Say we have two polynomials, $a(x)$ of degree $n$ and $b(x)$ of degree $m$, where $n > m$. Euclid's algorithm lets us compute the GCD of these two polynomials using Euclidean division (detailed at: Euclid's algorithm).
What is the maximum number of steps / Euclidean divisions that Euclid's algorithm will need to get the GCD? Is it logarithmic in the degree, similar to Euclid's algorithm for the integers? Or is it linear in the degree (please provide a sample $a(x)$ and $b(x)$ if so).