This paper investigates the volume polynomials of Euclidean tetrahedra under prescribed edge-length equalities. It identifies 25 edge-length symmetry classes by grouping edge-equality patterns up to vertex relabeling. For each class, the Cayley–Menger determinant is specialized to obtain a homogeneous cubic polynomial in the distinct squared edge lengths. Candidate formulas are discovered computationally by generating valid tetrahedra, fitting polynomial coefficients using least squares, and reconstructing exact rational coefficients. Each formula is then verified symbolically and factored over the corresponding rational polynomial ring. The results show that 16 of the 25 specialized volume polynomials are reducible, while nine are irreducible. The study further demonstrates that factorization depends not only on the multiplicities of equal edge lengths but also on their geometric arrangement, particularly the adjacency and opposition of equal edges. Finally, it interprets polynomial factors in terms of degenerate boundary configurations and suggests extending the computational method to higher-dimensional simplices.