当m取什么值时,x的3次方+y的3次方+z的3次方+mxyz(xyz不等于0)能被x+y+z整除

可以这样想，就是设法提出x+y+z出来，采用降次的方法，过程如下：
x＾3+y＾3+z＾3
=x＾2(x+y+z)+y＾2(x+y+z)+z＾2(x+y+z)-x＾2(y+z)-y＾2(x+z)-z＾2(x+y)
=(x＾2+y＾2+z＾2)(x+y+z)-x＾2(y+z)-y＾2(x+z)-z＾2(x+y)
=(x＾2+y＾2+z＾2)(x+y+z)-[xy(x+y+z)+xz（x+y+z)+yz(x+y+z)-3xyz]
=(x＾2+y＾2+z＾2)(x+y+z)-（xy+xz+yz)(x+y+z）+3xyz
故当m=-3时，代数式能被x+y+z整除

xyz≠0
x^3+y^3+z^3+mxyz
=(x^3+y^3+z^3+3x^2y+3xy^2+3y^2z+3yz^2+3x^2z+3xz^2+6xyz)-(3x^2y+3xy^2+3y^2z+3yz^2+3x^2z+3xz^2+6xyz)+ mxyz
=(x+y+z)^3-(3x^2y+3xy^2+3xyz-3xyz)-(3y^2z+3yz^2+3xyz-3xyz)-(3x^2z+3xz^2+3xyz-3xyz)-6xyz+ mxyz
=(x+y+z)^3-3xy(x+y+z)-3yz(x+y+z)-3xz(x+y+z)+3*3xyz-6xyz+mxyz
=(x+y+z)^3-3xy(x+y+z)-3yz(x+y+z)-3xz(x+y+z)+(m+3)xyz
当m=-3时，x^3+y^3+z^3+mxyz =(x+y+z)^3-3xy(x+y+z)-3yz(x+y+z)-3xz(x+y+z)，可以被（x+y+z)整除
整除的结果是：(x+y+z)^2-3xy-3yz-3xz