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 Z^{ } , a is odd number , a>1 , and a_1\neq 1

are irreducible over the ring of integers Z ?

Note that general form of f_{n} is :

f_{n}=a_{n}x^{n} a_{n-1}x^{n-1} \cdots a_{1}x a_{0}

Eisenstein's criterion (http://en.wikipedia.org/wiki/Eisenstein's_criterion) , Cohn's criterion (http://en.wikipedia.org/wiki/Cohn's_irreducibility_criterion) , and Perron's criterion (http://rms.unibuc.ro/bulletin/pdf/53-3/perron.pdf) 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.