KöMaL 2020 B5129

Two players are taking turns in selecting one of the coefficients $a$, $b$ and $c$ of the polynomial $x^3+ax^2+bx+c$, and giving it some integer value of their choice. Prove that the starting player can achieve that (after the three steps) all three roots of the polynomial should be integers (i.e. that the polynomial can be expressed as a product of three polynomials of integer coefficients).