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 with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to if we restrict the coefficients to be real. Let and be the vector space of all polynomials of degree or less with real coefficients. In this article, we give explicit forms of polynomials in such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on which preserve real roots of polynomials in a certain subset of .
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