 Solution

The Lagrange basis polynomials satisfy ${\ell }_i(x_j)={\delta }_{ij}$: for $j\ne i$, the product has a zero factor, whereas for $j=i$ every factor is one. The polynomialhas degree of a polynomial at most $n$ and has $n+1$ distinct roots of a polynomial. The root bound for a polynomial implies $q=0$. Hence the interpolation identity holds for every polynomial of degree at most $n$.