I was reading about interpolation and approximation in Numerical Methods and came across this statement in my course material, "for n data points, there is one and only one polynomial of order (n − 1) that passes through all the points" for example, we have 3 data points on a straight line then how can a second order polynomial satisfy it?
2026-04-03 12:30:31.1775219431
Interpolation and approximation from Numerical Methods
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The statement could be rephrased with "of order at most $n-1$" or "with possibly zero coefficients". What matters is that these polynomials have exactly $n$ degrees of freedom, and more importantly, that Lagrangian interpolation is always possible and unique.
For completeness, the text might also specify $n$ data points "with different abscissas".