Ask Me Help Desk - Are these polynomials irreducible over Z ?

Is it true that polynomials of the form :

$f_n= x^n x^{n-1} \cdots x^{k 1} ax^k ax^{k-1} \cdots a$

where $\gcd(n 1,k 1)=1$ , $a\in \mathbb{Z^{ }}$ , $a$ is odd number , $a>1$ , and $a_1\neq 1$

are irreducible over the ring of integers $\mathbb{Z}$ ?

Eisenstein's criterion , Cohn's criterion , and Perron's criterion cannot be applied to the polynomials of this form.

Example :

The polynomial $x^4 x^3 x^2 3x 3$ is irreducible over the integers but none of the criteria above can be applied on this polynomial.