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Ejercicio 104

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CAPITULO X

Descomposición Factorial
Ejercicio 104
Descomponer en 2 factores:
  1. 1+ ( x+y ) 3 =[ 1+( x+y ) ] [ 1–( x+y ) + ( x+y ) 2 ] =( x+y+1 ) ( 1–x–y+ x 2 +2xy+ y 2 )
  2. 1– ( a+b ) 3 =[ 1–( a+b ) ] [ 1+( a+b ) + ( a+b ) 2 ] =( 1–a–b ) ( 1+a+b+ a 2 +2ab+ b 2 )
  3. 27+ ( m–n ) 3 =[ 3+( m–n ) ] [ 3 2 –3( m–n ) + ( m–n ) 2 ] =( 3+m–n ) ( 9–3m+3n+ m 2 –2mn+ n 2 )
  4. ( x–y ) 3 –8 =[ ( x–y ) –2 ] [ ( x–y ) 2 +2( x–y ) +4 ] =( x–y–2 ) ( x 2 –2xy+ y 2 +2x–2y+4 )
  5. ( x+2y ) 3 +1 =[ ( x+2y ) +1 ] [ ( x+2y ) 2 –( x+2y ) +1 ] =( x+2y+1 ) ( x 2 +4xy+4 y 2 –x–2y+1 )
  6. 1– ( 2a–b ) 3 =[ 1–( 2a–b ) ] [ 1+( 2a–b ) + ( 2a–b ) 2 ] =( 1–2a+b ) ( 1+2a–b+4 a 2 –4ab+ b 2 )
  7. a 3 + ( a+1 ) 3 =[ a+( a+1 ) ] [ a 2 –a( a+1 ) + ( a+1 ) 2 ] =( 2a+1 ) ( a 2 – a 2 –a+ a 2 +2a+1 ) =( 2a+1 ) ( a 2 –a+1 )
  8. 8 a 3 – ( a–1 ) 3 =[ 2a–( a–1 ) ] [ ( 2a ) 2 +2a( a–1 ) + ( a–1 ) 2 ] =( 2a–a+1 ) ( 4 a 2 +2 a 2 –2a+ a 2 –2a+1 ) =( a+1 ) ( 7 a 2 –4a+1 )
  9. 27 x 3 – ( x–y ) 3 =[ 3x–( x–y ) ] [ ( 3x ) 2 +3x( x–y ) + ( x–y ) 2 ] =( 3x–x+y ) ( 9 x 2 +3 x 2 –3xy+ x 2 –2xy+ y 2 ) =( 2x+y ) ( 13 x 2 –5xy+ y 2 )
  10. ( 2a–b ) 3 –27 =[ ( 2a–b ) –3 ] [ ( 2a–b ) 2 +3( 2a–b ) + 3 2 ] =( 2a–b–3 ) ( 4 a 2 –4ab+ b 2 +6a–3b+9 )
  11. x 6 – ( x+2 ) 3 =[ x 2 –( x+2 ) ] [ ( x 2 ) 2 + x 2 ( x+2 ) + ( x+2 ) 2 ] =( x 2 –x–2 ) ( x 4 + x 4 +2 x 2 + x 2 +4x+4 ) =( x 2 –x–2 ) ( 2 x 4 +3 x 2 +4x+4 )
  12. ( a+1 ) 3 + ( a–3 ) 3 =[ ( a+1 ) +( a–3 ) ] [ ( a+1 ) 2 –( a+1 ) ( a–3 ) + ( a–3 ) 2 ] =( a+1+a–3 ) [ a 2 +2a+1–( a 2 –2a–3 ) + a 2 –6a+9 ] =( 2a–2 ) [ 2 a 2 –4a+10– a 2 +2a+3 ] =2( a–1 ) ( a 2 –2a+13 )
  13. ( x–1 ) 3 – ( x+2 ) 3 =[ ( x–1 ) –( x+2 ) ] [ ( x–1 ) 2 +( x–1 ) ( x+2 ) + ( x+2 ) 2 ] =[ x –1– x –2 ] [ x 2 –2x+1+ x 2 +x–2+ x 2 +4x+4 ] =–3( 3 x 2 +3x+3 ) =–9( x 2 +x+1 )
  14. ( x–y ) 3 – ( x+y ) 3 =[ ( x–y ) –( x+y ) ] [ ( x–y ) 2 +( x–y ) ( x+y ) + ( x+y ) 2 ] =[ x –y– x –y ] [ x 2 – 2xy + y 2 + x 2 – y 2 + x 2 + 2xy + y 2 ] =–2y( 3 x 2 + y 2 )
  15. ( m–2 ) 3 + ( m–3 ) 3 =[ ( m–2 ) +( m–3 ) ] [ ( m–2 ) 2 –( m–2 ) ( m–3 ) + ( m–3 ) 2 ] =[ m–2+m–3 ] [ m 2 –4m+4–( m 2 –5m+6 ) + m 2 –6m+9 ] =( 2m–5 ) [ 2 m 2 –10m+13– m 2 +5m–6 ] =( 2m–5 ) ( m 2 –5m+7 )
  16. ( 2x–y ) 3 + ( 3x+y ) 3 =[ ( 2x–y ) +( 3x+y ) ] [ ( 2x–y ) 2 –( 2x–y ) ( 3x+y ) + ( 3x+y ) 2 ] =[ 2x– y +3x+ y ] [ 4 x 2 –4xy+ y 2 –( 6 x 2 –xy– y 2 ) +9 x 2 +6xy+ y 2 ] =5x[ 13 x 2 +2xy+2 y 2 –6 x 2 +xy+ y 2 ] =5x( 7 x 2 +3xy+3 y 2 )
  17. 8 ( a+b ) 3 + ( a–b ) 3 =[ 2( a+b ) +( a–b ) ] { [ 2( a+b ) ] 2 –2( a+b ) ( a–b ) + ( a–b ) 2 } =[ 2a+2b+a–b ] { 4 ( a+b ) 2 –2( a 2 – b 2 ) + a 2 –2ab+ b 2 } =( 3a+b ) { 4( a 2 +2ab+ b 2 ) –2 a 2 +2 b 2 + a 2 –2ab+ b 2 } =( 3a+b ) { 4 a 2 +8ab+4 b 2 – a 2 +3 b 2 –2ab } =( 3a+b ) ( 3 a 2 +6ab+7 b 2 )
  18. 64 ( m+n ) 3 –125 =[ 4( m+n ) –5 ] { [ 4( m+n ) ] 2 +20( m+n ) + 5 2 } =[ 4m+4n–5 ] { 16 ( m+n ) 2 +20m+20n+25 } =( 4m+4n–5 ) { 16( m 2 +2mn+ n 2 ) +20m+20n+25 } =( 4m+4n–5 ) ( 16 m 2 +32mn+16 n 2 +20m+20n+25 )
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