Summary: For \(n \in {\mathbb N}\), \(L > 0\), and \(p \geq 1\) let \(\kappa_p(n,L)\) be the largest possible value of \(k\) for which there is a polynomial \(P \not \equiv 0\) of the form \[ P(x) = \sum_{j=0}^n{a_jx^j}, |a_0| \geq L \Bigl( \sum_{j=1}^n{|a_j|^p} \Bigr)^{1/p}, \; a_j \in {\mathbb C}, \] such that \((x-1)^k\) divides \(P(x)\). For \(n \in {\mathbb N}\), \(L > 0\), and \(q \geq 1\) let \(\mu_q(n,L)\) be the smallest value of \(k\) for which there is a polynomial \(Q\) of degree \(k\) with complex coefficients such that \[ |Q(0)| > \frac 1L \Bigl( \sum_{j=1}^n{|Q(j)|^q} \Bigr)^{1/q}. \] We find the size of \(\kappa_p(n,L)\) and \(\mu_q(n,L)\) for all \(n \in {\mathbb N}\), \(L > 0\), and \({1 \leq p,q \leq \infty}\). The result about \(\mu_\infty(n,L)\) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special case.