What is the congruence class of $x^3\mod x^3+x+1$?

Question

I have a given Polynom congruence with a Polynom $x^3+x+1$ ... so the set of the congruence classes is $\{0, 1,x,x+1,x^2,x^2+1,x^2+x,x^2+x+1\}$

But what would look this like? $$x^3\mod x^3+x+1\equiv ?$$
I think the result must be one of the congruence classes of the set, mentioned above, but I cannot figure out which one.

Answer 1

You're working over $\Bbb Z/(2)$. Note that $-1=1$ there, so from $x^3+x+1\equiv 0$ you get $x^3\equiv -x-1\equiv x+1$, that is, the class of $x^3$ is that of $1+x$.