Theorem: P_2(1/sqrt(3)) = 0 (Legendre root for 2-point Gauss quadrature)

Theorem (Legendre-root-2pt canonical): 2-point Gauss-Legendre nodes on (-1, 1) are roots of P_2(x) = (3 x^2 - 1)/2, factoring as (sqrt(3) x - 1)(sqrt(3) x + 1)/2 with roots x = +/- 1/sqrt(3). P_2(1/sqrt(3)) = (3 (1/3) - 1)/2 = 0…