If one root of the polynomial equation $$Ax^3 + Bx^2 + Cx + D = 0$$, $$A \ne 0$$, is the arithmetic mean of the other two, find a simplest relation between $$A$$, $$B$$, $$C$$ and $$D$$.

PLACEHOLDER ANSWER - the answer is a relation. Let the roots be $$\alpha - k$$, $$\alpha$$, $$\alpha + k$$; their sum $$3\alpha = -\frac{B}{A}$$ gives $$\alpha = -\frac{B}{3A}$$. Since $$\alpha$$ is a root, $$A\alpha^3 + B\alpha^2 + C\alpha + D = 0$$. Substituting and multiplying by $$27A^2$$ gives $$2B^3 - 9ABC + 27A^2 D = 0$$.