Abstract: To classify cubic polynomials with multiple roots in the sense of topological conjugation, this study specifically analyses the following two classes of cubic polynomials. One class is the cubic polynomials with a triple root and the other class is the cubic polynomials with a double root and a single root. By constructing the conjugate functions, the conclusions for the classification of cubic polynomials with multiple roots are presented. Also, the proofs of the conclusions are given. The results show that the cubic polynomials with a triple root can be categorized into three classes if the first coefficient is positive, and into one class if the first coefficient is negative. If the first coefficient is positive, the cubic polynomials which have a double root of 0 and a single root a \ne 0 can be divided into two classes. Similarly, the cubic polynomials which have a double root a \ne 0 and a single root 0 can also be divided into two classes.