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Ejercicio 127

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CAPITULO XIV

Operaciones con Fracciones
Ejercicio 127
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  1. 1 a+1 + 1 a–1 = a– 1 +a+ 1 ( a+1 ) ( a–1 ) = 2a a 2 –1
  2. 2 x+4 + 1 x–3 = 2( x–3 ) +x+4 ( x+4 ) ( x–3 ) = 2x–6+x+4 ( x+4 ) ( x–3 ) = 3x–2 ( x+4 ) ( x–3 )
  3. 3 1–x + 6 2x+5 = 3( 2x+5 ) +6( 1–x ) ( 1–x ) ( 2x+5 ) = 6x +15+6– 6x ( 1–x ) ( 2x+5 ) = 21 ( 1–x ) ( 2x+5 )
  4. x x–y + x x+y = x( x+y ) +x( x–y ) ( x–y ) ( x+y ) = x[ x+ y +x– y ] x 2 – y 2 = 2 x 2 x 2 – y 2
  5. m+3 m–3 + m+2 m–2 = ( m–2 ) ( m+3 ) +( m+2 ) ( m–3 ) ( m–3 ) ( m–2 ) = m 2 + m –6+ m 2 – m –6 ( m–3 ) ( m–2 ) = 2 m 2 –12 ( m–3 ) ( m–2 ) = 2( m 2 –6 ) ( m–3 ) ( m–2 )
  6. x+y x–y + x–y x+y = ( x+y ) 2 + ( x–y ) 2 ( x–y ) ( x+y ) = x 2 + 2xy + y 2 + x 2 – 2xy + y 2 x 2 – y 2 = 2 x 2 +2 y 2 x 2 – y 2 = 2( x 2 + y 2 ) x 2 – y 2
  7. x x 2 –1 + x+1 ( x–1 ) 2 = x ( x–1 ) ( x+1 ) + x+1 ( x–1 ) 2 = x( x–1 ) + ( x+1 ) 2 ( x+1 ) ( x–1 ) 2 = x 2 –x+ x 2 +2x+1 ( x+1 ) ( x–1 ) 2 = 2 x 2 +x+1 ( x+1 ) ( x–1 ) 2
  8. 2 x–5 + 3x x 2 –25 = 2 x–5 + 3x ( x–5 ) ( x+5 ) = 2( x+5 ) +3x ( x–5 ) ( x+5 ) = 2x+10+3x ( x–5 ) ( x+5 ) = 5x+10 ( x–5 ) ( x+5 ) = 5( x+2 ) ( x–5 ) ( x+5 )
  9. 1 3x–2y + x–y 9 x 2 –4 y 2 = 1 3x–2y + x–y ( 3x–2y ) ( 3x+2y ) = 3x+2y+x–y ( 3x–2y ) ( 3x+2y ) = 4x+y ( 3x–2y ) ( 3x+2y )
  10. x+a x+3a + 3 a 2 – x 2 x 2 –9 a 2 = x+a x+3a + 3 a 2 – x 2 ( x+3a ) ( x–3a ) = ( x+a ) ( x–3a ) +3 a 2 – x 2 ( x+3a ) ( x–3a ) = x 2 –2ax– 3 a 2 + 3 a 2 – x 2 ( x+3a ) ( x–3a ) = –2ax x 2 –9 a 2 = 2ax 9 a 2 – x 2
  11. a 1– a 2 + a 1+ a 2 =a[ 1+ a 2 +1– a 2 ( 1– a 2 ) ( 1+ a 2 ) ] =a[ 2 1– a 4 ] = 2a 1– a 4
  12. 2 a 2 –ab + 2 ab+ b 2 =2[ 1 a( a–b ) + 1 b( a+b ) ] =2[ b( a+b ) +a( a–b ) ab( a+b ) ( a–b ) ] =2[ ab + b 2 + a 2 – ab ab( a 2 – b 2 ) ] = 2( a 2 + b 2 ) ab( a 2 – b 2 )
  13. ab 9 a 2 – b 2 + a 3a+b =a[ b ( 3a+b ) ( 3a–b ) + 1 3a+b ] =a[ b +3a– b ( 3a+b ) ( 3a–b ) ] = 3 a 2 9 a 2 – b 2
  14. 1 a 2 – b 2 + 1 ( a–b ) 2 = 1 ( a+b ) ( a–b ) + 1 ( a–b ) 2 = a– b +a+ b ( a+b ) ( a–b ) 2 = 2a ( a+b ) ( a–b ) 2
  15. 3 x 2 + y 2 + 2 ( x+y ) 2 = 3 ( x+y ) 2 +2( x 2 + y 2 ) ( x 2 + y 2 ) ( x+y ) 2 = 3( x 2 +2xy+ y 2 ) +2 x 2 +2 y 2 ( x 2 + y 2 ) ( x+y ) 2 = 3 x 2 +6xy+3 y 2 +2 x 2 +2 y 2 ( x 2 + y 2 ) ( x+y ) 2 = 5 x 2 +6xy+5 y 2 ( x 2 + y 2 ) ( x+y ) 2
  16. x a 2 –ax + a+x ax + a ax– x 2 = x a( a–x ) + a+x ax + a x( a–x ) = x 2 +( a+x ) ( a–x ) + a 2 ax( a–x ) = x 2 + a 2 – x 2 + a 2 ax( a–x ) = 2 a 2 a x( a–x ) = 2a x( a–x )
  17. 3 2x+4 + x–1 2x–4 + x+8 x 2 –4 = 3 2( x+2 ) + x–1 2( x–2 ) + x+8 ( x+2 ) ( x–2 ) = 3( x–2 ) +( x–1 ) ( x+2 ) +2( x+8 ) 2( x+2 ) ( x–2 ) = 3x–6+ x 2 +x–2+2x+16 2( x+2 ) ( x–2 ) = x 2 +6x+8 2( x+2 ) ( x–2 ) = ( x+4 ) ( x+2 ) 2 ( x+2 ) ( x–2 ) = x+4 2( x–2 )
  18. 1 x+ x 2 + 1 x– x 2 + x+3 1– x 2 = 1 x( 1+x ) + 1 x( 1–x ) + x+3 ( 1–x ) ( 1+x ) = 1– x +1+ x +x( x+3 ) x( 1–x ) ( 1+x ) = 2+ x 2 +3x x( 1–x ) ( 1+x ) = x 2 +3x+2 x( 1–x ) ( 1+x ) = ( x+2 ) ( x+1 ) x( 1–x ) ( x+1 ) = x+2 x( 1–x )
  19. x–y x+y + x+y x–y + 4xy x 2 – y 2 = x–y x+y + x+y x–y + 4xy ( x+y ) ( x–y ) = ( x–y ) 2 + ( x+y ) 2 +4xy ( x+y ) ( x–y ) = x 2 – 2xy + y 2 + x 2 + 2xy + y 2 +4xy ( x+y ) ( x–y ) = 2 x 2 +4xy+2 y 2 ( x+y ) ( x–y ) = 2( x 2 +2xy+ y 2 ) ( x+y ) ( x–y ) = 2 ( x+y ) 2 ( x+y ) ( x–y ) = 2( x+y ) x–y
  20. 1 a–5 + a a 2 –4a–5 + a+5 a 2 +2a+1 = 1 a–5 + a ( a–5 ) ( a+1 ) + a+5 ( a+1 ) 2 = ( a+1 ) 2 +a( a+1 ) +( a+5 ) ( a–5 ) ( a–5 ) ( a+1 ) 2 = a 2 +2a+1+ a 2 +a+ a 2 –25 ( a–5 ) ( a+1 ) 2 = 3 a 2 +3a–24 ( a–5 ) ( a+1 ) 2 = 3( a 2 +a–8 ) ( a–5 ) ( a+1 ) 2
  21. 3 a + 2 5a–3 + 1–85a 25 a 2 –9 = 3 a + 2 5a–3 + 1–85a ( 5a–3 ) ( 5a+3 ) = 3( 5a–3 ) ( 5a+3 ) +2a( 5a+3 ) +a( 1–85a ) a( 5a–3 ) ( 5a+3 ) = 3( 25 a 2 –9 ) +10 a 2 +6a+a–85 a 2 a( 5a–3 ) ( 5a+3 ) = 75 a 2 –27+7a– 75 a 2 a( 25 a 2 –9 ) = 7a–27 a( 25 a 2 –9 )
  22. x+1 10 + x–3 5x–10 + x–2 2 = x+1 10 + x–3 5( x–2 ) + x–2 2 = ( x+1 ) ( x–2 ) +2( x–3 ) +5 ( x–2 ) 2 10( x–2 ) = x 2 –x–2+2x–6+5( x 2 –4x+4 ) 10( x–2 ) = x 2 +x–8+5 x 2 –20x+20 10( x–2 ) = 6 x 2 –19x+12 10( x–2 )
  23. x+5 x 2 +x–12 + x+4 x 2 +2x–15 + x–3 x 2 +9x+20 = x+5 ( x+4 ) ( x–3 ) + x+4 ( x+5 ) ( x–3 ) + x–3 ( x+5 ) ( x+4 ) = ( x+5 ) 2 + ( x+4 ) 2 + ( x–3 ) 2 ( x+4 ) ( x–3 ) ( x+5 ) = x 2 +10x+25+ x 2 +8x+16+ x 2 –6x+9 ( x+4 ) ( x–3 ) ( x+5 ) = 3 x 2 +12x+50 ( x+4 ) ( x–3 ) ( x+5 )
  24. 1 x–2 + 1–2 x 2 x 3 –8 + x x 2 +2x+4 = 1 x–2 + 1–2 x 2 ( x–2 ) ( x 2 +2x+4 ) + x x 2 +2x+4 = x 2 +2x+4+1–2 x 2 +x( x–2 ) ( x–2 ) ( x 2 +2x+4 ) = x 2 + 2x +4+1– 2 x 2 + x 2 – 2x x 3 –8 = 5 x 3 –8
  25. 2 a+1 + a ( a+1 ) 2 + a+1 ( a+1 ) 3 = 1 a+1 [ 2+ a a+1 + a+1 ( a+1 ) 2 ] = 1 a+1 [ 2 ( a+1 ) 2 +a( a+1 ) +a+1 ( a+1 ) 2 ] = 1 a+1 [ 2( a 2 +2a+1 ) + a 2 +a+a+1 ( a+1 ) 2 ] = 1 a+1 [ 2 a 2 +4a+2+ a 2 +2a+1 ( a+1 ) 2 ] = 1 a+1 [ 3 a 2 +6a+3 ( a+1 ) 2 ] = 3 a+1 [ a 2 +2a+1 ( a+1 ) 2 ] = 3 a+1 [ ( a+1 ) 2 ( a+1 ) 2 ] = 3 a+1
  26. 2x 3 x 2 +11x+6 + x+1 x 2 –9 + 1 3x+2 = 2x 3 x 2 +9x+2x+6 + x+1 ( x–3 ) ( x+3 ) + 1 3x+2 = 2x 3x( x+3 ) +2( x+3 ) + x+1 ( x–3 ) ( x+3 ) + 1 3x+2 = 2x ( 3x+2 ) ( x+3 ) + x+1 ( x–3 ) ( x+3 ) + 1 3x+2 = 2x( x–3 ) +( x+1 ) ( 3x+2 ) +( x+3 ) ( x–3 ) ( 3x+2 ) ( x+3 ) ( x–3 ) = 6 x 2 –x–7 ( 3x+2 ) ( x+3 ) ( x–3 )
  27. x 2 –4 x 3 +1 + 1 x+1 + 3 x 2 –x+1 = x 2 –4 ( x+1 ) ( x 2 –x+1 ) + 1 x+1 + 3 x 2 –x+1 = x 2 –4+ x 2 –x+1+3( x+1 ) ( x+1 ) ( x 2 –x+1 ) = 2 x 2 – 4 –x+ 1 +3x+ 3 ( x+1 ) ( x 2 –x+1 ) = 2 x 2 +2x ( x+1 ) ( x 2 –x+1 ) = 2x ( x+1 ) ( x+1 ) ( x 2 –x+1 ) = 2x x 2 –x+1
  28. 1 x–1 + 1 ( x–1 ) ( x+2 ) + x+1 ( x–1 ) ( x+2 ) ( x+3 ) = 1 x–1 [ 1+ 1 x+2 + x+1 ( x+2 ) ( x+3 ) ] = 1 x–1 [ ( x+2 ) ( x+3 ) +x+3+x+1 ( x+2 ) ( x+3 ) ] = 1 x–1 [ x 2 +5x+6+2x+4 ( x+2 ) ( x+3 ) ] = 1 x–1 [ x 2 +7x+10 ( x+2 ) ( x+3 ) ] = 1 x–1 [ ( x+2 ) ( x+5 ) ( x+2 ) ( x+3 ) ] = x+5 ( x–1 ) ( x+3 )
  29. x–2 2 x 2 –5x–3 + x–3 2 x 2 –3x–2 + 2x–1 x 2 –5x+6 = x–2 2 x 2 +x–6x–3 + x–3 2 x 2 –4x+x–2 + 2x–1 ( x–3 ) ( x–2 ) = x–2 x( 2x+1 ) –3( 2x+1 ) + x–3 2x( x–2 ) +( x–2 ) + 2x–1 ( x–3 ) ( x–2 ) = x–2 ( x–3 ) ( 2x+1 ) + x–3 ( 2x+1 ) ( x–2 ) + 2x–1 ( x–3 ) ( x–2 ) = ( x–2 ) 2 + ( x–3 ) 2 +( 2x+1 ) ( 2x–1 ) ( x–3 ) ( 2x+1 ) ( x–2 ) = x 2 –4x+4+ x 2 –6x+9+4 x 2 –1 ( x–3 ) ( 2x+1 ) ( x–2 ) = 6 x 2 –10x+12 ( x–3 ) ( 2x+1 ) ( x–2 ) = 2( 3 x 2 –5x+6 ) ( x–3 ) ( 2x+1 ) ( x–2 )
  30. a–2 a–1 + a+3 a+2 + a+1 a–3 = ( a–2 ) ( a+2 ) ( a–3 ) +( a–1 ) ( a–3 ) ( a+3 ) +( a–1 ) ( a+1 ) ( a+2 ) ( a–1 ) ( a+2 ) ( a–3 ) = ( a 2 –4 ) ( a–3 ) +( a–1 ) ( a 2 –9 ) +( a 2 –1 ) ( a+2 ) ( a–1 ) ( a+2 ) ( a–3 ) = a 3 –4a–3 a 2 +12+ a 3 –9a– a 2 +9+ a 3 +2 a 2 –a–2 ( a–1 ) ( a+2 ) ( a–3 ) = 3 a 3 –2 a 2 –14a+19 ( a–1 ) ( a+2 ) ( a–3 )
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