The Cauchy-Mirimanoff Polynomials are a class of polynomials that naturally arise in various classical studies of Fermat's Last Theorem. Originally conjectured to be irreducible over 100 years ago, the irreducibility of the Cauchy-Mirimanoff polynomials is still an open conjecture. This dissertation takes a new approach to the study of the Cauchy-Mirimanoff Polynomials. The reciprocal transform of a self-reciprocal polynomial is defined, and the reciprocal transforms of the Cauchy-Mirimanoff Polynomials are found and studied. Particular attention is given to the Cauchy-Mirimanoff Polynomials with index three times a power of a prime, and it is shown that the Cauchy-Mirimanoff Polynomials of index three times a prime are irreducible.