# The Coefficient Characterization of Polynomials with Golden Ratio Roots

#### Abstract

The golden ratio and the Fibonacci numbers have a very noticeable presence in many mathematical applications. They are intricately connected in many explorations that involve the optimization of every-day systems (both natural and man-made). With such a prominent existence, it is helpful to discover ways to characterize the polynomial models that have the golden ratio and its conjugate as roots. As it turns out, these polynomials can be completely characterized by the constant and linear terms of the polynomial, which are both functions of the Fibonacci and Lucas numbers. These polynomials can then be used to derive many new and well-known identities of Fibonacci and Lucas numbers as well as the golden ratio.

*This paper has been withdrawn.*

The Coefficient Characterization of Polynomials with Golden Ratio Roots

Room E101, Center for Natural Sciences, Illinois Wesleyan University

The golden ratio and the Fibonacci numbers have a very noticeable presence in many mathematical applications. They are intricately connected in many explorations that involve the optimization of every-day systems (both natural and man-made). With such a prominent existence, it is helpful to discover ways to characterize the polynomial models that have the golden ratio and its conjugate as roots. As it turns out, these polynomials can be completely characterized by the constant and linear terms of the polynomial, which are both functions of the Fibonacci and Lucas numbers. These polynomials can then be used to derive many new and well-known identities of Fibonacci and Lucas numbers as well as the golden ratio.