We study a random algebraic polynomial , where the coefficients A₀,A₁,… form a sequence of centred Gaussian random variables. Moreover, we assume that the increments Δ_j=A_j-A_j-1, j=0,1,2,… are independent, assuming A_-1=0. The coefficients can be considered as n consecutive observations of a Brownian motion. We obtain the asymptotic behaviour of the expected number of times that such a random polynomial assumes the real value K, where K is any non-zero real constant. It is shown that the results are valid even for K→∞, as long as K=o(n^1/4).