Abstract
In this paper, we suggest an iterative method for solving nonlinear equations that can be used in the physical sciences. This response is broken down into three parts. Our methodology is inspired by both the standard Taylor's method and an earlier Halley's method. Three evaluations of the given function and two evaluations of its first derivative are all that are needed for each iteration with this method. Because of this, the unique methodology can complete its goal far more quickly than many of the other methods currently in use. We looked at several additional practical research models, including population growth, blood rheology, and neurophysiology. Polynomiographs can be used to show the convergence zones of certain polynomials with complex values. Polynomiographs are produced as a byproduct, and these end up having an appealing look and being artistically engaging. The twisting of polynomiographs is symmetric when the parameters are all real and asymmetric when some of the parameters are imaginary.