Let $f(x)$ be a polynomial in $(\mathbb{Z}/\mathbb{2Z})[x]$ of degree $2$ or $3$. Prove that $f(x)$ is irreducible if and only if $f(x)$ does not have a root in $\mathbb{Z}/\mathbb{2Z}.$
I know that $f(x)$ is irreducible if and only if $F[x]/(f(x))$ is a field.
Any suggestions/hints will be appreciated.