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分解因式x^4+y^4+z^4+1+8xyz-2(y^2z^2+z^2x^2+x^2y^2+x^2+y^2+z^2)
题目详情
分解因式x^4+y^4+z^4+1+8xyz-2(y^2z^2+z^2x^2+x^2y^2+x^2+y^2+z^2)
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答案和解析
(x^4)+(y^4)+(z^4)+1+8xyz-2(y?+z?+x?+x?+y?+z?)
=x^4+y^4+z^4-2x?-2x?+2y?-4y?-2x?-2y?-2z?+8xyz+1
=[(x?-y?-z?)]?-[4y?-8xyz+4x?]+2[x?-y?-z?]+1
=[x?-y?-z?+1]?-[2(yz-x)]?
=(x?-y?-z?+1+2yz-2x)(x?-y?-z?+1-2yz+2x)
=[(x-1)?-(y-z)?][(x+1)?-(y+z)?]
=(x+y-z-1)(x-y+z-1)(x+y+z+1)(x-y-z+1)
=x^4+y^4+z^4-2x?-2x?+2y?-4y?-2x?-2y?-2z?+8xyz+1
=[(x?-y?-z?)]?-[4y?-8xyz+4x?]+2[x?-y?-z?]+1
=[x?-y?-z?+1]?-[2(yz-x)]?
=(x?-y?-z?+1+2yz-2x)(x?-y?-z?+1-2yz+2x)
=[(x-1)?-(y-z)?][(x+1)?-(y+z)?]
=(x+y-z-1)(x-y+z-1)(x+y+z+1)(x-y-z+1)
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