Ejercicio 109

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CAPITULO X

Descomposición Factorial
Ejercicio 109
Descomponer en cinco factores:
  1. x 9 x y 8 =x( x 8 y 8 ) =x( x 4 + y 4 ) ( x 4 y 4 ) =x( x 4 + y 4 ) ( x 2 + y 2 ) ( x 2 y 2 ) =x( x 4 + y 4 ) ( x 2 + y 2 ) (x+y ) (xy )
  2. x 5 40 x 3 +144x =x( x 4 40 x 2 +144 ) . 144 72 36 1 | 2 2 36 =x( x 4 36 x 2 4 x 2 +144 ) =x[ x 2 ( x 2 36 ) 4( x 2 36 ) ] =x( x 2 36 ) ( x 2 4 ) =x(x+6 ) (x6 ) (x+2 ) (x2 )
  3. a 6 + a 3 b 3 a 4 a b 3 =a[ a 5 + a 2 b 3 a 3 b 3 ] =a[ a 2 ( a 3 + b 3 ) ( a 3 + b 3 ) ] =a( a 3 + b 3 ) ( a 2 1 ) =a(a+b ) ( a 2 ab+ b 2 ) (a+1 ) (a1 )
  4. 4 x 4 8 x 2 +4 =4( x 4 2 x 2 +1 ) =4 ( x 2 1 ) 2 =4 [(x1 ) (x+1 ) ] 2 =4 (x1 ) 2 (x+1 ) 2 =4(x1 ) (x1 ) (x+1 ) (x+1 )
  5. a 7 a b 6 =a( a 6 b 6 ) =a( a 3 + b 3 ) ( a 3 b 3 ) =a(a+b ) ( a 2 ab+ b 2 ) (ab ) ( a 2 +ab+ b 2 )
  6. 2 a 4 2 a 3 4 a 2 2 a 2 b 2 +2a b 2 +4 b 2 =2 a 2 ( a 2 a2 ) 2 b 2 ( a 2 a2 ) =( a 2 a2 ) (2 a 2 2 b 2 ) =a( a 2 b 2 ) ( a 2 a2 ) =a( a 2 b 2 ) ( a 2 2a+a2 ) =a(a+b ) (ab ) [a(a2 ) +(a2 ) ] =a(a+b ) (ab ) (a2 ) (a+1 )
  7. x 6 +5 x 5 81 x 2 405x = x 6 81 x 2 +5 x 5 405x = x 2 ( x 4 81 ) +5x( x 4 81 ) =( x 4 81 ) ( x 2 +5x ) =x( x 2 +9 ) ( x 2 9 ) (x+5 ) =x( x 2 +9 ) (x+3 ) (x3 ) (x+5 )
  8. 33 a 6 =3(1 a 6 ) =3(1 a 3 ) (1+ a 3 ) =3(1a ) (1+a+ a 2 ) (1+a ) (1a+ a 2 )
  9. 4a x 2 ( a 2 2ax+ x 2 ) a 3 +2 a 2 xa x 2 =4a x 2 (ax ) 2 a( a 2 2ax+ x 2 ) =4a x 2 (ax ) 2 a (ax ) 2 = (ax ) 2 [4a x 2 a ] =a (ax ) 2 [4 x 2 1 ] =a(ax ) (ax ) (2x1 ) (2x+1 )
  10. x 7 + x 4 81 x 3 81 = x 4 ( x 3 +1 ) 81( x 3 +1 ) =( x 3 +1 ) ( x 4 81 ) =(x+1 ) ( x 2 x+1 ) ( x 2 +9 ) ( x 2 9 ) =(x+1 ) ( x 2 x+1 ) ( x 2 +9 ) (x+3 ) (x3 )
    Descomponer en seis factores
  11. x 17 x =x( x 16 1 ) =x( x 8 +1 ) ( x 8 1 ) =x( x 8 +1 ) ( x 4 +1 ) ( x 4 1 ) =x( x 8 +1 ) ( x 4 +1 ) ( x 2 +1 ) ( x 2 1 ) =x( x 8 +1 ) ( x 4 +1 ) ( x 2 +1 ) (x+1 ) (x1 )
  12. 3 x 6 75 x 4 48 x 2 +1200 =3 x 6 48 x 2 75 x 4 +1200 =3 x 2 ( x 4 16 ) 75( x 4 16 ) =( x 4 16 ) (3 x 2 75 ) =3( x 2 +4 ) ( x 2 4 ) ( x 2 25 ) =3( x 2 +4 ) (x+2 ) (x2 ) (x+5 ) (x5 )
  13. a 6 x 2 x 2 + a 6 xx = x 2 ( a 6 1 ) +x( a 6 1 ) =( a 6 1 ) ( x 2 +x ) =x( a 3 +1 ) ( a 3 1 ) (x+1 ) =x(a+1 ) ( a 2 a+1 ) (a1 ) ( a 2 +a+1 ) (x+1 )
  14. ( a 2 ax ) ( x 4 82 x 2 +81 ) =a(ax ) ( x 4 x 2 81 x 2 +81 ) . 81 1 | 81 =a(ax ) [ x 2 ( x 2 1 ) 81( x 2 1 ) ] =a(ax ) ( x 2 1 ) ( x 2 81 ) =a(ax ) (x+1 ) (x1 ) (x+9 ) (x9 )