We know that for two non-zero polynomial forms $f,g\in F[X]$ over the field $F$, their greatest common divisor $d$ exists and is unique (when ignoring those multiplied by constants), and it is the last non-zero remainder in the Euclidean algorithm; and there exists $u,v\in F[x]$ such that $$ uf+vg=d; $$ this is called the Bézout's Identity for Polynomial Ring.  On my textbook, there is a corollary of this that states $f,g\in F[x]$ are coprime  polynomials (have only the invertible elements in $F[x]$, i.e. elements of $F*$, as common divisors) if and only if there exists $u,v\in F[X]$ such that $$ uf+vg=1. $$ The $(\implies)$ direction of this corollary is trivial to me, but I think I need a little help on the $(\impliedby)$. Any kind of help is appreciated. Thanks.
We were given the following problem as an assignment, we proved the converse in class for prime numbers; however, we can't use the assumption that $\operatorname{gcd}(a,b) = 1$.  So I am struggling to think of my first possible move.  We are given $ar + bs = 1$, the only think I can think of is let $d$ be a common divisor or $ar + bs$ and then show that the $d = 1$.  However, I am not really sure how to do that.  If anyone could just give me a hint or two to nudge me in the right direction I would be appreciative.
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