This is an iterative method using the tangent line approximation. $$x_n+1 = x_n - \fracf(x_n)f'(x_n)$$
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If $n+1$ data points are known, there exists a unique polynomial of degree $n$ passing through them. The Lagrange formula is: $$P(x) = \sum_i=0^n y_i L_i(x)$$ Where $L_i(x)$ are basis polynomials dependent only on the $x$-coordinates.