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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 ) (x–y )
  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 ) (x–6 ) (x+2 ) (x–2 )
  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 ) (a–1 )
  4. 4 x 4 –8 x 2 +4 =4( x 4 –2 x 2 +1 ) =4 ( x 2 –1 ) 2 =4 [(x–1 ) (x+1 ) ] 2 =4 (x–1 ) 2 (x+1 ) 2 =4(x–1 ) (x–1 ) (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 ) (a–b ) ( 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 –a–2 ) –2 b 2 ( a 2 –a–2 ) =( a 2 –a–2 ) (2 a 2 –2 b 2 ) =a( a 2 – b 2 ) ( a 2 –a–2 ) =a( a 2 – b 2 ) ( a 2 –2a+a–2 ) =a(a+b ) (a–b ) [a(a–2 ) +(a–2 ) ] =a(a+b ) (a–b ) (a–2 ) (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 ) (x–3 ) (x+5 )
  8. 3–3 a 6 =3(1– a 6 ) =3(1– a 3 ) (1+ a 3 ) =3(1–a ) (1+a+ a 2 ) (1+a ) (1–a+ a 2 )
  9. 4a x 2 ( a 2 –2ax+ x 2 ) – a 3 +2 a 2 x–a x 2 =4a x 2 (a–x ) 2 –a( a 2 –2ax+ x 2 ) =4a x 2 (a–x ) 2 –a (a–x ) 2 = (a–x ) 2 [4a x 2 –a ] =a (a–x ) 2 [4 x 2 –1 ] =a(a–x ) (a–x ) (2x–1 ) (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 ) (x–3 )
    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 ) (x–1 )
  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 ) (x–2 ) (x+5 ) (x–5 )
  13. a 6 x 2 – x 2 + a 6 x–x = 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 ) (a–1 ) ( a 2 +a+1 ) (x+1 )
  14. ( a 2 –ax ) ( x 4 –82 x 2 +81 ) =a(a–x ) ( x 4 – x 2 –81 x 2 +81 ) . 81 1 | 81 =a(a–x ) [ x 2 ( x 2 –1 ) –81( x 2 –1 ) ] =a(a–x ) ( x 2 –1 ) ( x 2 –81 ) =a(a–x ) (x+1 ) (x–1 ) (x+9 ) (x–9 )
Categories: Capítulo X