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Ejercicio 65

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CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 65
Escribir, por simple inspección, el resultado de:
  1. ( x+y+z ) ( x+y–z ) =[ ( x+y ) +z ] [ ( x+y ) –z ] = ( x+y ) 2 – ( z ) 2 = x 2 +2xy+ y 2 – z 2
  2. ( x–y+z ) ( x+y–z ) =[ x–( y–z ) ] [ x+( y–z ) ] = ( x ) 2 – ( y–z ) 2 = x 2 –( y 2 –2yz+ z 2 ) = x 2 – y 2 +2yz– z 2
  3. ( x+y+z ) ( x–y–z ) =[ x+( y+z ) ] [ x–( y+z ) ] = ( x ) 2 – ( y+z ) 2 = x 2 –( y 2 +2yz+ z 2 ) = x 2 – y 2 –2yz– z 2
  4. ( m+n+1 ) ( m+n–1 ) =[ ( m+n ) +1 ] [ ( m+n ) –1 ] = ( m+n ) 2 – ( 1 ) 2 = m 2 +2m+ n 2 –1
  5. ( m–n–1 ) ( m–n+1 ) =[ ( m–n ) –1 ] [ ( m–n ) +1 ] = ( m–n ) 2 – ( 1 ) 2 = m 2 –2m+ n 2 –1
  6. ( x+y–2 ) ( x–y+2 ) =[ x+( y–2 ) ] [ x–( y–2 ) ] = ( x ) 2 – ( y–2 ) 2 = x 2 –( y 2 –4y+4 ) = x 2 – y 2 +4y–4
  7. ( n 2 +2n+1 ) ( n 2 –2n–1 ) =[ n 2 +( 2n+1 ) ] [ n 2 –( 2n+1 ) ] = ( n 2 ) 2 – ( 2n+1 ) 2 = n 4 –( 4 n 2 +4n+1 ) = n 4 –4 n 2 –4n–1
  8. ( a 2 –2a+3 ) ( a 2 +2a+3 ) =[ ( a 2 +3 ) –2a ] [ ( a 2 +3 ) +2a ] = ( a 2 +3 ) 2 – ( 2a ) 2 = a 4 +6 a 2 +9–4 a 2 = a 4 +2 a 2 +9
  9. ( m 2 –m+–1 ) ( m 2 +m+–1 ) =[ ( m 2 –1 ) –m ] [ ( m 2 –1 ) +m ] = ( m 2 –1 ) 2 – ( m ) 2 = m 4 –2 m 2 +1– m 2 = m 4 –3 m 2 +1
  10. ( 2a–b–c ) ( 2a–b+c ) =[ ( 2a–b ) –c ] [ ( 2a–b ) +c ] = ( 2a–b ) 2 – ( c ) 2 =4 a 2 +4ab+ b 2 – c 2
  11. ( 2x+y–z ) ( 2x–y+z ) =[ 2x+( y–z ) ] [ 2x–( y–z ) ] = ( 2x ) 2 – ( y–z ) 2 =4 x 2 –( y 2 –2yz+ z 2 ) =4 x 2 – y 2 +2yz– z 2
  12. ( x 2 –5x+6 ) ( x 2 +5x–6 ) =[ x 2 –( 5x–6 ) ] [ x 2 +( 5x–6 ) ] = ( x 2 ) 2 – ( 5x–6 ) 2 = x 4 –( 25 x 2 –60x+36 ) = x 4 –25 x 2 +60x–36
  13. ( a 2 –ab+ b 2 ) ( a 2 + b 2 +ab ) =[ ( a 2 + b 2 ) –ab ] [ ( a 2 + b 2 ) +ab ] = ( a 2 + b 2 ) 2 – ( ab ) 2 = a 4 +2 a 2 b 2 + b 4 – a 2 b 2 = a 4 + a 2 b 2 + b 4
  14. ( x 3 – x 2 –x ) ( x 3 + x 2 +x ) =[ x 3 –( x 2 +x ) ] [ x 3 +( x 2 +x ) ] = ( x 3 ) 2 – ( x 2 +x ) 2 = x 6 –( x 4 +2 x 3 + x 2 ) = x 6 – x 4 –2 x 3 – x 2
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