We theoretically explore the conditions for generating optical bistability (OB) in a heterodimer comprised of a semiconductor quantum dot (SQD) and a metallic nanoshell (MNS). The MNS is made of a metallic nanosphere as a core and a dielectric material as a shell. For the specific hybrid system considered, the bistable effect appears only if the frequency of the pump field is equal to (or slightly less than) the exciton frequency for a proper shell thickness. Bistability phase diagrams, when plotted, show that the dipole-induced bistable region can be greatly broadened by changing the shell thickness of the MNS in a strong exciton-plasmon coupling regime. In particular, we demonstrate that the multipole polarization not only narrows the bistable zone but also enlarges the corresponding thresholds for a given intermediate scaled pumping intensity. On the other hand, when the SQD couples strongly with the MNS, the multipole polarization can also significantly broaden the bistable region and induce a great suppression of the FWM (four-wave mixing) signal for a fixed shell thickness. These interesting findings offer a fresh understanding of the bistability conditions in an SQD/MNS heterodimer, and may be useful in the fabrication of high-performance and low-threshold optical bistable nanodevices.