How does one prove that $x^3 + x^2 + 1$ is irreducible over $\mathbb{F}_{2}$?

Question

Where $\mathbb{F}_{2}$ is the finite field with two elements.

Thanks in advance.

Answer 1

Hint: It is enough to show that it does not have roots since a reduced polynomial of degree 3 has a root.

Answer 2

Assume it is reducible. Since it has degree $3$, one of the factors would have to be linear, therefore there would be a root of this polynomial. Now check, by plugging in directly, whether any of the elements of $\mathbb{F}_2$ is a root.