Show that $(x^3+x+1)$ is prime in $\mathbb{Z}/(2)[x]$

Question

I want to show that the ideal $(x^3+x+1)$ is prime in $\mathbb{Z}/(2)[x]$. I know that $\mathbb{Z}/(2)[x]$ is isomorphic with $\mathbb{Z}[x]/(2)$ and also know that since $\mathbb{Z}/(2)[x]$ is commutative it would be enough to show that for every $p(x),q(x)\in\mathbb{Z}/(2)[x]$, if $p(x)q(x)\in(x^3+x+1)$ then either $p(x)\in(x^3+x+1)$ or $q(x)\in(x^3+x+1)$, or equivalently if $p(x)q(x)=(x^3+x+1)r(s)$ then either $x^3+x+1$ divides $p(x)$ or divides $q(x)$. I have also proved (using a simple proof by cases) that $x^3+x+1$ cannot be factorized in $\mathbb{Z}/(2)[x]$, but I don't know how I should proceed further.