Real root polynomials and real root preserving transformations (Q2038321)

Summary: 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 \(n\geq 1\) 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 \(P_n\).