4 point quadratic curve

Question

I can define a curve that passes through 3 points using a quadratic equation:

ax2 + bx + c = 0

I would like to know is it possible to define a curve that passes through 4 points using:

ax3 + bx2 + cx + d = 0

Cheers

Answer 1

The answer was already in the comments upon migration: Use a Lagrange polynomial. The restriction "in most cases" is unnecessary; the Lagrange polynomial is completely general and yields a polynomial which interpolates the points as long as no two of them have the same $x$ coordinate; if they do, there can be no univariate function, polynomial or otherwise, that interpolates them.