Factor the polynomial $x^4 + 2x − 4$ in $\mathbb{Z}_5[x]$.

Question

I'm confused as to how this is different from factoring in the reals?

Would I start this by writing $x^4+2x-4 \equiv 0 \pmod 5$? What changes?

Answer 1

Hint: in the given field we have it that 4 is a root of the polynomial. What can you say about the factoring given that there is a known root?

Answer 2

see: $x^4 + 2x - 4 = (x+1)(x^3 -x^2 + x+ 1)$ and $0,1,2,3,4$ are not of $mod \;5$ root of $(x^3 -x^2 + x+ 1).$ Hence $x^4 + 2x - 4 = (x+1)(x^3 -x^2 + x+ 1)$ gives the factorization.