How do you factor x^10-9x^8+x^7+4x^6-9x^5-36x^4+4x^3-36x?
x^10-9x^8+x^7+4x^6-9x^5-36x^4+4x^3-36x
= x ( x^9 - 9x^7 + x^6 + 4x^5 - 9 x^4 - 36 x^3 + 4x^2 - 36)
= x ( x-3) ( x^8 + 3x^7 + x^5 + 7x^4 + 12 x^3 + 4x + 12)
= x ( x-3) ( x+3) ( x^7 + x^4 + 4x^3 +4)
= x ( x-3) (x+3) [ x^4( x^3+1) + 4 ( x^3 +1)]
= x( x-3) (x+3) ( x^4 + 4) ( x^3 +1)
= x( x-3) ( x+3) ( x^4 + 4) ( x+1) (x^2 - x +1)
*** If you want: ( I think it's not necessary to do this step, because x^4 + 4 >0)
x^4 + 4 = ( x^2) ^ 2 + 2^2 + 2.2.x^2 - 2.2.x^2
= ( x^2 + 2)^2 - ( 2x) ^ 2
= ( x^2 + 2x + 2) ( x^2 - 2x +2)
Thus:
= x( x-3)( x+3)( x+1)(x^2 - x +1)( x^2 + 2x + 2)( x^2 - 2x +2)
x^10 - 9x^8 + x^7 + 4x^6 - 9x^5 - 36x^4 + 4x^3 - 36x
= (x^10 + x^7 + 4x^6 + 4x^3) - (9x^8 + 9x^5 + 36x^4 + 36x)
= x^3 (x^7 + x^4 + 4x^3 + 4) - 9x (x^7 + x^4 + 4x^3 + 4)
= x(x^2 - 9) [x^7 + x^4 + 4(x^3 + 1)]
= x(x - 3)(x + 3)[ x^4(x^3 + 1) + 4(x^3 + 1) ]
= x(x-3)(x+3)(x^3+1)(x^4+4)
= x(x-3)(x+3)(x+1)(x^2-x+1)(x^4 + 4x^2 + 4 - 4x^2)
= x(x-3)(x+3)(x+1)(x^2-x+1)[(x^2 + 2)^2 - (2x)^2]
= x(x-3)(x+3)(x+1)(x^2-x+1)(x^2 + 2x + 2)(x^2 - 2x + 2)
what? try to make it more clear.
do u mean x(pwr)10? or x^10?
Comments
x^10-9x^8+x^7+4x^6-9x^5-36x^4+4x^3-36x
= x ( x^9 - 9x^7 + x^6 + 4x^5 - 9 x^4 - 36 x^3 + 4x^2 - 36)
= x ( x-3) ( x^8 + 3x^7 + x^5 + 7x^4 + 12 x^3 + 4x + 12)
= x ( x-3) ( x+3) ( x^7 + x^4 + 4x^3 +4)
= x ( x-3) (x+3) [ x^4( x^3+1) + 4 ( x^3 +1)]
= x( x-3) (x+3) ( x^4 + 4) ( x^3 +1)
= x( x-3) ( x+3) ( x^4 + 4) ( x+1) (x^2 - x +1)
*** If you want: ( I think it's not necessary to do this step, because x^4 + 4 >0)
x^4 + 4 = ( x^2) ^ 2 + 2^2 + 2.2.x^2 - 2.2.x^2
= ( x^2 + 2)^2 - ( 2x) ^ 2
= ( x^2 + 2x + 2) ( x^2 - 2x +2)
Thus:
x^10-9x^8+x^7+4x^6-9x^5-36x^4+4x^3-36x
= x( x-3)( x+3)( x+1)(x^2 - x +1)( x^2 + 2x + 2)( x^2 - 2x +2)
x^10 - 9x^8 + x^7 + 4x^6 - 9x^5 - 36x^4 + 4x^3 - 36x
= (x^10 + x^7 + 4x^6 + 4x^3) - (9x^8 + 9x^5 + 36x^4 + 36x)
= x^3 (x^7 + x^4 + 4x^3 + 4) - 9x (x^7 + x^4 + 4x^3 + 4)
= x(x^2 - 9) [x^7 + x^4 + 4(x^3 + 1)]
= x(x - 3)(x + 3)[ x^4(x^3 + 1) + 4(x^3 + 1) ]
= x(x-3)(x+3)(x^3+1)(x^4+4)
= x(x-3)(x+3)(x+1)(x^2-x+1)(x^4 + 4x^2 + 4 - 4x^2)
= x(x-3)(x+3)(x+1)(x^2-x+1)[(x^2 + 2)^2 - (2x)^2]
= x(x-3)(x+3)(x+1)(x^2-x+1)(x^2 + 2x + 2)(x^2 - 2x + 2)
what? try to make it more clear.
do u mean x(pwr)10? or x^10?