Vertical-line factor from a degree-zero currying with a coefficient root

Let $h$ be a bivariate real polynomial whose currying $\mathrm{Curry}_0(h)$ (viewed as a univariate polynomial over the coefficient ring) has $\mathrm{natDegree} = 0$, and let $x \in \mathbb{R}$ be a real root of its constant coefficient, i.e. $\mathrm{eval}(x, (\mathrm{Curry}_0(h)).\mathrm{coeff}\,0) = 0$. Then the vertical-line polynomial $X_1 - x$ divides $h$:

This extracts a vertical-line factor from a degenerate (constant-in-the-eliminated-variable) currying, and is used to classify the fibers over which specialization vanishes identically.