Find the multiplicative inverse of $x^4+x^3+1$ in $\mathbb{Z}_2[x]/\langle x^5+x^2+1 \rangle$.
What I've tried is as follows:
We know $\alpha^5+\alpha^2+1=0$. Then $\alpha^4+\alpha^3+1=\alpha^4+\alpha^3+\alpha^2+1=\alpha^2(\alpha^2+1)(\alpha+1)$.
From $\alpha^5+\alpha^2+1=0$ we have, $\alpha^2(\alpha^3+1)=1$, so now we know the multiplicative inverse of $\alpha^2$, but we had no success finding the inverse of $\alpha^2+1$ nor $\alpha+1$. All of this was to avoid writing $q(x)$ as a general polynomial in $\mathbb{Z}_2[x]/\langle x^5+x^2+1 \rangle$ and then solving that equation.
Are we on the right track? Is there any better approach? Thanks in advance for any help and any advice will be welcomed.