Let $ \mathbb{F} $ be a field, consider the polynomial ring $ \mathbb{F} \left[ x\right] $ and suppose that the polynomials $f,g \in \mathbb{F} \left[ x\right]$ have degrees $2,2^n$, respectively, where $n$ is a positive integer. Moreover, assume that they are both *monic* and have *exactly one* common root.

Question.Is there a known closed form for this common root in terms of the coefficients of $f$ and $g$?

For $n \in \{1,2,3\}$ the root is a rational function with integer coefficients in the coefficients of $f,g$.

Fundamental problems in algorithmic algebra. $\endgroup$2more comments