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Ejercicio 96

Gestor Baldor
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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 +9–4 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 +1–4 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 –2x–1 ) ( x 2 +2x–1 )
  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 +25–9 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 –3x–5 ) ( 2 x 2 +3x–5 )
  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 –mn–3 n 2 ) ( 4 m 2 +mn–3 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 –5xy–7 y 2 ) ( 6 x 2 +5xy–7 y 2 )
  13. 81 m 8 +2 m 4 +1 =81 m 8 +2 m 4 +16 m 4 +1–16 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 +100–25 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 –5c–10 ) ( c 2 +5c–10 )
  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 ] =( 7–6n+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 –7xy–9 y 2 ) ( 5 x 2 +7xy–9 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. 4–108 x 2 +121 x 4 =4–108 x 2 +64 x 2 +121 x 4 –64 x 2 =( 4–44 x 2 +121 x 4 ) –64 x 2 = ( 2–11 x 2 ) 2 –64 x 2 =[ ( 2–11 x 2 ) –8x ] [ ( 2–11 x 2 ) +8x ] =( 2–8x–11 x 2 ) ( 2+8x–11 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 ] =( 12–7 n 3 +3 n 6 ) ( 12+7 n 3 +3 n 6 )
  22. 16–9 c 4 + c 8 =16–9 c 4 + c 4 + c 8 – c 4 =( 16–8 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 ] =( 15–5m+ m 2 ) ( 15+5m+ m 2 )
  25. 1–126 a 2 b 4 +169 a 4 b 8 =1–126 a 2 b 4 +100 a 2 b 4 +169 a 4 b 8 –100 a 2 b 4 =( 1–26 a 2 b 4 +169 a 4 b 8 ) –100 a 2 b 4 = ( 1–13 a 2 b 4 ) 2 –100 a 2 b 4 =[ ( 1–13 a 2 b 4 ) –10a b 2 ] [ ( 1–13 a 2 b 4 ) +10a b 2 ] =( 1–10a 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 )
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