Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 19D / a / Solution

The Lagrange interpolation polynomial basis for nodes $0,2,3$ givesEach polynomial is one at its own node and zero at the other two. For a polynomial $f$ of degree at most two, the difference between $f$ and this interpolation polynomial has degree at most two and three distinct roots, so it is identically zero. This also proves uniqueness of the quadratic coefficients.