<p>This study introduces a novel one-step iterative root-finding method with memory, designed to solve nonlinear equations and offering enhanced convergence behavior. The with-memory method is constructed by approximating the nonlinear function using a bivariate polynomial curve that satisfies certain tangency conditions at multiple points, effectively improving the convergence performance. A thorough local convergence analysis is conducted to validate the theoretical robustness of the approach. Moreover, the stability of the proposed method is investigated through a multidimensional dynamical analysis using fixed-point operators derived from various quadratic and cubic polynomials applied within the iterative method. The dynamical plane constructions associated with the method provide visual insights into its behavior, revealing convergence exclusively to the roots of the given polynomials. Numerical experiments on many standard test nonlinear problems confirm the method’s superior accuracy and efficiency compared to some existing techniques, confirming its applicability for broad application in scientific and engineering computations.</p>