The polynomials $p(x) = 5x^3 – 27x^2 + 45x – 21$ and $q(x) = x^4 – 5x^3 + 8x^2 – 5x + 3$ both interpolate the points $(1,2) , (2,1) , (3,6), (4,47)$. Even though these polynomials are of different degree, I do not understand how this is possible when the Lagrange interpolation theorem states there is only one polynomial which should interpolate these points.
[Math] Uniqueness in Lagrange Interpolation Theorem
interpolationpolynomials
Best Answer
Working in the Newton basis makes this clearer. The Lagrange interpolation theorem says that your cubic polynomial is the unique polynomial interpolant whose degree is at most $3$. It can be written in the Newton basis as:
$$c_1 + c_2 (x-1) + c_3 (x-1)(x-2) + c_4 (x-1)(x-2)(x-3)$$
for some $c_1 , \dots , c_4$. If I now consider a polynomial of the form
$$c_1 + c_2 (x-1) + c_3 (x-1)(x-2) + c_4 (x-1)(x-2)(x-3) + c_5 (x-1)(x-2)(x-3)(x-4)$$
then no matter what $c_5$ is, this will also be an interpolant. (Why?) I can also replace $c_5$ by any polynomial I want, and it will still be an interpolant.