Polynomials in $\mathbb Z_2[x]$ Determine which polynomials in $\mathbb Z_2[x]$ have roots in $\mathbb Z_2$.
I don't know how to solve this, but found that that it may be helpful to use the factor theorem, but since I haven't studied that theorem, I can't use it. Is there another way to prove this?

Let $f\in \mathbb Z_2[x]$. Clearly if the constant term of $f$ is $0$, then $0$ is a root of $f$. Otherwise $f(0)=1$. If $f$ has an even number of non-zero terms then $f(1)=0$, and otherwise $f(1)=1$. Thus $f$ has a root in $\mathbb Z_2$ if and only if $f$ has a constant term of $0$ or an even number of non-zero terms.
