Show that every Legendre polynomial $p_{n}$ has $n$ distinct roots in $[-1,1]$, where $n$ is the degree of $p_{n}$.

Let $x_{1}, \ldots, x_{n}$ be distinct numbers in $[-1,1]$. Show that there are unique real numbers $A_{1}, \ldots, A_{n}$ such that the formula

$$\int_{-1}^{1} P(t) d t=\sum_{i=1}^{n} A_{i} P\left(x_{i}\right)$$

holds for every polynomial $P$ of degree less than $n$.

Now suppose that the above formula in fact holds for every polynomial $P$ of degree less than $2 n$. Show that then $x_{1}, \ldots, x_{n}$ are the roots of $p_{n}$. Show also that $\sum_{i=1}^{n} A_{i}=2$ and that all $A_{i}$ are positive.