Is it true that polynomials of the form :

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

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

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

Note that general form of$\displaystyle f_n$ is :$\displaystyle f_n=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ , so condition $\displaystyle a_1 \neq 1$ is equivalent to the condition$\displaystyle k \geq 1$ . Also polynomial can be rewritten into form :

$\displaystyle f_n=\frac{x^{n+1}+(a-1)x^{k+1}-a}{x-1}$

Eisenstein's criterion ,

Cohn's criterion , and

Perron's criterion cannot be applied to the polynomials of this form.

Example :

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