  This paper seeks to further explore the distribution of the real roots of
random polynomials with non-centered coefficients. We focus on polynomials
where the typical values of the coefficients have power growth and count the
average number of real zeros. Almost all previous results require coefficients
with zero mean, and it is non-trivial to extend these results to the general
case. Our approach is based on a novel comparison principle that reduces the
general situation to the mean-zero setting. As applications, we obtain new
results for the Kac polynomials, hyperbolic random polynomials, their
derivatives, and generalizations of these polynomials. The proof features new
logarithmic integrability estimates for random polynomials (both local and
global) and fairly sharp estimates for the local number of real zeros.
