Analysis of the Distribution and Asymptotic Approximations of Roots of the Polynomial Equation $$z^{n+1}=(1+z{)}^n,n\in \mathbb{N}$$ in the Complex Plane

We study the spatial distribution of the positive, negative and non-real complex roots $z_n$ of the sequence the $(n+1)$th degree polynomial equation $$z^{n+1}=(1+z{)}^n,n\in \mathbb{N}\mathrm{.}$$ We establish asymptotic approximations to the sequence of the negative, the positive and the non-real complex roots of the equation as $n\to \mathrm{\infty }$. In addition, we discuses the possible areas of applications of the current problem.