Irreducible Factorization of $x^4 - 1$ over $\mathbb{F_3}$.

Question

In class, we showed that

$$x^4 - 1 = (x-1)(x+1)(x^2 + 1)$$

over $\mathbb{F_3}$ is a factorization of irreducible polynomials, but it was also an exercise to show that

$$x^4 - 1 = (x-1)(x-2)(x^2 + 1)$$

over the same field. I don't get it. Are those two factorizations the same?

Answer 1

Yes, because $+2 \equiv -1 \mod 3$.