Question

# Prove that n distinct point-value pairs are necessary to uniquely specify a polynomial of degree-bound n, that is, if fewer than n distinct point-value pairs are given, they fail to specify a unique polynomial of degree-bound n.

Solution

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Let we have $n-1$ distinct points, hence we can get a unique polynomial representation for a polynomial up to degree ($n-1$), satisfied by the $n-1$ points, according to the mentioned theorem in the hint.

Now if we add a new point to our ($n-1$) points we can get a new unique representation for a polynomial up to degree $n$, satisfied by thhe $n$ points.

Again if we add a new point (different from the point just we chose) to our ($n-1$) points we can get a new unique representation for a polynomial up to degree $n$, satisfied by thhe $n$ points.

Since our $n-1$ points give two different representation of two different polynomial, $n-1$ points can't give a unique representation for a polynomial of degree $n$.

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