Lagrange Interpolation Formula
For any distinct complex numbers and any complex numbers , there exists a unique polynomial of degree less than or equal to such that for all integers , , and this polynomial is
While this formula may appear intimidating, it's actually not so difficult to see what is going on: for each term in the sum, we are finding a polynomial of degree that goes through the points and for . When we add them all together, we end up with a polynomial that interpolates the desired points.
This formula is useful for many olympiad problems, especially since such a polynomial is unique.
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