The problem:
Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$.
My attempt:
I know that the polynomials $x$ and $x+1$ have roots $[0]$ and $[1]$ respectively in $\mathbb{Z}_2[x]/(x^4+x+1)$.
Polynomials of degree 2 that are reducible in $\mathbb{Z}_2[x]$ clearly have roots so it remains to check the ones that are irreducible. The only irreducible polynomial of second degree in $\mathbb{Z}_2[x]$ is $x^2+x+1$ and it has root $[x^2+x+1]$ since $[x^2+x+1]^2+[x^2+x+1]+1=[x^4+x^2+1+x^2+x+1+1]=[x^4+x+1]=[0]$.
In degree four is where I have trouble. The reducible ones clearly have roots, and I've narrowed down the irreducible ones to $x^4+x+1$, $x^4+x^3+1$ and $x^4+x^3+x^2+x+1$. The first one is clear since it has the root $[x]$. I'm confused at the last two. I know I could brute force the answer by plugging in each possible $[a_3x^3+a_2x^2+a_1x+a_0]$ into $x^4+x+1$ and check by long division if the resulting polynomial is congruent to $[0]$, but somehow I suspect that isn't the intended method. However, I can't think of anything better either.
Would someone care to give me a hint? Thanks!
Hints. If $a=x\bmod (x^4+x+1)$, then try $a^{-1}$ for $x^4+x^3+1$, and $a^{-1}+1$ for $x^4+x^3+x^2+x+1$.