Consider the general problem: Find the real zeros of an arbitrary polynomial.
Here is my plan of attack. I will introduce some points on the curve, see how it behaves at those points, and use that information to find points closer to the zeros.
How should I introduce points? What behavior interests me? How do I isolate an interval that must contain a zero?
To understand how this might be done, I begin with a particular case which lends itself to the discovery of answers:

To give the polynomial from arbitrary points P1, P2, P3, P4:
Let the points be given by
And vector Y be the function values of the four points:
The ith row of matrix M be x_i evaluated at each of the four functions:
The unknowns are the coefficients:
For each point, I have
The system of four points can be written as the matrix multiplication
Hence, the coefficients are
I can now pass back and forth between arbitrary points and coefficients. But I would like to do better than this. I want to choose points which give me information about the function and its zeros...
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To find the inverse of M, Set M = I, and perform Gaussian elimination on the two at once. When the left hand side is I, the right hand side is the inverse matrix.