(a) Prove that if the roots of \[x^3 + ax^2 + bx + c = 0\]form an arithmetic sequence, then $2a^3 + 27c = 9ab.$
(b) Prove that if $2a^3 + 27c = 9ab,$ then the roots of \[x^3 + ax^2 + bx + c = 0\]form an arithmetic sequence.
If a = b = c = 0, then all the roots are 0.
