Ejercicio 96

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CAPITULO X

Descomposición Factorial
Ejercicio 96
Factorar o descomponer en dos factores:
  1. a 4 + a 2 +1 = a 4 + a 2 + a 2 +1 a 2 =( a 4 +2 a 2 +1 ) a 2 = ( a 2 +1 ) 2 a 2 =[( a 2 +1 ) a ] [( a 2 +1 ) +a ] =( a 2 a+1 ) ( a 2 +a+1 )
  2. m 4 + m 2 n 2 + n 4 = m 4 + m 2 n 2 + m 2 n 2 + n 4 m 2 n 2 =( m 4 +2 m 2 n 2 + n 4 ) m 2 n 2 = ( m 2 + n 2 ) 2 m 2 n 2 =[( m 2 + n 2 ) mn ] [( m 2 + n 2 ) +mn ] =( m 2 + n 2 mn ) ( m 2 + n 2 +mn )
  3. x 8 +3 x 4 +4 = x 8 +3 x 4 + x 4 +4 x 4 =( x 8 +4 x 4 +4 ) x 4 = ( x 4 +2 ) 2 x 4 =[( x 4 +2 ) x 2 ] [( x 4 +2 ) + x 2 ] =( x 4 x 2 +2 ) ( x 4 + x 2 +2 )
  4. a 4 +2 a 2 +9 = a 4 +2 a 2 +4 a 2 +94 a 2 =( a 4 +6 a 2 +9 ) 4 a 2 = ( a 2 +3 ) 2 4 a 2 =[( a 2 +3 ) 2a ] [( a 2 +3 ) +2a ] =( a 2 2a+3 ) ( a 2 +2a+3 )
  5. a 4 3 a 2 b 2 + b 4 = a 4 3 a 2 b 2 + a 2 b 2 + b 4 a 2 b 2 =( a 4 2 a 2 b 2 + b 4 ) a 2 b 2 = ( a 2 b 2 ) 2 a 2 b 2 =[( a 2 b 2 ) ab ] [( a 2 b 2 ) +ab ] =( a 2 ab b 2 ) ( a 2 +ab b 2 )
  6. x 4 6 x 2 +1 = x 4 6 x 2 +4 x 2 +14 x 2 =( x 4 2 x 2 +1 ) 4 x 2 = ( x 2 1 ) 2 4 x 2 =[( x 2 1 ) 2x ] [( x 2 1 ) +2x ] =( x 2 2x1 ) ( x 2 +2x1 )
  7. 4 a 4 +3 a 2 b 2 +9 b 4 =4 a 4 +3 a 2 b 2 +9 a 2 b 2 +9 b 4 9 a 2 b 2 =(4 a 4 +12 a 2 b 2 +9 b 4 ) 9 a 2 b 2 = (2 a 2 +3 b 2 ) 2 9 a 2 b 2 =[(2 a 2 +3 b 2 ) 3ab ] [(2 a 2 +3 b 2 ) +3ab ] =(2 a 2 3ab+3 b 2 ) (2 a 2 +3ab+3 b 2 )
  8. 4 x 4 29 x 2 +25 =4 x 4 29 x 2 +9 x 2 +259 x 2 =(4 x 4 20 x 2 +25 ) 9 x 2 = (2 x 2 5 ) 2 9 x 2 =[(2 x 2 5 ) 3x ] [(2 x 2 5 ) +3x ] =(2 x 2 3x5 ) (2 x 2 +3x5 )
  9. x 8 +4 x 4 y 4 +16 y 8 = x 8 +4 x 4 y 4 +4 x 4 y 4 +16 y 8 4 x 4 y 4 =( x 8 +8 x 4 y 4 +16 y 8 ) 4 x 4 y 4 = ( x 4 +4 y 4 ) 2 4 x 4 y 4 =[( x 4 +4 y 4 ) 2 x 2 y 2 ] [( x 4 +4 y 4 ) +2 x 2 y 2 ] =( x 4 2 x 2 y 2 +4 y 4 ) ( x 4 +2 x 2 y 2 +4 y 4 )
  10. 16 m 4 25 m 2 n 2 +9 n 4 =16 m 4 25 m 2 n 2 + m 2 n 2 +9 n 4 m 2 n 2 =(16 m 4 24 m 2 n 2 +9 n 4 ) m 2 n 2 = (4 m 2 3 n 2 ) 2 m 2 n 2 =[(4 m 2 3 n 2 ) mn ] [(4 m 2 3 n 2 ) +mn ] =(4 m 2 mn3 n 2 ) (4 m 2 +mn3 n 2 )
  11. 25 a 4 +54 a 2 b 2 +49 b 4 =25 a 4 +54 a 2 b 2 +16 a 2 b 2 +49 b 4 16 a 2 b 2 =(25 a 4 +70 a 2 b 2 +49 b 4 ) 16 a 2 b 2 = (5 a 2 +7 b 2 ) 2 16 a 2 b 2 =[(5 a 2 +7 b 2 ) 4ab ] [(5 a 2 +7 b 2 ) +4ab ] =(5 a 2 4ab+7 b 2 ) (5 a 2 +4ab+7 b 2 )
  12. 36 x 4 109 x 2 y 2 +49 y 4 =36 x 4 109 x 2 y 2 +25 x 2 y 2 +49 y 4 25 x 2 y 2 =(36 x 4 84 x 2 y 2 +49 y 4 ) 25 x 2 y 2 = (6 x 2 7 y 2 ) 2 25 x 2 y 2 =[(6 x 2 7 y 2 ) 5xy ] [(6 x 2 7 y 2 ) +5xy ] =(6 x 2 5xy7 y 2 ) (6 x 2 +5xy7 y 2 )
  13. 81 m 8 +2 m 4 +1 =81 m 8 +2 m 4 +16 m 4 +116 m 4 =(81 m 8 +18 m 4 +1 ) 16 m 4 = (9 m 4 +1 ) 2 16 m 4 =[(9 m 4 +1 ) 4 m 2 ] [(9 m 4 +1 ) +4 m 2 ] =(9 m 4 4 m 2 +1 ) (9 m 4 +4 m 2 +1 )
  14. c 4 45 c 2 +100 = c 4 45 c 2 +25 c 2 +10025 c 2 =( c 4 20 c 2 +100 ) 25 c 2 = ( c 2 10 ) 2 25 c 2 =[( c 2 10 ) 5c ] [( c 2 10 ) +5c ] =( c 2 5c10 ) ( c 2 +5c10 )
  15. 4 a 8 53 a 4 b 4 +49 b 8 =4 a 8 53 a 4 b 4 +25 a 4 b 4 +49 b 8 25 a 4 b 4 =(4 a 8 28 a 4 b 4 +49 b 8 ) 25 a 4 b 4 = (2 a 4 7 b 4 ) 2 25 a 4 b 4 =[(2 a 4 7 b 4 ) +5 a 2 b 2 ] [(2 a 4 7 b 4 ) 5 a 2 b 2 ] =(2 a 4 +5 a 2 b 2 7 b 4 ) (2 a 4 5 a 2 b 2 7 b 4 )
  16. 49+76 n 2 +64 n 4 =49+76 n 2 +36 n 2 +64 n 4 36 n 2 =(49+112 n 2 +64 n 4 ) 36 n 2 = (7+8 n 2 ) 2 36 n 2 =[(7+8 n 2 ) 6n ] [(7+8 n 2 ) +6n ] =(76n+8 n 2 ) (7+6n+8 n 2 )
  17. 25 x 4 139 x 2 y 2 +81 y 4 =25 x 4 139 x 2 y 2 +49 x 2 y 2 +81 y 4 49 x 2 y 2 =(25 x 4 90 x 2 y 2 +81 y 4 ) 49 x 2 y 2 = (5 x 2 9 y 2 ) 2 49 x 2 y 2 =[(5 x 2 9 y 2 ) 7xy ] [(5 x 2 9 y 2 ) +7xy ] =(5 x 2 7xy9 y 2 ) (5 x 2 +7xy9 y 2 )
  18. 49 x 8 +76 x 4 y 4 +100 y 8 =49 x 8 +76 x 4 y 4 +64 x 4 y 4 +100 y 8 64 x 4 y 4 =(49 x 8 +140 x 4 y 4 +100 y 8 ) 64 x 4 y 4 = (7 x 4 +10 y 4 ) 2 64 x 4 y 4 =[(7 x 4 +10 y 4 ) 8 x 2 y 2 ] [(7 x 4 +10 y 4 ) +8 x 2 y 2 ] =(7 x 4 8 x 2 y 2 +10 y 4 ) (7 x 4 +8 x 2 y 2 +10 y 4 )
  19. 4108 x 2 +121 x 4 =4108 x 2 +64 x 2 +121 x 4 64 x 2 =(444 x 2 +121 x 4 ) 64 x 2 = (211 x 2 ) 2 64 x 2 =[(211 x 2 ) 8x ] [(211 x 2 ) +8x ] =(28x11 x 2 ) (2+8x11 x 2 )
  20. 121 x 4 133 x 2 y 4 +36 y 8 =121 x 4 133 x 2 y 4 + x 2 y 4 +36 y 8 x 2 y 4 =(121 x 4 132 x 2 y 4 +36 y 8 ) x 2 y 4 = (11 x 2 6 y 4 ) 2 x 2 y 4 =[(11 x 2 6 y 4 ) x y 2 ] [(11 x 2 6 y 4 ) +x y 2 ] =(11 x 2 x y 2 6 y 4 ) (11 x 2 +x y 2 6 y 4 )
  21. 144+23 n 6 +9 n 12 =144+23 n 6 +49 n 6 +9 n 12 49 n 6 =144+72 n 6 +9 n 12 49 n 6 = (12+3 n 6 ) 2 49 n 6 =[(12+3 n 6 ) 7 n 3 ] [(12+3 n 6 ) +7 n 3 ] =(127 n 3 +3 n 6 ) (12+7 n 3 +3 n 6 )
  22. 169 c 4 + c 8 =169 c 4 + c 4 + c 8 c 4 =(168 c 4 + c 8 ) c 4 = (4 c 4 ) 2 c 4 =[(4 c 4 ) c 2 ] [(4 c 4 ) + c 2 ] =(4 c 2 c 4 ) (4+ c 2 c 4 )
  23. 64 a 4 169 a 2 b 4 +81 b 8 =64 a 4 169 a 2 b 4 +25 a 2 b 4 +81 b 8 25 a 2 b 4 =(64 a 4 144 a 2 b 4 +81 b 8 ) 25 a 2 b 4 = (8 a 2 9 b 4 ) 2 25 a 2 b 4 =[(8 a 2 9 b 4 ) 5a b 2 ] [(8 a 2 9 b 4 ) +5a b 2 ] =(8 a 2 5a b 2 9 b 4 ) (8 a 2 +5a b 2 9 b 4 )
  24. 225+5 m 2 + m 4 =225+5 m 2 +25 m 2 + m 4 25 m 2 =(225+30 m 2 + m 4 ) 25 m 2 = (15+ m 2 ) 2 25 m 2 =[(15+ m 2 ) 5m ] [(15+ m 2 ) +5m ] =(155m+ m 2 ) (15+5m+ m 2 )
  25. 1126 a 2 b 4 +169 a 4 b 8 =1126 a 2 b 4 +100 a 2 b 4 +169 a 4 b 8 100 a 2 b 4 =(126 a 2 b 4 +169 a 4 b 8 ) 100 a 2 b 4 = (113 a 2 b 4 ) 2 100 a 2 b 4 =[(113 a 2 b 4 ) 10a b 2 ] [(113 a 2 b 4 ) +10a b 2 ] =(110a b 2 13 a 2 b 4 ) (1+10a b 2 13 a 2 b 4 )
  26. x 4 y 4 +21 x 2 y 2 +121 = x 4 y 4 +21 x 2 y 2 + x 2 y 2 +121 x 2 y 2 =( x 4 y 4 +22 x 2 y 2 +121 ) x 2 y 2 = ( x 2 y 2 +11 ) 2 x 2 y 2 =[( x 2 y 2 +11 ) +xy ] [( x 2 y 2 +11 ) xy ] =( x 2 y 2 +xy+11 ) ( x 2 y 2 xy+11 )
  27. 49 c 8 +75 c 4 m 2 n 2 +196 m 4 n 4 =49 c 8 +75 c 4 m 2 n 2 +121 c 4 m 2 n 2 +196 m 4 n 4 121 c 4 m 2 n 2 =49 c 8 +196 c 4 m 2 n 2 +196 m 4 n 4 121 c 4 m 2 n 2 = (7 c 4 +14 m 2 n 2 ) 2 121 c 4 m 2 n 2 =[(7 c 4 +14 m 2 n 2 ) 11 c 2 mn ] [(7 c 4 +14 m 2 n 2 ) +11 c 2 mn ] =(7 c 4 11 c 2 mn+14 m 2 n 2 ) (7 c 4 +11 c 2 mn+14 m 2 n 2 )
  28. 81 a 4 b 8 292 a 2 b 4 x 8 +256 x 16 =81 a 4 b 8 292 a 2 b 4 x 8 +4 a 2 b 4 x 8 +256 x 16 4 a 2 b 4 x 8 =(81 a 4 b 8 288 a 2 b 4 x 8 +256 x 16 ) 4 a 2 b 4 x 8 = (9 a 2 b 4 16 x 8 ) 2 4 a 2 b 4 x 8 =[(9 a 2 b 4 16 x 8 ) 2a b 2 x 4 ] [(9 a 2 b 4 16 x 8 ) +2a b 2 x 4 ] =(9 a 2 b 4 2a b 2 x 4 16 x 8 ) (9 a 2 b 4 +2a b 2 x 4 16 x 8 )