I have 4 data points:
- (204, 5422892)
- (205, 5722486)
- (207, 6343357)
- (213, 8386502)
I have information that these data points were generated with a cubic polynomial
$y = ax ^ 3 + bx ^ 2 + cx + d$
with $a$, $b$, $c$, and $d$ being rational (but not necessarily integral) coefficients, and with the resulting $y$ values being floats that were floored to an integer (the $x$ values are known to be true integers, not floored floats). I also have information that the coefficients are relatively "nice", "simple", or "short" numbers with only a few digits like $50$ or $1/2$ rather than anything that is a long string of many digits.
I need to determine what those coefficients are. I tried submitting the query
"find cubic polynomial that fits (204, 5422892), (205, 5722486), (207, 6343357), (213, 8386502)"
to Wolfram Alpha, but Wolfram Alpha gives me the answer
$(589/36) x ^ 3 - (116363/18) x ^ 2 + (10691263/12) x - 46197659$
which has coefficients that are rational but are not "nice" (in the sense used above). I think the difficulty is that Wolfram Alpha is using Lagrange interpolation to try to find a cubic polynomial that is an absolute exact fit to the given data points, but there is actually flooring to an integer being applied to the $y$ values.
So can anyone help me...
(1) find the coefficients for this specific case of the 4 data points given above?
(2) show me how to, in general, given any 4 data points that are known to be generated with a cubic polynomial with rational coefficients with the $y$ values then floored to an integer, find those coefficients? (I am aware of Lagrange interpolation, but I think that is what Wolfram Alpha is doing above and is what is giving wrong results due to not being able to take into account flooring.)