<p>Several third-order iterative methods for finding multiple roots of nonlinear equations are developed and systematically compared with well-known existing schemes. The proposed approach is based on a flexible framework that generates candidate methods through different choices of correction terms in the Taylor expansion. These methods leverage knowledge of the root multiplicity and include generalizations of Newton, Halley, and Euler–Cauchy-type methods. Both qualitative and quantitative analyses are conducted. Basins of attraction are employed to visualize convergence behavior in the complex plane, while metrics such as iteration count, divergence rate, and computational efficiency are systematically recorded. Although the framework can generate many candidate schemes, only a subset of them exhibits competitive numerical performance. Among the methods considered, several demonstrate performance comparable to classical approaches, and one method consistently achieves the best overall performance, including the shortest average CPU time across the tested examples. The results indicate that careful parameter selection within the proposed iterative framework can lead to effective methods for solving nonlinear equations with multiple roots. Possible applications include engineering models such as supersonic flow and complex fluid modeling, as illustrated by the chemical engineering examples considered in this work.</p>