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A polynomial f(x) with rational coefficients is solvable by radicals if its roots (in the field of complex numbers C) can be expressed in terms of its coefficients using the basic operations and radicals. It is known that for quintic polynomials there is no generic formula for the roots. That is, some quintic polynomials are solvable and some are not. In this paper, we address the mathematical theory that makes the formula for the roots of a polynomial. Primarily we will focus on our methodology of generating and examining quintic polynomials. In one case study, we will examine quintic polynomials that may have roots that can be expressed in a radical form. In the other case study we show the methodology of generating polynomials that do have solutions which can be expressed in a radical form from the coefficients of the polynomials.
Tivenan, Stephen, "Exploration of Solvable Quintic Polynomials" (2020). Student Research Submissions. 326.