Real Root Polynomials and Real Root Preserving Transformations

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 $n\ge 1$ 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 $P_n$ .