Irreducibility of a polynomial over $\mathbb{Q}$

Question

I need to show that the polynomial $x^5-5x^4-6x+2$ is irreducible. Given the context the question was presented, there is supposed to be a trick to make the solution arrive quickly, which is what I'm interested in. Its not difficult with a little time to rule out 1 or 4 degree divisors, but I'm struggling to show there are no 2 or 3 degree divisors.

Answer 1

It suffices to show the polynomial is irreducible when taken as a polynomial with coefficients in $\mathbb{Z}/(3).$ By substitution we see it has no roots, so we are done if we check no irreducible quadratic divides it. The only irreducible quadratics (up to associates) are $$x^2+1 \ , \ x^2+x+2 \ , \ x^2+2x+2 .$$ The division algorithm always gives a non-zero remainder for these polynomials, so your polynomial is irreducible.