Summary: For any fixed field \(K \in \{\mathbb{Q}_2, \mathbb{Q}_3, \mathbb{Q}_5, \ldots \}\), we prove that all univariate polynomials \(f\) with exactly 3 (resp. 2) monomial terms, degree \(d\), and all coefficients in \(\{\pm 1, \ldots, \pm H \}\), can be solved over \(K\) within deterministic time \(\log^{4 + o (1)}(d H) \log^3 d\) (resp. \(\log^{2 + o (1)}(d H))\) in the classical Turing model: Our underlying algorithm correctly counts the number of roots of \(f\) in \(K\), and for each such root generates an approximation in \(\mathbb{Q}\) with logarithmic height \(O(\log^2(d H) \log d)\) that converges at a rate of \(O ((1 / p)^{2^i})\) after \(i\) steps of Newton iteration. We also prove significant speed-ups in certain settings, a minimal spacing bound of \(p^{- O (p \log_p^2 (d H) \log d)}\) for distinct roots in \(\mathbb{C}_p\), and even stronger root repulsion when there are nonzero degenerate roots in \(\mathbb{C}_p\): \(p\)-adic distance \(p^{- O (\log_p (d H))}\). On the other hand, we prove that there is an explicit family of tetranomials with distinct nonzero roots in \(\mathbb{Z}_p\) indistinguishable in their first \(\Omega(d \log_p H)\) most significant base-\(p\) digits. So speed-ups for \(t\)-nomials with \(t \geq 4\) will require evasion or amortization of such worst-case instances.