The Polubarinova–Galin equation describes a parametric conformal map for the zero-surface-tension Hele-Shaw flow driven by injection or suction. In this paper, we classify degree three polynomial solutions to the Polubarinova–Galin equation into three categories: global solutions, solutions which can be continued after blow-up and solutions which cannot be continued after blow-up. The coefficient region of the initial functions in each category is obtained.