Statement

Let ${\mathbb{K}}$ be a field. Let $a,b,c \in {\mbox{${\mathbb{K}}$}}[x]$ be three non-zero univariate polynomials in $x$ with coefficients in ${\mathbb{K}}$ . We are interested in computing the polynomials $u, v \in {\mbox{${\mathbb{K}}$}}[x]$ such that

$$a u + b v = c$$ (2)

holds. Let $g \in {\mbox{${\mathbb{K}}$}}[x]$ be the gcd of $a, b$ and let $s, t \in {\mbox{${\mathbb{K}}$}}[x]$ be such that

$$a s + b t = g$$ (3)

holds. Remember that $g,s,t$ can be computed by the Extended Euclidean Algorithm.