 Coefficient forms from three polynomial evaluations

For $p(X)=a+bX+c{X}^2$, the three point evaluation functionals at $0,1,-1$ form a basis of the dual space of real polynomials of degree at most two. The displayed recovery formulas express the coordinate functionals, hence the dual basis to $1,X,X^2$, as linear combinations of these evaluations. More generally distinct-node polynomial interpolation identifies values with coefficient data through an invertible Vandermonde matrix.