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Ejercicio 94

Gestor Baldor
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DESCARGA
CAPITULO X

Descomposición Factorial
Ejercicio 94
Descomponer en dos factores y simplificar, si es posible:
  1. ( x+y ) 2 – a 2 =[ ( x+y ) +a ] [ ( x+y ) –a ] =( x+y+a ) ( x+y–a )
  2. 4– ( a+1 ) 2 =[ 2+( a+1 ) ] [ 2–( a+1 ) ] =( 2+a+1 ) ( 2–a–1 ) =( a+3 ) ( 1–a )
  3. 9– ( m+n ) 2 =[ 3+( m+n ) ] [ 3–( m+n ) ] =( 3+m+n ) ( 3–m–n )
  4. ( m–n ) 2 –16 =[ ( m–n ) –4 ] [ ( m–n ) +4 ] =( m–n–4 ) ( m–n+4 )
  5. ( x–y ) 2 –4 z 2 =[ ( x–y ) +2z ] [ ( x–y ) –2z ] =( x–y+2z ) ( x–y–2z )
  6. ( a+2b ) 2 –1 =[ ( a+2b ) –1 ] [ ( a+2b ) +1 ] =( a+2b–1 ) ( a+2b+1 )
  7. 1– ( x–2y ) 2 =[ 1–( x–2y ) ] [ 1+( x–2y ) ] =( 1–x+2y ) ( 1+x–2y )
  8. ( x+2a ) 2 –4 x 2 =[ ( x+2a ) –2x ] [ ( x+2a ) +2x ] =( x+2a–2x ) ( x+2a+2x ) =( 2a–x ) ( 2a+3x )
  9. ( a+b ) 2 – ( c+d ) 2 =[ ( a+b ) +( c+d ) ] [ ( a+b ) –( c+d ) ] =( a+b+c+d ) ( a+b–c–d )
  10. ( a–b ) 2 – ( c–d ) 2 =[ ( a–b ) +( c–d ) ] [ ( a–b ) –( c–d ) ] =( a–b+c–d ) ( a–b–c+d )
  11. ( x+1 ) 2 –16 x 2 =[ ( x+1 ) –4x ] [ ( x+1 ) +4x ] =( x+1–4x ) ( x+1+4x ) =( 1–3x ) ( 5x+1 )
  12. 64 m 2 – ( m–2n ) 2 =[ 8m–( m–2n ) ] [ 8m+( m–2n ) ] =( 8m–m+2n ) ( 8m+m–2n ) =( 7m+2n ) ( 9m–2n )
  13. ( a–2b ) 2 – ( x+y ) 2 =[ ( a–2b ) –( x+y ) ] [ ( a–2b ) +( x+y ) ] =( a–2b–x–y ) ( a–2b+x+y )
  14. ( 2a–c ) 2 – ( a+c ) 2 =[ ( 2a–c ) +( a+c ) ] [ ( 2a–c ) –( a+c ) ] =( 2a– c +a+ c ) ( 2a–c–a–c ) =3a( a–2c )
  15. ( x+1 ) 2 –4 x 2 =[ ( x+1 ) –2x ] [ ( x+1 ) +2x ] =( x+1–2x ) ( x+1+2x ) =( 1–x ) ( 3x+1 )
  16. 36 x 2 – ( a+3x ) 2 =[ 6x–( a+3x ) ] [ 6x+( a+3x ) ] =( 6x–a–3x ) ( 6x+a+3x ) =( 3x–a ) ( 9x+a )
  17. a 6 – ( a–1 ) 2 =[ a 3 –( a–1 ) ] [ a 3 +( a–1 ) ] =( a 3 –a+1 ) ( a 3 +a–1 )
  18. ( a–1 ) 2 – ( m–2 ) 2 =[ ( a–1 ) –( m–2 ) ] [ ( a–1 ) +( m–2 ) ] =( a–1–m+2 ) ( a–1+m–2 ) =( a–m+1 ) ( a+m–3 )
  19. ( 2x–3 ) 2 – ( x–5 ) 2 =[ ( 2x–3 ) +( x–5 ) ] [ ( 2x–3 ) –( x–5 ) ] =( 2x–3+x–5 ) ( 2x–3–x+5 ) =( 3x–8 ) ( x+2 )
  20. 1– ( 5a+2x ) 2 =[ 1+( 5a+2x ) ] [ 1–( 5a+2x ) ] =( 1+5a+2x ) ( 1–5a–2x )
  21. ( 7x+y ) 2 –81 =[ ( 7x+y ) –9 ] [ ( 7x+y ) +9 ] =( 7x+y–9 ) ( 7x+y+9 )
  22. m 6 – ( m 2 –1 ) 2 =[ m 3 –( m 2 –1 ) ] [ m 3 +( m 2 –1 ) ] =( m 3 – m 2 +1 ) ( m 3 + m 2 –1 )
  23. 16 a 10 – ( 2 a 2 +3 ) 2 =[ 4 a 5 –( 2 a 2 +3 ) ] [ 4 a 5 +( 2 a 2 +3 ) ] =( 4 a 5 –2 a 2 –3 ) ( 4 a 5 +2 a 2 +3 )
  24. ( x–y ) 2 – ( c+d ) 2 =[ ( x–y ) –( c+d ) ] [ ( x–y ) +( c+d ) ] =( x–y–c–d ) ( x–y+c+d )
  25. ( 2a+b–c ) 2 – ( a+b ) 2 =[ ( 2a+b–c ) –( a+b ) ] [ ( 2a+b–c ) +( a+b ) ] =( 2a+ b –c–a– b ) ( 2a+b–c+a+b ) =( a–c ) ( 3a+2b–c )
  26. 100– ( x–y+z ) 2 =[ 10–( x–y+z ) ] [ 10+( x–y+z ) ] =( 10–x+y–z ) ( 10+x–y+z )
  27. x 2 – ( y–x ) 2 =[ x–( y–x ) ] [ x+( y–x ) ] =( x–y+x ) ( x +y– x ) =y( 2x–y )
  28. ( 2x+3 ) 2 – ( 5x–1 ) 2 =[ ( 2x+3 ) –( 5x–1 ) ] [ ( 2x+3 ) +( 5x–1 ) ] =( 2x+3–5x+1 ) ( 2x+3+5x–1 ) =( 4–3x ) ( 7x+2 )
  29. ( x–y+z ) 2 – ( y–z+2x ) 2 =[ ( x–y+z ) –( y–z+2x ) ] [ ( x–y+z ) +( y–z+2x ) ] =( x–y+z–y+z–2x ) ( x–y+z+y–z+2x ) =( x–y+z–y+z–2x ) ( x– y + z + y – z +2x ) =3x( 2z–2y–x )
  30. ( 2x+1 ) 2 – ( x+4 ) 2 =[ ( 2x+1 ) –( x+4 ) ] [ ( 2x+1 ) +( x+4 ) ] =( 2x+1–x–4 ) ( 2x+1+x+4 ) =( x–3 ) ( 3x+5 )
  31. ( a+2x+1 ) 2 – ( x+a–1 ) 2 =[ ( a+2x+1 ) –( x+a–1 ) ] [ ( a+2x+1 ) +( x+a–1 ) ] =( a +2x+1–x– a +1 ) ( a+2x+ 1 +x+a– 1 ) =( x+2 ) ( 2a+3x )
  32. 4 ( x+a ) 2 –49 y 2 =[ 2( x+a ) –7y ] [ 2( x+a ) +7y ] =( 2x+2a–7y ) ( 2x+2a+7y )
  33. 25 ( x–y ) 2 –4 ( x+y ) 2 =[ 5( x–y ) –2( x+y ) ] [ 5( x–y ) +2( x+y ) ] =( 5x–5y–2x–2y ) ( 5x–5y+2x+2y ) =( 3x–7y ) ( 7x–3y )
  34. 36 ( m+n ) 2 –121 ( m–n ) 2 =[ 6( m+n ) –11( m–n ) ] [ 6( m+n ) +11( m–n ) ] =( 6m+6n–11m+11n ) ( 6m+6n+11m–11n ) =( 17n–5m ) ( 17m–5n )
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