CAPITULO X
Descomposición Factorial
Descomposición Factorial
- Ejercicio 109
Descomponer en cinco factores:
- 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 )
- 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 )
- 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 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 )
- 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 )
- 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 )
- 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 )
- 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 )
- 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 )
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Descomponer en seis factores - 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 )
- 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 )
- 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 )
- ( 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 )