In Gaussian quadrature formula of integration we need to have the zeros of Legendre's polynomials. Although we may find the zeros numerically, I got closed form formulas such as $$\pm\frac{1}{3}\sqrt{5+2\sqrt\frac{10}{7}}$$for some roots of the 5th order Legendre's polynomials. Do you know about any textbook talking about the detailed proof of this or can you prove it?
2026-03-25 05:07:39.1774415259
How to derive closed form expression for the zeros of Legendre's Polynomials (Gaussian-Quadrature formulas)?
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The polynomial is $\frac{63}{8}x^5-\frac{35}{4}x^3+\frac{15}{8}x = x(\frac{63}{8}x^4-\frac{35}{4}x^2+\frac{15}{8})$, which is the product of $x$ and a polynomial of degree $2$ (with respect to $x^2$). Equate each factor to zero, and use the quadratic formula to find $x^2$ in the second resulting equation.