BALKAN-SL 2009 A7

Let $n\geq 2$ be a positive integer and \begin{align*} P(x) = c_0 X^n + c_1 X^{n-1} + \ldots + c_{n-1} X +c_n \end{align*} be a polynomial with integer coefficients, such that $\mid c_n \mid$ is a prime number and \begin{align*} |c_0| + |c_1| + \ldots + |c_{n-1}| < |c_n| \end{align*} Prove that the polynomial $P(X)$ is irreducible in the $\mathbb{Z}[x]$