Finding the gcd of polynomials $p,q$ and writing $\gcd(p,q)=d_1(x)p+d_2(x)q$

Question

I'm given $p(x)=x^3+4x^2+x-6, q(x)=x^5-6x+5$. I'm asked for the GCD in $\mathbb{Q}[x]$; which I know to be $x-1$ by factoring. However; writing $x-1$ as a combination of p and q (that is; finding $d_1(x),d_2(x) \in \mathbb{Q}[x]$ s.t. $qd_1(x)+pd_2(x)=x-1$)seems to require Euclid algorithm (this is how I would proceed if these where say, integers). This gets impossibly messy; I've tried 3 times now and none of my solutions checked out. Is there some more sophisticated trick or method I'm missing here?

Answer 1

The Euclidean algorithm works perfectly fine. Two polynomial long divisions yield: \begin{array}{rrrrr} 1\times&(x^5-6x+5)&-&(x^2-4x+15)\times&(x^3+4x^2+x-6)&=&-50x^2-45x+95\\ 50\times&(x^3+4x^2+x-6)&-&(-x-\tfrac{31}{10})\times&(-50x^2-45x+95)&=&\tfrac{11}{2}(x-1).\\ \end{array} Substituting back then yields $$x-1=\frac{10x+31}{55}q+\frac{-10x^3+9x^2-26x+35}{55}p.$$