Nathan Easley

Document Type

Honors Thesis


Most advanced college students are familiar with the fact that an equation p(x) = 0, where p(x) is a polynomial of degree n with real coefficients, will have at most n solutions. When the coefficients are complex, the Fundamental Theorem of Algebra (FTA) says that there are exactly n solutions, counting multiplicity. For example, x^3 - x = 0 has exactly three solutions, 0, 1, and -1. This thesis investigates how many solutions polynomial equations have in other number systems, particularly in hyperbolic and parabolic numbers. Our methods involve looking for how many solutions simple equations, such as x^2 - k = 0, possess for different choices of k in these distinct number systems. We then consider general polynomials. Overall, our results demonstrate that there can be more or less solutions than the degree of the polynomial, contradicting the outcomes expected from the FTA.

Publication Date






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