求证:(x的3次方+y的3次方+z的3次方)-3xyz能被x+y+z整除,其中x,y,z均为正整数

Question

求证:(x的3次方+y的3次方+z的3次方)-3xyz能被x+y+z整除,其中x,y,z均为正整数

Answer 1

x^3+y^3+z^3-3xyz
=x^3+x^2y+x^2z+y^2x+y^3+y^2z+z^2x+z^2y+z^3-x^2y-y^2x-xyz-xyz-y^2z-yz^2-x^2z-xyz-z^2x
=x^2(x+y+z)+y^2(x+y+z)+z^2(x+y+z)-xy(x+y+z)-yz(x+y+z)-zx(x+y+z)
=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)
所以能被x+y+z整除
如果本题有什么不明白可以追问,