解 分三步分解
[1],(yz^5+zx^5+xy^5)-(zy^5+xz^5+yx^5)
=-yz(y^4-z^4)+zx(x^4-z^4)-xy(x^4-y^4)
=-yz(y-z)(y+z)*(y^2+z^2)+zx(x-y+y-z)(z+x)(z^2+x^2)-xy(x-y)(x+y)(x^2+y^2)
=-x(x-y)(y-z)[(y+z)(y^2+z^2)+x(y^2+z^2+yz)+x^2*(y+z)+x^3]+z(y-z)(x-y)[((x+y)(x^2+y^2)+z(x^2+y^2+xy)+z^2*(x+y)+z^3]
=(y-z)(x-y)(z-x)[(z^3+x^3+zx(z+x)+y(z^2+x^2+zx)+zx(z+x)+y^2*(z+x)+xyz+y^3-xyz+xyz-zx(z+x)]
=(y-z)(x-y)(z-x)[x^3+y^3+z^3+yz(y+z)+zx(z+x)+xy(x+y)+xyz]
[2],(y^2*z^4+z^2*x^4+x^2*y^4)-(y^4*z^2+z^4*x^2+x^4*y^2)
=-y^2*z^2(y-z)(y+z)+z^2*x^2(z+x)(x-y+y-z)-x^2*y^2(x-y)(x+y)
=z^2*(y-z)*(x-y)[x^2+xy+y^2+z(x+y)]-x^2*(y-z)(x-y)[z^2+yz+y^2+x(y+z)]
=(y-z)(x-y)(z-x)[xyz+y^2(z+x)+zx(z+x)+y(z^2+x^2+zx)]
=(y-z)(x-y)(z-x)[2xyz+yz(y+z)+zx(z+x)+xy(x+y)]
[3],xyx(yz^2+zx^2+xy^2)-xyz(zy^2+yx^2+xz^2)
=xyz[-yz(y-z)+zx(x-y+y-z)-xy(x-y)]
=xyz[z(y-z)(x-y)-x(x-y)(y-z)]
=xyz(y-z)(x-y)(z-x)
所以上式分解为
(y-z)(x-y)(z-x)[x^3+y^3+z^3+yz(y+z)+zx(z+x)+xy(x+y)+xyz-2xyz-yz(y+z)-zx(z+x)-xy(x+y)+xyz]
=(y-z)(x-y)(z-x)(x^3+y^3+z^3)