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Ejercicio 68

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CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES
Ejercicio 68
Miscelanea
Escribir, por simple inspección, el resultado de:
  1. ( x+2 ) 2 = ( x ) 2 +2( x ) ( 2 ) + ( 2 ) 2 = x 2 +4x+4
  2. ( x+2 ) ( x+3 ) = ( x ) 2 +( 2+3 ) ( x ) +( 2 × 3 ) = x 2 +5x+6
  3. ( x+1 ) ( x–1 ) = ( x ) 2 – ( 1 ) 2 = x 2 –1
  4. ( x–1 ) 2 = ( x ) 2 –2( x ) ( 1 ) + ( 1 ) 2 = x 2 –2x+1
  5. ( n+3 ) ( n+5 ) = ( n ) 2 +( 3+5 ) ( n ) +[ 3 × 5 ] = n 2 +8n+15
  6. ( m–3 ) ( m+3 ) = ( m ) 2 – ( 3 ) 2 = m 2 –9
  7. ( a+b+1 ) ( a+b–1 ) =[ ( a+b ) +1 ] [ ( a+b ) –1 ] = ( a+b ) 2 – ( 1 ) 2 = a 2 +2ab+ b 2 –1
  8. ( 1+b ) 3 = ( 1 ) 3 +3 ( 1 ) 2 ( b ) +3( 1 ) ( b ) 2 + ( b ) 3 =1+3b+3 b 2 + b 3
  9. ( a 2 +4 ) ( a 2 –4 ) = ( a 2 ) 2 – ( 4 ) 2 = a 4 –16
  10. ( 3ab–5 x 2 ) 2 = ( 3ab ) 2 –2( 3ab ) ( 5 x 2 ) + ( 5 x 2 ) 2 =9 a 2 b 2 –30ab x 2 +25 x 4
  11. ( ab+3 ) ( 3–ab ) =( 3+ab ) ( 3–ab ) = ( 3 ) 2 – ( ab ) 2 =9– a 2 b 2
  12. ( 1–4ax ) 2 = ( 1 ) 2 +2( 1 ) ( 4ax ) + ( 4ax ) 2 =1–8ax+16 a 2 x 2
  13. ( a 2 +8 ) ( a 2 –7 ) = ( a 2 ) 2 +( 8–7 ) ( a 2 ) +[ 8( –7 ) ] = a 4 + a 2 –56
  14. ( x+y+1 ) ( x–y–1 ) =[ x+( y+1 ) ] [ x–( y+1 ) ] = ( x ) 2 – ( y+1 ) 2 = x 2 –( y 2 +2y+1 ) = x 2 – y 2 –2y–1
  15. ( 1–a ) ( a+1 ) =( 1–a ) ( 1+a ) =1– a 2
  16. ( m–8 ) ( m+12 ) = ( m ) 2 +( –8+12 ) ( m ) +[ ( –8 ) × 12 ] = m 2 +4m–96
  17. ( x 2 –1 ) ( x 2 +3 ) = ( x 2 ) 2 +( –1+3 ) ( x 2 ) +[ ( –1 ) × 3 ] = x 4 +2 x 2 –3
  18. ( x 3 +6 ) ( x 3 –8 ) = ( x 3 ) 2 +( 6–8 ) ( x 3 ) +[ 6( –8 ) ] = x 6 –2 x 3 –48
  19. ( 5 x 3 +6 m 4 ) 2 = ( 5 x 3 ) 2 +2( 5 x 3 ) ( 6 m 4 ) + ( 6 m 4 ) 2 =25 x 6 +60 x 3 m 4 +36 m 8
  20. ( x 4 –2 ) ( x 4 +5 ) = ( x 4 ) 2 +( –2+5 ) ( x 4 ) +[ ( –2 ) × 5 ] = x 8 +3 x 4 –10
  21. ( 1–a+b ) ( b–a–1 ) =[ ( b–a ) +1 ] [ ( b–a ) –1 ] = ( b–a ) 2 – ( 1 ) 2 = b 2 –2ab+ a 2 –1
  22. ( a x + b n ) ( a x – b n ) = ( a x ) 2 – ( b n ) 2 = a 2x – b 2n
  23. ( x a+1 –8 ) ( x a+1 +9 ) = ( x a+1 ) 2 +( –8+9 ) ( x a+1 ) +[ ( –8 ) × 9 ] = x 2a+2 + x a+1 –72
  24. ( a 2 b 2 + c 2 ) ( a 2 b 2 – c 2 ) = ( a 2 b 2 ) 2 – ( c 2 ) 2 = a 4 b 2 – c 4
  25. ( 2a+x ) 3 = ( 2a ) 3 +3 ( 2a ) 2 ( x ) +3( 2a ) ( x ) 2 + ( x ) 3 =8 a 3 +12 a 2 x+6a x 2 + x 3
  26. ( x 2 –11 ) ( x 2 –2 ) = ( x 2 ) 2 +( –11–2 ) ( x 2 ) +[ ( –11 ) ( –2 ) ] = x 4 –13 x 2 +22
  27. ( 2 a 3 –5 b 4 ) 2 = ( 2 a 3 ) 2 +2( 2 a 3 ) ( 5 b 4 ) + ( 5 b 4 ) 2 =4 a 6 –20 a 3 b 4 +25 b 8
  28. ( a 3 +12 ) ( a 3 –15 ) = ( a 3 ) 2 +( 12–15 ) ( a 3 ) +[ 12( –15 ) ] = a 6 –3 a 3 –180
  29. ( m 2 –m+n ) ( n+m+ m 2 ) =[ ( m 2 +n ) –m ] [ ( m 2 +n ) +m ] = ( m 2 +n ) 2 – ( m ) 2 = m 4 +2mn+ n 2 – m 2
  30. ( x 4 +7 ) ( x 4 –11 ) = ( x 4 ) 2 +( 7–11 ) ( x 4 ) +[ 7( –11 ) ] = x 8 –4 x 4 –77
  31. ( 11–ab ) 2 = ( 11 ) 2 +2( 11 ) ( ab ) + ( ab ) 2 =121–22ab+ a 2 b 2
  32. ( x 2 y 3 –8 ) ( x 2 y 3 +6 ) = ( x 2 y 3 ) 2 +( –8+6 ) ( x 2 y 3 ) +[ ( –8 ) × 6 ] = x 4 y 6 –2 x 2 y 3 –48
  33. ( a+b ) ( a–b ) ( a 2 – b 2 ) =( a 2 – b 2 ) ( a 2 – b 2 ) = ( a 2 – b 2 ) 2 = ( a 2 ) 2 –2( a 2 ) ( b 2 ) + ( b 2 ) 2 = a 4 –2 a 2 b 2 + b 4
  34. ( x+1 ) ( x–1 ) ( x 2 –2 ) =( x 2 –1 ) ( x 2 –2 ) = ( x 2 ) 2 +( –1–2 ) ( x 2 ) +[ ( –1 ) ( –2 ) ] = x 4 –3 x 2 +2
  35. ( a+3 ) ( a 2 +9 ) ( a–3 ) =( a 2 –9 ) ( a 2 +9 ) = ( a 2 ) 2 – ( 9 ) 2 = a 4 –81
  36. ( x+5 ) ( x–5 ) ( x 2 +1 ) =( x 2 –25 ) ( x 2 +1 ) = ( x 2 ) 2 +( –25+1 ) ( x 2 ) +[ ( –25 ) × 1 ] = x 4 –24x–25
  37. ( a+1 ) ( a–1 ) ( a+2 ) ( a–2 ) =( a 2 –1 ) ( a 2 –4 ) = ( a 2 ) 2 +( –1–4 ) ( a 2 ) +[ ( –1 ) ( –4 ) ] = a 4 –5 a 2 +4
  38. ( a+2 ) ( a–3 ) ( a–2 ) ( a+3 ) =( a 2 –4 ) ( a 2 –9 ) = ( a 2 ) 2 +( –4–9 ) ( a 2 ) +[ ( –4 ) ( –9 ) ] = a 4 –13 a 2 +36
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