= Solution
Put $P(t)=\prod_i(t-x_i)$. For every <polynomial> $q$ of degree at most $n-1$, $Pq$ has degree at most $2n-1$ and vanishes at all nodes; thus $\int_{-1}^1Pq=0$. The monic degree-$n$ <orthogonal polynomial> for this integral is unique: the difference of two such monic <polynomials> has degree below $n$ and is orthogonal to itself, hence is zero. Consequently $P$ is the monic normalization of the degree-$n$ <Legendre polynomial>, proving \b[the given nodes are exactly its roots]. Together the three results characterize this <Gaussian quadrature> rule.
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