The force on a charged dust grain in a plasma due to polarization of thermal ions and degenerate electrons around the grain is derived in the limits of weakly relativistic and ultra-relativistic degeneracy of electrons.It is found that in both these cases,the magnitude of the polarization force is enhanced compared to that in classical plasmas.The influence of this force on dust-acoustic (DA) modes is examined and discussed.It is shown that the DA wave frequency in degenerate plasmas is significantly reduced compared to the classical DA mode.