Many polynomial equations cannot be solved by a simple formula like the quadratic formula, yet their roots still follow precise symmetry patterns. In this talk, we study the family f(x) = x¹⁰ + ax⁵ + b and ask: as the parameters (a) and (b) vary, what symmetry patterns can its ten complex roots exhibit?

These patterns are described by the Galois group, which measures how the roots can be rearranged without changing the algebraic relationships they satisfy. Although ten roots could in principle exhibit many different symmetry behaviors, the special structure of this polynomial, particularly the presence of the x⁵ term, strongly limits what can occur.

By combining theoretical arguments with computer algebra experiments in Mathematica, GAP, and Pari/GP, we show that only four symmetry types arise for irreducible polynomials in this family. Moreover, we determine concrete conditions on (a) and (b) that distinguish among these four possibilities. This classification highlights how the form of a polynomial shapes the symmetries of its solutions.

This is joint research done in collaboration with C. Awtrey and F. Patane.