I have the following past exam question:
Calculate $\operatorname{gcd}(x^3 + 2x^2 + 2,2x^2 + 1)$ in $\mathbb{F}_3$
Now I haven't encountered this sort of gcd before(usually I am trying to solve linear Diophantine equations with Euclidian algorithm).
I have computed $\cfrac{x^3 + 2x^2 + 2}{2x^2 + 1}$
Which yielded $1 + \frac x2 + \frac{2-x}{4x^2 + 2}$
But I don't think this helps, and factorizing $2x^2 + 1$ is complex, so I am not sure what to do. Any tips would be appreciated.
Your divisions should not have any denominator because you should be using the Euclidean algorithm to find a quotient and remainder. I'm sure you have done this with both integers and polynomials at school. In this case, you have to remember to do arithmetic with the coefficients modulo $3$. Hint: the first step is $$x^3 + 2x^2 + 2=(2x^2 + 1)(2x+1)+(x+1)\ .$$