<p>This paper investigates the factorization of Cauchy-polynomial matrices, a structured class that generalizes Cauchy-Vandermonde matrices by incorporating polynomial bases different from the monomial basis. We analyze the impact of different polynomial bases, including <i>q</i>-Bernstein, <i>h</i>-Bernstein, and Said-Ball polynomials, on numerical accuracy. A key focus is the derivation of formulas for their determinants, satisfying the No Inaccurate Cancellation condition and ensuring high relative accuracy. The sensitivity of the algorithm under deviations of the input data is analyzed, and an upper running error bound is provided. Theoretical findings are supported by numerical experiments, demonstrating the superior accuracy of the proposed determinant formulas compared to standard computational methods, even when perturbations are considered.</p>