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CAPITULO X

Descomposición Factorial
Ejercicio 108
Descomponer en cuatro factores:
  1. 1 a 8 =(1+ a 4 ) (1 a 4 ) =(1+ a 4 ) (1+ a 2 ) (1 a 2 ) =(1+ a 4 ) (1+ a 2 ) (1+a ) (1a )
  2. a 6 1 =( a 3 1 ) ( a 3 +1 ) =(a1 ) ( a 2 +a+1 ) (a+1 ) ( a 2 a+1 )
  3. x 4 41 x 2 +400 = x 4 41 x 2 +400+ x 2 x 2 = x 4 40 x 2 +400 x 2 = ( x 2 20 ) 2 x 2 =[( x 2 20 ) x ] [( x 2 20 ) +x ] =( x 2 x20 ) ( x 2 +x20 ) =( x 2 5x+4x20 ) ( x 2 4x+5x20 ) =[x(x5 ) +4(x5 ) ] [x(x4 ) +5(x4 ) ] =(x5 ) (x+4 ) (x4 ) (x+5 )
  4. a 4 2 a 2 b 2 + b 4 = ( a 2 b 2 ) 2 = [(ab ) (a+b ) ] 2 = (ab ) 2 (a+b ) 2 =(ab ) (ab ) (a+b ) (a+b )
  5. x 5 + x 3 2x =x( x 4 + x 2 2 ) . 2 1 | 2 =x( x 4 x 2 +2 x 2 2 ) =x[ x 2 ( x 2 1 ) +( x 2 1 ) ] =x( x 2 1 ) ( x 2 +1 ) =x(x1 ) (x+1 ) ( x 2 +1 )
  6. 2 x 4 +6 x 3 2x6 =2 x 3 (x+3 ) 2(x+3 ) =(x+3 ) (2 x 3 2 ) =2(x+3 ) ( x 3 1 ) =2(x+3 ) (x1 ) ( x 2 +x+1 )
  7. 3 x 4 243 =3( x 4 81 ) =3( x 2 +9 ) ( x 2 9 ) =3( x 2 +9 ) (x3 ) (x+3 )
  8. 16 x 4 8 x 2 y 2 + y 4 = (4 x 2 y 2 ) 2 = [(2x+y ) (2xy ) ] 2 = (2x+y ) 2 (2xy ) 2 =(2x+y ) (2x+y ) (2xy ) (2xy )
  9. 9 x 4 +9 x 3 y x 2 xy =9 x 3 (x+y ) x(x+y ) =(x+y ) (9 x 3 x ) =x(x+y ) (9 x 2 1 ) =x(x+y ) (3x1 ) (3x+1 )
  10. 12a x 4 +33a x 2 9a =3a(4 x 4 +11 x 2 3 ) . 12 1 | 12 =3a(4 x 4 x 2 +12 x 2 3 ) =3a[ x 2 (4 x 2 1 ) +3(4 x 2 1 ) ] =3a( x 2 +3 ) (4 x 2 1 ) =3a( x 2 +3 ) (2x1 ) (2x+1 )
  11. x 8 y 8 =( x 4 + y 4 ) ( x 4 y 4 ) =( x 4 + y 4 ) ( x 2 + y 2 ) ( x 2 y 2 ) =( x 4 + y 4 ) ( x 2 + y 2 ) (x+y ) (xy )
  12. x 6 7 x 3 8 = x 6 8 x 3 + x 3 8 . 8 1 | 8 = x 3 ( x 3 8 ) +( x 3 8 ) =( x 3 8 ) ( x 3 +8 ) =(x2 ) ( x 2 +2x+4 ) (x+2 ) ( x 2 2x+4 )
  13. 64 x 6 =(8 x 3 ) (8+ x 3 ) =(2x ) (4+2x+ x 2 ) (2+x ) (42x+ x 2 )
  14. a 5 a 3 b 2 a 2 b 3 + b 5 = a 3 ( a 2 b 2 ) b 3 ( a 2 b 2 ) =( a 2 b 2 ) ( a 3 b 3 ) =(ab ) (a+b ) (ab ) ( a 2 +ab+ b 2 )
  15. 8 x 4 +6 x 2 2 =2(4 x 4 +3 x 2 1 ) . 4 1 | 4 =2(4 x 4 x 2 +4 x 2 1 ) =2[ x 2 (4 x 2 1 ) +(4 x 2 1 ) ] =2(4 x 2 1 ) ( x 2 +1 ) =2(2x+1 ) (2x1 ) ( x 2 +1 )
  16. a 4 25 a 2 +144 = a 4 9 a 2 16 a 2 +144 . 144 72 36 18 9 1 | 2 2 2 2 9 = a 2 ( a 2 9 ) 16( a 2 9 ) =( a 2 9 ) ( a 2 16 ) =(a9 ) (a+3 ) (a+4 ) (a4 )
  17. a 2 x 3 a 2 y 3 +2a x 3 2a y 3 = a 2 ( x 3 y 3 ) +2a( x 3 y 3 ) =( a 2 +2a ) ( x 3 y 3 ) =a(a+2 ) (xy ) ( x 2 +xy+ y 2 )
  18. a 4 +2 a 3 a 2 2a = a 4 a 2 +2 a 3 2a = a 2 ( a 2 1 ) +2a( a 2 1 ) =( a 2 +2a ) ( a 2 1 ) =a(a+2 ) (a1 ) (a+1 )
  19. 12 a 3 + a 6 = (1 a 3 ) 2 = [(1a ) (1+a+ a 2 ) ] 2 = (1a ) 2 (1+a+ a 2 ) 2
  20. m 6 729 =( m 3 +27 ) ( m 3 27 ) =(m+3 ) ( m 2 3m+9 ) (m3 ) ( m 2 +3m+9 )
  21. x 5 x =x( x 4 1 ) =x( x 2 +1 ) ( x 2 1 ) =x( x 2 +1 ) (x1 ) (x+1 )
  22. x 5 x 3 y 2 + x 2 y 3 y 5 = x 3 ( x 2 y 2 ) + y 3 ( x 2 y 2 ) =( x 2 y 2 ) ( x 3 + y 3 ) =(x+y ) (xy ) (x+y ) ( x 2 xy+ y 2 )
  23. a 4 b a 3 b 2 a 2 b 3 +a b 4 = a 3 b(ab ) a b 3 (ab ) =(ab ) ( a 3 ba b 3 ) =ab(ab ) ( a 2 b 2 ) =ab(ab ) (a+b ) (ab )
  24. 5 a 4 3125 =5( a 4 625 ) =5( a 2 +25 ) ( a 2 25 ) =5( a 2 +25 ) (a5 ) (a+5 )
  25. ( a 2 +2a ) 2 2( a 2 +2a ) 3 = ( a 2 +2a ) 2 3( a 2 +2a ) +( a 2 +2a ) 3 . 3 1 | 3 =( a 2 +2a ) [( a 2 +2a ) 3 ] +[( a 2 +2a ) 3 ] =[( a 2 +2a ) 3 ] [( a 2 +2a ) +1 ] =( a 2 +2a3 ) ( a 2 +2a+1 ) =( a 2 a+3a3 ) (a+1 ) 2 =[a(a1 ) +3(a1 ) ] (a+1 ) 2 =(a+3 ) (a1 ) (a+1 ) 2 =(a+3 ) (a1 ) (a+1 ) (a+1 )
  26. a 2 x 3 +2a x 3 8 a 2 16a =a x 3 (a+2 ) 8a(a+2 ) =(a+2 ) (a x 3 8a ) =a(a+2 ) ( x 3 8 ) =a(a+2 ) (x2 ) ( x 2 +2x+4 )
  27. 1 a 6 b 6 =(1 a 3 b 3 ) (1+ a 3 b 3 ) =(1ab ) (1+ab+ a 2 b 2 ) (1+ab ) (1ab+ a 2 b 2 )
  28. 5a x 3 +10a x 2 5ax10a =5a x 3 5ax+10a x 2 10a =5ax( x 2 1 ) +10a( x 2 1 ) =( x 2 1 ) (5ax+10a ) =5a(x1 ) (x+1 ) (x+2 )
  29. a 2 x 2 + b 2 y 2 b 2 x 2 a 2 y 2 = a 2 x 2 a 2 y 2 b 2 x 2 + b 2 y 2 = a 2 ( x 2 y 2 ) b 2 ( x 2 y 2 ) =( x 2 y 2 ) ( a 2 b 2 ) =(x+y ) (xy ) (a+b ) (ab )
  30. x 8 + x 4 2 = x 8 x 4 +2 x 4 2 . 2 1 | 2 = x 4 ( x 4 1 ) +2( x 4 1 ) =( x 4 1 ) ( x 4 +2 ) =( x 2 1 ) ( x 2 +1 ) ( x 4 +2 ) =(x1 ) (x+1 ) ( x 2 +1 ) ( x 4 +2 )
  31. a 4 + a 3 9 a 2 9a = a 3 (a+1 ) 9a(a+1 ) =(a+1 ) ( a 3 9a ) =a(a+1 ) ( a 2 9 ) =a(a+1 ) (a3 ) (a+3 )
  32. a 2 x 2 + a 2 x6 a 2 x 2 x+6 = a 2 ( x 2 +x6 ) ( x 2 +x6 ) =( x 2 +x6 ) ( a 2 1 ) =( x 2 2x+3x6 ) ( a 2 1 ) =[x(x2 ) +3(x2 ) ] ( a 2 1 ) =(x+3 ) (x2 ) (a+1 ) (a1 )
  33. 16 m 4 25 m 2 +9 =16 m 4 25 m 2 +9+ m 2 m 2 =16 m 4 24 m 2 +9 m 2 = (4 m 2 3 ) 2 m 2 =[(4 m 2 3 ) m ] [(4 m 2 3 ) +m ] =(4 m 2 m3 ) (4 m 2 +m3 ) =(4 m 2 4m+3m3 ) (4 m 2 +4m3m3 ) =[4m(m1 ) +3(m1 ) ] [4m(m+1 ) 3(m+1 ) ] =(m1 ) (4m+3 ) (m+1 ) (4m3 )
  34. 3ab x 2 12ab+3b x 2 12b =3ab x 2 +3b x 2 12ab12b =3b x 2 (a+1 ) 12b(a+1 ) =(a+1 ) (3b x 2 12b ) =3b(a+1 ) ( x 2 4 ) =3b(a+1 ) (x2 ) (x+2 )
  35. 3 a 2 m+9am30m+3 a 2 +9a30 =3m( a 2 +3a10 ) +3( a 2 +3a10 ) . 10 5 1 | 2 5 =( a 2 +3a10 ) (3m+3 ) =3( a 2 2a+5a10 ) (m+1 ) =3[a(a2 ) +5(a2 ) ] (m+1 ) =3(a2 ) (a+5 ) (m+1 )
  36. a 3 x 2 5 a 3 x+6 a 3 + x 2 5x+6 = a 3 ( x 2 5x+6 ) +( x 2 5x+6 ) . 6 3 1 | 2 3 =( x 2 2x3x+6 ) ( a 3 +1 ) =[x(x2 ) 3(x2 ) ] ( a 3 +1 ) =(x3 ) (x2 ) ( a 3 +1 ) =(x3 ) (x2 ) (a+1 ) ( a 2 a+1 )
  37. x 2 ( x 2 y 2 ) (2x1 ) ( x 2 y 2 ) =( x 2 y 2 ) [ x 2 (2x1 ) ] =(xy ) (x+y ) ( x 2 2x+1 ) =(xy ) (x+y ) (x1 ) 2 =(xy ) (x+y ) (x1 ) (x1 )
  38. a( x 3 +1 ) +3ax(x+1 ) =a[( x 3 +1 ) +3x(x+1 ) ] =a[(x+1 ) ( x 2 x+1 ) +3x(x+1 ) ] =a[(x+1 ) {( x 2 x+1 ) +3x } ] =a[(x+1 ) { x 2 x+1+3x } ] =a(x+1 ) ( x 2 +2x+1 ) =a(x+1 ) (x+1 ) 2 =a(x+1 ) (x+1 ) (x+1 )