All Possible Polynomials With Max Degree 2 in GF(3)

Question

How many max degree 2 polynomials (not necessarily irreducibles) are there with coefficients in $\{0, 1, 2\}$? There are $3$ choices for coefficients in $3$ positions. For example: $[]x^2 + []x + []$. I am using this symble $[]$ as the placeholder for coefficients. This gives me $27$ polynomials. Then there are three $`+'$ signs which could be negatives. This confused me. So does it mean we have $8 \times 27$ polynomials? Is there any closed form formula for finding this number?

Answer 1

Given a coefficient $-ax^n$ I can always rewrite it in the form $+bx^n$ where $b\in\{0,1,2\}$ via the division algorithm (adding or subtracting $3x^n\equiv0$ repeatedly). Thus the signs do not give new polynomials, and there are $27$ distinct up-to-quadratic polynomials.

In general, in $GF(n)$ there are $n^{d+1}$ distinct polynomials up to degree $d$.