Solving nonlinear equations efficiently in the presence of degeneracy remains a challenge in numerical analysis and scientific computing. Classical methods, including Newton’s method and polynomial-based algorithms, often exhibit stagnation near singularities, limiting their effectiveness in practical applications. This paper introduces a p-factor interpolation method, a structured modification of the classical Newton interpolation polynomial that systematically incorporates higher-order derivative information to improve the accuracy of root approximations near degenerate points. We prove that, under mild smoothness and p-regularity conditions, the proposed method achieves second-order accuracy in approximating solutions of the nonlinear equation \(f(x)=0\) even when lower-order derivatives vanish. We further demonstrate the natural integration of the p-factor function within a trust-region framework, where it enhances local model accuracy near singularities while preserving computational simplicity. The method has a potential to enable larger, more effective steps within trust regions and reduce iteration counts compared to classical approaches, offering a practical enhancement for solving nonlinear equations and optimization problems in the presence of degeneracy.