Real Root Polynomials and Real Root Preserving Transformations

Suchada Pongprasert,Puttarawadee Sriphrom,Kanyarat Chaengsisai,Wuttichai Kaewleamthong,Chin-Chia Wu

doi:10.1155/2021/5585480

Suchada Pongprasert, Puttarawadee Sriphrom + Show 3 more

Open Access

https://doi.org/10.1155/2021/5585480

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Abstract

Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. The fundamental theorem of algebra tells us that every nonconstant polynomial p with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to p if we restrict the coefficients to be real. Let n ≥ 1 and P n be the vector space of all polynomials of degree n or less with real coefficients. In this article, we give explicit forms of polynomials in P n such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on P n which preserve real roots of polynomials in a certain subset of P n .

Highlights

Polynomials are the simplest and most commonly used in mathematics
It is natural to ask which forms of polynomials guarantee all real roots and when we map a polynomial to another polynomial, how linear transformations that preserve real roots of polynomials look like. e obvious result is proved in [6] that only nonzero multiples of the identity transformation preserve roots of all polynomials in the vector space of all polynomials of degree n or less with real coefficients, Pn
En, we prove that they preserve real roots of all polynomials in S

Summary

Introduction

Polynomials are the simplest and most commonly used in mathematics. We can approximate functions from real-world situation models by polynomials, and they give us results that are “close enough” to what we would get by using the actual functions and are a lot easier to use. e roots of a polynomial which are the x-intercepts of the graph are key information when it comes to draw the polynomial graph, and they have real-world meaning when they are real numbers. We can approximate functions from real-world situation models by polynomials, and they give us results that are “close enough” to what we would get by using the actual functions and are a lot easier to use. It is natural to ask which forms of polynomials guarantee all real roots and when we map a polynomial to another polynomial, how linear transformations that preserve real roots of polynomials look like.

Preliminary

Polynomials

Real Root Polynomials

Real Root Preserving Transformations

Conclusion