MathDB

A7

Part of 2009 Balkan MO Shortlist

Problems(1)

Prove that the given polynomial is irreducible

Source: Balkan MO ShortList 2009 A7

4/6/2020
Let n≥2n\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 ∣cn∣\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)P(X) is irreducible in the Z[x]\mathbb{Z}[x]