Say I have the data:
$x=[ 1, 2, 3.3, 4, 5.5, 8, 9, 10.2, 11, 45 ]$
$y=[ 9,27,64,91,164,330,462,540,630,10218]$
The data is subjective though.
How would one approximate a valid polynomial for this data?
Where the original function for finding this "fake" data was:
$5x^2+2x+3$
And the polynomial from the approximation is:
$5.0080x^2 + 1.6369x+3.1341$
Obviously I tweaked the fake data a slight bit to make it closer to a real scenario.
My initial thought was to brute-forcing using computers, because of their to do countless calculations per second, but there must be a better way?
If you have more data points than the degree of the polynomial, you can do a least squares fit. The fact that it is a polynomial is not important. The important point is that the fitting function (your polynomial) depends linearly on the parameters (the coefficients). $x^2$ is not linear, but no matter. You could as well fit a Fourier series, where the basis functions are sines and cosines instead of monomials. Any linear analysis text will explain the process, which can be done using the normal equations. I like chapter 15 of Numerical Recipes. Obsolete versions are free online