Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ so that it has exactly 4 distinct roots and factorize it as product of irreducible factors.
I'm really struggling in finding such polynomial, so basically I need to find an $f$ which has a $c$ root that doesn't belong to $\mathbb Z_5$. So does this polynomial is the one I am looking for? If not is there any way I could get to such $f$?
$f = x^5+\sqrt{2}x^4-5x^3-5\sqrt{2}x^2+4x+4\sqrt{2} = (x+1)(x-1)(x+2)(x-2)(x+\sqrt{2})$