(i) Let $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ be any set of $n$ distinct numbers. Show that there exist numbers $A_{1}, A_{2}, \ldots, A_{n}$ such that the formula

$$\int_{-1}^{1} p(x) d x=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$$

is valid for every polynomial $p$ of degree $\leqslant n-1$.

(ii) For $n=0,1,2, \ldots$, let $p_{n}$ be the Legendre polynomial, over $[-1,1]$, of degree $n$. Suppose that $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ are the roots of $p_{n}$, and $A_{1}, A_{2}, \ldots, A_{n}$ are the numbers corresponding to $x_{1}, x_{2}, \ldots, x_{n}$ as in (i).

[You may assume without proof that for $n \geqslant 1, p_{n}$ has $n$ distinct roots in $[-1,1] .$ ]

Prove that the integration formula in (i) is now valid for any polynomial $p$ of degree $\leqslant 2 n-1$.

(iii) Is it possible to choose $n$ distinct points $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ and corresponding numbers $A_{1}, A_{2}, \ldots, A_{n}$ such that the integration formula in (i) is valid for any polynomial $p$ of degree $\leqslant 2 n$ ? Justify your answer.