Why is the polynomial $f(x) =9x^{6} - 6x^{5} + x^{4} +4x^{3} - x^{2} - x + 1$ irreducible? I checked it on wolframalpha and it is indeed irreducible, but I don't know if I can prove it in some nice way.
I tried looking at polynomials $f(x+1)$ and $f(x-1)$ and use Eisenstein's criterion on it -- however, these polynomials did not fulfil the assumptions of the Eisenstein's criterion. I do not know of any other irreducibility criterion.
This is not an exercise or homework or any other similar projects -- I just started wondering that if one could prove irreducibility without the help of computational softwares.
The polynomial is already irreducible over the field $\Bbb F_2$, hence also irreducible over $\Bbb Z$ and $\Bbb Q$. Of course, this also amounts to a (small) computation. You don't need a computer, though.
By the rational root theorem, we do not have a rational root, so there is no linear factor. But a possible decomposition could be $(x^4+\cdots)(x^2+\cdots)$ or $(x^3+\cdots)(x^3+\cdots)$ over $\Bbb F_2$. A short comparison of coefficients shows that both are not possible.