Silly question: are polynomials "X" and "X^2" reducible?

Question

Since f(0)=0 holds for both, are they both reducible polynomials?

I'm asking because I'm working on this question where they're asking you to factor the following in irreducible factors.

X^5 - X^4 - X^3 + X^2 + X ∈ F3[X]

The solution is X(X^2 + X + 2)^2, so that includes factor X, which I believe is reducible... Am I missing something here?

Answer 1

As any other polynomial of degree one (over some field) , $x$ is irreducible, whereas $x^2=x\cdot x$ is reducible

Answer 2

$x$ is a linear polynomial so it is irreducible. $x^2$, on the other hand, is not irreducible because it can be factored as the difference of squares: $x^2 \iff x^2 - 0 \iff x^2 - 0^2 \iff (x-0)(x+0)$