The following problem is investigated: let \(f\) be a polynomial given by the expansion \(f(z)=a_0 p_0(z)+a_1p_1(z)+\cdots+a_np_n(z)\) in terms of orthogonal polynomials. What can be said about the zeros of \(f\) in terms of the zeros of the orthogonal polynomials \(p_j\) and the Fourier coefficients \(a_j\)? The main result is a condition on the (real) Fourier coefficients \(a_j\) which implies that \(f\) has at least \(k\) distinct real zeros of odd multiplicity. If the condition does not hold, then a region is given outside of which \(f\) has at most \((n-k)/2\) pairs of conjugate zeros. As a corollary, a strip is found where the polynomial has at least \(k\) zeros. Another result, of independent interest, is an upper bound for the ratio \(\|f\|/ \|fg\|\), where \(f\) and \(g\) are polynomials of degree \(n\) and \(k\), respectively, and the norms are in \(L^2(\sigma)\), where \(\sigma\) is the orthogonality measure for the polynomials \(p_k\).