Extended $\gcd$ of $f(x)$ and $g(x)$

Question

I need to compute the extended gcd of $f(x)$ and $g(x)$, where

\begin{align*} f(x) &= x^5 + 3x^3 + x^2 + 2x + 1 \\ g(x) &= x^4 - 5x^3 - 5x^2 - 5x - 6 \end{align*}

How can I do that?

Answer 1

It starts out like this: \begin{align*} \underline{x^5 + 3x^3 + x^2 + 2x + 1} &= \\ (x + 5)(\underline{x^4 - 5x^3 - 5x^2 - 5x - 6}) &+ \underline{33x^3 + 31x^2 + 21x - 29} \\ \underline{x^4 - 5x^3 - 5x^2 - 5x - 6} &= \\ (\frac{x}{33} - \frac{196}{33^2} )(\underline{33x^3 + 31x^2 + 21x - 29}) &+ \underline{-\frac{62}{1089} x^2 - \frac{124}{363} x + \frac{850}{1089}} \end{align*}

As you can see, it gets a bit messy. But if you can see what is going on and continue from here, you will get to the answer.