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