  In this paper we introduce an algebraic form of the dispersion relation for a
non ideal inhomogeneous dusty plasma in order to improve drastically the
calculation of the drift instability growth rate. This method makes use of the
multipole approximation of the Z dispersion function, previously published, and
valid for the entire range. A careful analysis of the solutions spectra of this
kind of polynomial equation permits us to calculate easily the growth rate of
the drift instability for the ion-dust and dust acoustic mode. The value of the
parallel to magnetic field wavelength for which the instability reaches the
maximal value is carefully localized and discussed. The unstable dust-ion and
dust acoustic mode are discriminated and analyzed in function of the density
gradient, Te/Ti - ratio, and dust grain radius.
