Let be the set of algebraic polynomials p nof order n with real coefficients and zero weighted mean value with respect to the ultraspherical weight on the interval [-1,1]: We study the problem on the smallest possible value of the measure of the set of points of the interval at which the polynomial is nonnegative. In this paper, the properties of an extremal polynomial of this problem are studied and an exact solution is presented for the case of cubic polynomials.