Monic irreducible polynomials in $\mathbb{F}_3[X]$

Question

I am trying to find monic irreducible polynomials in $\mathbb{F}_3[X]$ of degree 2 and for me it is enough to find a and b in $\mathbb{F}_3$ such that for P(X) = X²+aX+b then P(0), P(1), P(-1), P(2) and P(-2) are different from $0$. Thus I obtain the following polynomials: $X²+1\\ X²+X-1\\ X²-X-1\\ X²+2X-1\\ X²-2X-1\\ X²-2\\ X²+X+2\\ X²-X+2\\ X²+2X+2\\ X²-2X+2$

But according to my research, I'm wondering if it's totally right... Could you help me, or tell me where my reasoning is wrong! I thank you in advance!!

Answer 1

So I obtain : $X^2+1,X^2+X+2,X^2+2X+2$ Because P(0) ≠ 0 then b =1 or 2 and if b = 1 then a = 0 because P(1) and P(2) are différent from 0. And if b =2 then a = 1 or 2. Is it right ?