MathNet 2009-part-04 — 175 problems (EN–VI) Question 23

Let $p$ be a prime number and let $a$, $b$ and $c$ be integers divisible by $p$, such that the polynomial $q(x) = x^3 + a x^2 + b x + c$ has at least two different integer roots. Show that $p^2$ divides $b$ and $p^3$ divides $c$.