...It is not a very eficient way, as we must calculate al the derivatives of the polinomial, implying very high numbers (n!), that make the needed precision increase, making the computational complexity for high-degree polinomials poor.
But I have been able to make it work with a 40-degree polynomial, with two very proximal zeroes arround x=1000.
It needs a precision of arround 60 digits.
The problem comes with very-high degree polynomials.
The higher is the degree, the faster the execution time increases.
But for common examples, the application does the job well.