I have an element $$\frac{1}{p(x)} \in \frac{F[x]}{\langle f(x)\rangle }$$ (with gcd(f,p) = 1)
I want to write this in terms of the standard basis elements, so I perform Euclid's algorithm to find $a(x),b(x) \in F[x]$ such that
$$a(x)p(x) + b(x) f(x) = 1$$
Then we have $a(x)p(x) \cong 1 \mod \langle f(x) \rangle$
And so we find $$a(\overline{x}) = \frac{1}{p(\overline{x})}$$
My question is, how exactly do we use Euclid's algorithm in a concrete example to compute such $a(x)$ and $b(x)$?