Is there any method to determine a black box polynomial of real coefficients simply from inputs?

Question

I came across this question which asked to determine a polynomial with positive integer coefficients for which the minimal substitutions required turned out to be 1. I modified the problem by making it more general as in allowing the coefficients to be real. I have tried determining it by putting consecutive integer values from 0 until the successive difference reaches 0. This helps me to determine the degree of the polynomial and then creating a blueprint for the expressions of successive differences of the general polynomial of found degree, I can determine the unknown coefficients. This worked but someone showed me that if I get successive similar values, like getting 0 for consecutive integers, I can never know if the polynomial is just zero or just some polynomial of very high degree which vanishes at those particular integral inputs. I was also told that my method is similar to Newton's interpolation. Is there no method to determine the polynomial? Can there be no modifications/improvements to my method to solve this problem? P.S.: The degree of the polynomial is of course unknown, and I find it by observing at how many levels the successive difference becomes zero. I need deg(P(x)) + 2 substitutions to reach that point.