We prove that there is an absolute constant \(c > 0\) such that every polynomial P of the form \(\displaystyle P(z) = \sum _{j=0}^{n}{a_jz^j}\,, \quad |a_0| = 1\,, \quad |a_j| \leq M\,, \quad a_j \in \mathbb C\,, \quad M \geq 1\,, \) has at most \(cn^{1/2}(1+\log M)^{1/2}\) zeros in the interval \([-1,1]\) . This result is sharp up to the multiplicative constant \(c > 0\) and extends earlier results of Borwein, Erdélyi, and Kós from the case \(M=1\) to the case \(M \geq \) 1. This has also been proved recently with the factor \((1+\log M)\) rather than \((1+\log M)^{1/2}\) in the Appendix of a recent paper by Jacob and Nazarov by using a different method. We also prove that there is an absolute constant \(c > 0\) such that every polynomial P of the above form has at most \((c/a)(1+\log M)\) zeros in the interval \([-1+a,1-a]\) with \(a \in (0,1]\) . Finally we correct a somewhat incorrect proof of an earlier result by Borwein and Erdélyi by proving that there is a constant \(\eta > 0\) such that every polynomial P of the above form with \(M = 1\) has at most \(\eta n^{1/2}\) zeros inside any polygon with vertices on the unit circle, where the multiplicative constant \(\eta > 0\) depends only on the polygon.