Solution
= 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 <polynomial>
$$
q(x)=p(x)-\sum_{i=0}^np(x_i)\ell_i(x)
$$
has <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 \b[the interpolation identity holds for every polynomial of degree at most $n$].