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?
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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.
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$\begingroup$ Thank you! For other degrees of Legendre's polynomials let us say 13th is it possible to derive a closed form for the roots? Any textbook you know about it? $\endgroup$– AriaCommented Mar 23, 2019 at 18:13