What does $\mathbb Z_2[x]$ means?

Question

I know $\mathbb{Z}_2$ is the set of all integers modulo $2$. But $\mathbb{Z}_2[x]$ is the set of all polynomials. I am confused what it looks like.

Answer 1

The set of all polynomials with coefficients in $\mathbb{Z}_2$. As everything is congruent to either one or zero in $\mathbb{Z}_2$, we can also write such polynomials without using the minus symbol. Since $-1 \equiv 1$, we have things like $x^4-x^3+x^2-1$ equal to $x^4+x^3+x^2+1.$