a( a+x ) + 1 a+x = a+ x – x +ax ax( a+x ) = a+ax ax( a+x ) = a ( 1+x ) a x( a+x ) = 1+x x( a+x )
  11. 1 x+y – 1 x–y + 2y x 2 + y 2 = ( x–y ) ( x 2 + y 2 ) –( x+y ) ( x 2 + y 2 ) +2y( x+y ) ( x–y ) ( x+y ) ( x–y ) ( x 2 + y 2 ) = x 3 +x y 2 – x 2 y– y 3 –( x 3 +x y 2 + x 2 y+ y 3 ) +2y( x 2 – y 2 ) ( x+y ) ( x–y ) ( x 2 + y 2 ) = x 3 + x y 2 – x 2 y – y 3 – x 3 – x y 2 – x 2 y – y 3 + 2 x 2 y –2 y 3 ( x 2 – y 2 ) ( x 2 + y 2 ) = –4 y 3 x 4 – y 4 = 4 y 3 y 4 – x 4
  12. a–1 3a+3 – a–2 6a–6 + a 2 +2a–6 9 a 2 –9 = a–1 3( a+1 ) – a–2 6( a–1 ) + a 2 +2a–6 9( a 2 –1 ) = 1 3 [ a–1 a+1 – a–2 2( a–1 ) + a 2 +2a–6 3( a–1 ) ( a+1 ) ] = 1 3 [ 6 ( a–1 ) 2 –3( a–2 ) ( a+1 ) +2( a 2 +2a–6 ) 6( a–1 ) ( a+1 ) ] = 1 3 [ 6( a 2 –2a+1 ) –3( a 2 –a–2 ) +2 a 2 +4a–12 6( a–1 ) ( a+1 ) ] = 1 3 [ 6 a 2 –12a+ 6 –3 a 2 +3a+ 6 +2 a 2 +4a– 12 6( a–1 ) ( a+1 ) ] = 1 3 [ 5 a 2 –5a 6( a–1 ) ( a+1 ) ] = 1 3 [ 5a ( a–1 ) 6 ( a–1 ) ( a+1 ) ] = 5a 18( a+1 )
  13. 1 a 2 +2a–24 + 2 a 2 –2a–8 – 3 a 2 +8a+12 = 1 ( a+6 ) ( a–4 ) + 2 ( a–4 ) ( a+2 ) – 3 ( a+6 ) ( a+2 ) = a+2+2( a+6 ) –3( a–4 ) ( a+6 ) ( a–4 ) ( a+2 ) = a +2+ 2a +12– 3a +12 ( a+6 ) ( a–4 ) ( a+2 ) = 26 ( a+6 ) ( a–4 ) ( a+2 )
  14. x+y xy – x+2y xy+ y 2 – y x 2 +xy = x+y xy – x+2y y( x+y ) – y x( x+y ) = ( x+y ) 2 –x( x+2y ) – y 2 xy( x+y ) = x 2 + 2xy + y 2 – x 2 – 2xy – y 2 xy( x+y ) =0
  15. a 3 a 3 +1 + a+3 a 2 –a+1 – a–1 a+1 = a 3 ( a+1 ) ( a 2 –a+1 ) + a+3 a 2 –a+1 – a–1 a+1 = a 3 +( a+3 ) ( a+1 ) –( a–1 ) ( a 2 –a+1 ) ( a+1 ) ( a 2 –a+1 ) = a 3 + a 2 +4a+3–( a 3 – a 2 +a– a 2 +a–1 ) ( a+1 ) ( a 2 –a+1 ) = a 3 + a 2 +4a+3– a 3 +2 a 2 –2a+1 a 3 +1 = 3 a 2 +2a+4 a 3 +1
  16. 1 x–1 + 2x x 2 –1 – 3 x 2 x 3 –1 = 1 x–1 + 2x ( x–1 ) ( x+1 ) – 3 x 2 ( x–1 ) ( x 2 +x+1 ) = 1 x–1 [ 1+ 2x x+1 – 3 x 2 x 2 +x+1 ] = 1 x–1 [ ( x+1 ) ( x 2 +x+1 ) +2x( x 2 +x+1 ) –3 x 2 ( x+1 ) ( x+1 ) ( x 2 +x+1 ) ] = 1 x–1 [ x 3 + x 2 +x+ x 2 +x+1+ 2 x 3 + 2 x 2 +2x– 3 x 3 – 3 x 2 ( x+1 ) ( x 2 +x+1 ) ] = 1 x–1 [ x 2 +4x+1 ( x+1 ) ( x 2 +x+1 ) ] = x 2 +4x+1 ( x+1 ) ( x 3 –1 )
  17. a+b a 2 –ab+ b 2 – 1 a+b + 3 a 2 a 3 + b 3 = a+b a 2 –ab+ b 2 – 1 a+b + 3 a 2 ( a+b ) ( a 2 –ab+ b 2 ) = ( a+b ) 2 –( a 2 –ab+ b 2 ) +3 a 2 ( a+b ) ( a 2 –ab+ b 2 ) = a 2 +2ab+ b 2 – a 2 +ab– b 2 +3 a 2 ( a+b ) ( a 2 –ab+ b 2 ) = 3 a 2 +3ab ( a+b ) ( a 2 –ab+ b 2 ) = 3a ( a+b ) ( a+b ) ( a 2 –ab+ b 2 ) = 3a a 2 –ab+ b 2
  18. 2 x–2 + 2x+3 x 2 +2x+4 – 6x+12 x 3 –8 = 2 x–2 + 2x+3 x 2 +2x+4 – 6x+12 ( x–2 ) ( x 2 +2x+4 ) = 2( x 2 +2x+4 ) +( x–2 ) ( 2x+3 ) –( 6x+12 ) ( x–2 ) ( x 2 +2x+4 ) = 2 x 2 + 4x +8+2 x 2 +3x– 4x –6–6x–12 ( x–2 ) ( x 2 +2x+4 ) = 4 x 2 –3x–10 ( x–2 ) ( x 2 +2x+4 ) = 4 x 2 –8x+5x–10 ( x–2 ) ( x 2 +2x+4 ) = 4x( x–2 ) +5( x–2 ) ( x–2 ) ( x 2 +2x+4 ) = ( 4x+5 ) ( x–2 ) ( x–2 ) ( x 2 +2x+4 ) = 4x+5 x 2 +2x+4
  19. 3x+2 x 2 +3x–10 – 5x+1 x 2 +4x–5 + 4x–1 x 2 –3x+2 = 3x+2 ( x+5 ) ( x–2 ) – 5x+1 ( x+5 ) ( x–1 ) + 4x–1 ( x–2 ) ( x–1 ) = ( 3x+2 ) ( x–1 ) –( x–2 ) ( 5x+1 ) +( x+5 ) ( 4x–1 ) ( x+5 ) ( x–2 ) ( x–1 ) = 3 x 2 + 2x –3x+2–5 x 2 – x +10x+2+4 x 2 – x +20x–5 ( x+5 ) ( x–2 ) ( x–1 ) = 2 x 2 +27x–1 ( x+5 ) ( x–2 ) ( x–1 )
  20. 1 ( n–1 ) 2 + 1 n–1 – 1 ( n–1 ) 3 – 1 n = 1 ( n–1 ) 2 + 1 n–1 – 1 ( n–1 ) 3 – 1 n = n( n–1 ) +n ( n–1 ) 2 –n– ( n–1 ) 3 n ( n–1 ) 3 = n 2 –n+n( n 2 –2n+1 ) –n–( n 3 –3 n 2 +3n–1 ) n ( n–1 ) 3 = n 2 –2n+ n 3 –2 n 2 +n– n 3 +3 n 2 –3n+1 n ( n–1 ) 3 = 2 n 2 –4n+1 n ( n–1 ) 3
  21. 1 a 2 +5 – a 2 –5 ( a 2 +5 ) 2 + a 2 +5 a 4 –25 = 1 a 2 +5 – a 2 –5 ( a 2 +5 ) 2 + a 2 +5 ( a 2 +5 ) ( a 2 –5 ) = 1 a 2 +5 [ 1– a 2 –5 a 2 +5 + a 2 +5 a 2 –5 ] = 1 a 2 +5 [ ( a 2 +5 ) ( a 2 –5 ) – ( a 2 –5 ) 2 + ( a 2 +5 ) 2 ( a 2 +5 ) ( a 2 –5 ) ] = 1 a 2 +5 [ a 4 – 25 –( a 4 –10 a 2 +25 ) + a 4 +10 a 2 + 25 ( a 2 +5 ) ( a 2 –5 ) ] = 1 a 2 +5 [ 2 a 4 – a 4 +10 a 2 –25+10 a 2 ( a 2 +5 ) ( a 2 –5 ) ] = 1 a 2 +5 [ a 4 +20 a 2 –25 ( a 2 +5 ) ( a 2 –5 ) ] = a 4 +20 a 2 –25 ( a 2 +5 ) ( a 4 –25 )
  22. 1– x 2 9– x 2 – x 2 9+6x+ x 2 – 6x 9–6x+ x 2 = 1– x 2 ( 3–x ) ( 3+x ) – x 2 ( 3+x ) 2 – 6x ( 3–x ) 2 = ( 1– x 2 ) ( 3–x ) ( 3+x ) – x 2 ( 3–x ) 2 –6x ( 3+x ) 2 ( 3–x ) 2 ( 3+x ) 2 = ( 1– x 2 ) ( 9– x 2 ) – x 2 ( 9–6x+ x 2 ) –6x( 9+6x+ x 2 ) ( 3–x ) 2 ( 3+x ) 2 = 9– x 2 –9 x 2 + x 4 –9 x 2 + 6 x 3 – x 4 –54x–36 x 2 – 6 x 3 ( 3–x ) 2 ( 3+x ) 2 = 9–54x–55 x 2 ( 3–x ) 2 ( 3+x ) 2
  23. x 2x+2 – x+1 3x–3 + x–1 6x+6 – 5 18x–18 = x 2( x+1 ) – x+1 3( x–1 ) + x–1 6( x+1 ) – 5 18( x–1 ) = 9x( x–1 ) –6 ( x+1 ) 2 +3 ( x–1 ) 2 –5( x+1 ) 18( x–1 ) ( x+1 ) = 9 x 2 –9x–6( x 2 +2x+1 ) +3( x 2 –2x+1 ) –5x–5 18( x–1 ) ( x+1 ) = 9 x 2 –9x–6 x 2 –12x–6+3 x 2 –6x+3–5x–5 18( x–1 ) ( x+1 ) = 6 x 2 –32x–8 18( x–1 ) ( x+1 ) = 2 ( 3 x 2 –16x–4 ) ( x–1 ) ( x+1 ) = 3 x 2 –16x–4 9( x–1 ) ( x+1 )
  24. a+2 2a+2 – 7a 8 a 2 –8 – a–3 4a–4 = a+2 2( a+1 ) – 7a 8( a 2 –1 ) – a–3 4( a–1 ) = 1 2 [ a+2 a+1 – 7a 4( a+1 ) ( a–1 ) – a–3 2( a–1 ) ] = 1 2 [ 4( a+2 ) ( a–1 ) –7a–2( a–3 ) ( a+1 ) 4( a+1 ) ( a–1 ) ] = 1 2 [ 4( a 2 +a–2 ) –7a–2( a 2 –2a–3 ) 4( a+1 ) ( a–1 ) ] = 1 2 [ 4 a 2 +4a–8–7a–2 a 2 +4a+6 4( a+1 ) ( a–1 ) ] = 1 2 [ 2 a 2 +a–2 4( a+1 ) ( a–1 ) ] = 2 a 2 +a–2 8( a+1 ) ( a–1 )
  25. a–3 20a+10 + 2a+5 40a+20 – 4a–1 60a+30 = a–3 10( 2a+1 ) + 2a+5 20( 2a+1 ) – 4a–1 30( 2a+1 ) = 1 10( 2a+1 ) [ a–3+ 2a+5 2 – 4a–1 3 ] = 1 10( 2a+1 ) [ 6a–18+3( 2a+5 ) –2( 4a–1 ) 6 ] = 1 10( 2a+1 ) [ 6a–18+6a+15–8a+2 6 ] = 1 10( 2a+1 ) [ 4a–1 6 ] = 4a–1 60( 2a+1 )
  26. 2 2 x 2 +5x+3 – 1 2 x 2 –x–6 + 3 x 2 –x–2 = 2 2 x 2 +2x+3x+3 – 1 2 x 2 –4x+3x–6 + 3 ( x–2 ) ( x+1 ) = 2 2x( x+1 ) +3( x+1 ) – 1 2x( x–2 ) +3( x–2 ) + 3 ( x–2 ) ( x+1 ) = 2 ( 2x+3 ) ( x+1 ) – 1 ( 2x+3 ) ( x–2 ) + 3 ( x–2 ) ( x+1 ) = 2( x–2 ) –( x+1 ) +3( 2x+3 ) ( 2x+3 ) ( x+1 ) ( x–2 ) = 2x–4–x–1+6x+9 ( 2x+3 ) ( x+1 ) ( x–2 ) = 7x+4 ( 2x+3 ) ( x+1 ) ( x–2 )
  27. a–1 a–2 – a–2 a+3 + 1 a–1 = ( a+3 ) ( a–1 ) 2 – ( a–2 ) 2 ( a–1 ) +( a–2 ) ( a+3 ) ( a–2 ) ( a+3 ) ( a–1 ) = ( a+3 ) ( a 2 –2a+1 ) –( a 2 –4a+4 ) ( a–1 ) + a 2 +a–6 ( a–2 ) ( a+3 ) ( a–1 ) = a 3 –2 a 2 +a+3 a 2 –6a+3–( a 3 –4 a 2 +4a– a 2 +4a–4 ) + a 2 +a–6 ( a–2 ) ( a+3 ) ( a–1 ) = a 3 + a 2 –5a+3– a 3 +5 a 2 –8a+4+ a 2 +a–6 ( a–2 ) ( a+3 ) ( a–1 ) = 7 a 2 –12a+1 ( a–2 ) ( a+3 ) ( a–1 )
  28. 2+3a 2–3a – 2–3a 2+3a – a ( 2–3a ) 2 = ( 2+3a ) 2 ( 2–3a ) – ( 2–3a ) 3 –a( 2+3a ) ( 2+3a ) ( 2–3a ) 2 = ( 2+3a ) ( 4–9 a 2 ) –( 8–36a+54 a 2 –27 a 3 ) –2a–3 a 2 ( 2+3a ) ( 2–3a ) 2 = 8 –18 a 2 +12a– 27 a 3 – 8 +36a–54 a 2 + 27 a 3 –2a–3 a 2 ( 2+3a ) ( 2–3a ) 2 = 46a–75 a 2 ( 2+3a ) ( 2–3a ) 2
  29. 1 5+5a + 1 5–5a – 1 10+10 a 2 = 1 5( 1+a ) + 1 5( 1–a ) – 1 10( 1+ a 2 ) = 1 5 [ 1 1+a + 1 1–a – 1 2( 1+ a 2 ) ] = 1 5 [ 2( 1+ a 2 ) ( 1–a ) +2( 1+ a 2 ) ( 1+a ) –( 1–a ) ( 1+a ) 2( 1+ a 2 ) ( 1–a ) ( 1+a ) ] = 1 5 [ 2( 1+ a 2 –a– a 3 ) +2( 1+ a 2 +a+ a 3 ) –( 1– a 2 ) 2( 1+ a 2 ) ( 1–a ) ( 1+a ) ] = 1 5 [ 2+2 a 2 – 2a – 2 a 3 +2+2 a 2 + 2a + 2 a 3 –1+ a 2 2( 1+ a 2 ) ( 1– a 2 ) ] = 1 5 [ 5 a 2 +3 2( 1+ a 2 ) ( 1– a 2 ) ] = 5 a 2 +3 10( 1– a 4 )
  30. 1 3–3x – 1 3+3x + x 6+6 x 2 – x 2–2 x 2 = 1 3( 1–x ) – 1 3( 1+x ) + x 6( 1+ x 2 ) – x 2( 1– x 2 ) = 2( 1+ x 2 ) ( 1+x ) –2( 1+ x 2 ) ( 1–x ) +x( 1–x ) ( 1+x ) –3x( 1+ x 2 ) 6( 1+ x 2 ) ( 1–x ) ( 1+x ) = 2( 1+ x 2 +x+ x 3 ) –2( 1+ x 2 –x– x 3 ) +x( 1– x 2 ) –3x–3 x 3 6( 1+ x 2 ) ( 1– x 2 ) = 2 + 2 x 2 +2x+ 2 x 3 – 2 – 2 x 2 + 2x + 2 x 3 + x – x 3 – 3x – 3 x 3 6( 1– x 4 ) = 2x 6( 1– x 4 ) = 2 x ( 1– x 4 ) = x 3( 1– x 4 )
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