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(18)XY(X^2-Y^2)+YZ(Y^2-Z^2)+ZX(Z^2-X^2)

题目详情
(18)
XY(X^2-Y^2)+YZ(Y^2-Z^2)+ZX(Z^2-X^2)
▼优质解答
答案和解析
xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)
=x^3y-xy^3+y^3z-yz^3+z^3x-zx^3
=x^3(y-z)+y^3(z-x)+z^3(x-y)
因为x-y=(x-z)+(z-y),所以:
=x^3(y-z)+y^3(z-x)+z^3[(x-z)+(z-y)]
=(x^3-z^3)(y-z)+(y^3-z^3)(z-x)
=(x-z)(x^2+xz+z^2)(y-z)+(y-z)(y^2+yz+z^2)(z-x)
=(x-z)(y-z)(x^2+zx+z^2-y^2-yz-z^2)
=(x-z)(y-z)(x-y)(x+y+z)
另外,观察可知它是轮换式,根据轮换式的做法,
可以设为:=k(x-y)(y-z)(z-x)(x+y+z)
待定系数法~