I have an elementary question concerning zeros of polynomials, which must be well-known.

Fix a base field $ k$ (can assume to be characteristic zero if it makes a difference). Consider the affine space $ P_n \times \mathbb A^1_k $, where $ P_n $ denotes the space of polynomials of degree $ n $ over $ k$ (so $ P_n $ is an affine space of dimension $ n+1 $ given by the coefficients).

Given $ p \in P_n $ and $ z \in k $, let $ ord_z(p) $ denote the order of vanishing of the polynomial $ p $ at $z $.

Fix $ m \in \mathbb N $ and consider the closed subset of $ X_m \subset P_n \times \mathbb A^\times $ given as $$ X_m := \{ (p,s) : ord_s(p) + ord_0(p) \ge m \}$$ In other words, we study those polynomials whose vanishing orders at $ 0 $ and $ s $ add up to at least $ m$ (where $ s $ is non-zero). Then I take $ \overline{X_m} \subset P_n \times \mathbb A $.

Question: Is it true that $ \overline{X_m} \cap P_n \times \{0\} $ consists of those polynomials which vanish to order at least $ m $ at $ 0$?

This is just a set-theoretic question. Scheme-theoretically, this statement seems to be false.