1.xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)2.(y+z)(z+x)(x+y)+xyz是因式分解

Question

1.xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)
2.(y+z)(z+x)(x+y)+xyz
是因式分解

Answer 1

1.xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)
=xy(x^2-z^2+z^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)
=xy(x^2-z^2)+xy(z^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)
=(z^2-x^2)(zx-xy)+(y^2-z^2)(yz-xy)
=(z-x)(z-y)(x(z+x)-y(y+z))
=(z-x)(z-y)(z(x-y)+(x+y)(x-y))
=(z-x)(z-y)(x-y)(x+y+z)
2.(y+z)(z+x)(x+y)+xyz
=(x+y+z-x)(z+x)(x+y)+xyz
=(x+y+z)(z+x)(x+y)-x(z+x)(x+y)+xyz
=(x+y+z)(z+x)(x+y)-x^2z-x^3-x^2y
=(x+y+z)(z+x)(x+y)-x^2(x+y+z)
=(x+y+z)(xz+zy+xy)