Process for checking reducibility of polynomials of degree 2 or lower in $\mathbb{Z}_2$

Question

How might I verify that polynomials of degree 2 or lower and with coefficients from $\mathbb{Z}_2$ are reducible or not?

I figure I can write out all possible polynomials because there are only 2 possible coefficients and 3 possible exponents, but I want to know and understand a procedure to figuring out if it is reducible.

For example, can I just check for roots in $\mathbb{Z}_2$? E.g., $f(x) = x^2 + x + 1$, then try $f(0)$ and $f(1)$ to see if either equals 0? It seems like having roots would guarantee it is reducible because if the degree is 2 or less, any factor must be linear, which is equivalent to a root.

Answer 1

A polynomial $f\in K[X]$ of degree $2$ and $3$ over a field $K$ is reducible if and only if has a root in $K$. This answers your question for $K=\Bbb F_2$.

Reference: Irreducible polynomial means no roots?