NMC 2018 Problem 4

Let $f = f(x,y,z)$ be a polynomial in three variables $x$ , $y$ , $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$ . Show that there exist three polynomials $A$ , $B$ , $C$ in these same three variables such that $A + B + C = 0$ and $$f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x).$$ Is there any polynomial $f$ for which these $A$ , $B$ , $C$ are uniquely determined?