Calculate polynomial regression coefficients for MSE

To demonstrate the calculation of the coefficients for polynomial regression with MSE suppose we have 1-dimensional input and training data . For an order polynomial we are looking for coefficients for that roughly do the following

in a completely not mathematically sound way we can do the following rearrangement

completing our fuzzy maths. We use this as our definition of and moreover this has some nice properties that actually minimises with respect to MSE. This can expand out to be multi-variate by increasing the size of and respectively.