A given polynomial is transformed herein into a rational function. All the roots and multiplicities of the polynomial are then easily obtained from the poles and residues of this rational function, instead of solving for them directly using the original, high-degree multiple-root polynomial. The derived program, using only basic MATLAB built-in routines and existing double-precision arithmetic, amazingly gives the expected results for test polynomials of very high degree and multiplicity, even as high as p(x) = (x + 98.765)1234.