BG-MO 1989 Problem 3

Let $p$ be a real number and $f(x)=x^p-x+p$ . Prove that:

(a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$ ; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.