Location of zeros of families of trinomials (Q1576422)

Let \(n\) be a natural number \(\varphi= (1+\sqrt 5)/2\), and let \(a\) be a real number. In this paper the authors obtain explicit estimations, by using three different methods, of some roots of the following families of trinomials: \[ f_{n,a}(z)= z^{n+ 1}- az+ a-1, \] \[ G_n(z)= (\varphi- 1) z^{n+1}- \varphi z^n+ 1 \] and \[ H_n(z)= \varphi z^{n+ 1}-(\varphi- 1) z^n- 1. \] The first method is based on finding the approaching solutions of the equation. The second method is given by using the implicit function theorem, and the third method provides an expansion in hypergeometric series of solutions of a trinomial.