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 ≥ 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 .