Let \({\mathcal P}_ n\) be the class of algebraic polynomials \(P(x)= \sum^ n_{\nu= 0} a_ \nu x^ \nu\) of degree at most \(n\) and \(\| P\|_{d\sigma}= (f_{\mathbb{R}}| P(x)|^ 2 d\sigma(x))^{1/2}\), where \(d\sigma(x)\) is a nonnegative measure on \(\mathbb{R}\). The authors determine the best constant in the inequality \(| a_ \nu|\leq C_{n,\nu}(d\sigma)\| P\|_{d\sigma}\), for \(\nu= n\) and \(\nu= n-1\), when \(P\in {\mathcal P}_ n\) and such that \(P(\xi_ k)= 0\), \(k=1,\dots,m\). They also consider the cases when the measure \(d\sigma(x)\) corresponds to the classical orthogonal polynomials on the real line \(\mathbb{R}\).