Consider a sequence of polynomials P_n(x), n=1,2,..., with only nonpositive zeros and with deg P_n \infty. The method is based on establishing an equivalence between the limiting empirical zero distribution of P_n and an exponential profile associated with the coefficients of the polynomials. Using this equivalence we investigate how the exponential profile and the limiting zero distribution behave under certain operations on polynomials, including finite free convolutions, Hadamard products, and repeated differentiation. The talk is based on a joint work with Jonas Jalowy and Alexander Marynych, ArXiV: 2504.11593.