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What you ask for is not possible.
I assume that this is a homework question. There are good reasons to look for a faster implementation than Matlab's polyfit, but you do not ask for speed.
At first you need to construct the Vandermonde matrix. There are a lot of different algorithms to get it, and Google or WikiPedia know more details.
At second the least squares problem must be solved. V \ y with the Vandermonde matrix V and the y values is sufficient. A QR decomposition is more stable, so I suggest to try it and ask again in case or problems.
Be sure to use properly scaled inputs for x: A mean of 0 and a standard deviation of 1 is a fair choice to support the numerical stability of the algorithm. It is easy to consider the scaling afterwards.