Legendre is confusing please help asap tyvm

[Jolene]

 

[HallsofIvy]

Okay, what do you understand? Do you know what a "Legendre Polynomial" is? Do you know what a "quadrature rule" is? Do you understand that this question is about an approximate numerical integration,

 

[Jolene]

Okay, what do you understand? Do you know what a "Legendre Polynomial" is? Do you know what a "quadrature rule" is? Do you understand that this question is about an approximate numerical integration,

hm i know that usually a quadrature rule is integral f(x)dx from -1 to 1 ≈ summation(ci*f(xi)) and xi is the roots of the Legendre polynomial Ln+1(x). for ci, my notes says that its the integral of Ln,i(x)dx from -1 to 1 (but i am unsure about this part.
i am assuming that i need to find the roots of 5x^3-3x=0 (which is 0, 0.6^0.5 and -0.6^0.5) but idk what to do after ...