(a) Given a set of $n+1$ distinct real points $x_{0}, x_{1}, \ldots, x_{n}$ and real numbers $f_{0}, f_{1}, \ldots, f_{n}$, show that the interpolating polynomial $p_{n} \in \mathbb{P}_{n}[x], p_{n}\left(x_{i}\right)=f_{i}$, can be written in the form

$$p_{n}(x)=\sum_{k=0}^{n} a_{k} \prod_{j=0, j \neq k}^{n} \frac{x-x_{j}}{x_{k}-x_{j}}, \quad x \in \mathbb{R}$$

where the coefficients $a_{k}$ are to be determined.

(b) Consider the approximation of the integral of a function $f \in C[a, b]$ by a finite sum,

$$\int_{a}^{b} f(x) d x \approx \sum_{k=0}^{s-1} w_{k} f\left(c_{k}\right)$$

where the weights $w_{0}, \ldots, w_{s-1}$ and nodes $c_{0}, \ldots, c_{s-1} \in[a, b]$ are independent of $f$. Derive the expressions for the weights $w_{k}$ that make the approximation ( 1$)$ exact for $f$ being any polynomial of degree $s-1$, i.e. $f \in \mathbb{P}_{s-1}[x]$.

Show that by choosing $c_{0}, \ldots, c_{s-1}$ to be zeros of the polynomial $q_{s}(x)$ of degree $s$, one of a sequence of orthogonal polynomials defined with respect to the scalar product

$$\langle u, v\rangle=\int_{a}^{b} u(x) v(x) d x$$

the approximation (1) becomes exact for $f \in \mathbb{P}_{2 s-1}[x]$ (i.e. for all polynomials of degree $2 s-1)$.

(c) On the interval $[a, b]=[-1,1]$ the scalar product (2) generates orthogonal polynomials given by

$$q_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n}, \quad n=0,1,2, \ldots$$

Find the values of the nodes $c_{k}$ for which the approximation (1) is exact for all polynomials of degree 7 (i.e. $f \in \mathbb{P}_{7}[x]$ ) but no higher.