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