This is a typical symmetric polynomial factorization problem. $$x+y+z = 0 \implies f(x, y, z)=0 \implies f(x,y,z) = (x+y+z)(m(x^2+y^2+z^2)+n(xy+yz+zx))$$ where $m$ and $n$ are to-be-determined constants. Setting $(x, y, z)=(1,1,1)$ and $(0,0,1)$, respectively, yields $m=1$ and $n=-1$. Therefore, $$ f(x,y,z)=\boxed{(x+y+z)(x^2+y^2+z^2 -xy-yz-zx)}$$