Find the value(s) of coefficient such for two polynomials f (x) and g(x) are orthogonal to each other

a) Find the value(s) of coefficient b such that the two polynomials f (x) = x + 1 and g(x) = 1 + bx + x^2 are orthogonal to each other with respect to the L2-scalar product

b) Find all Legendre coefficients for f: [−1,1]→R: x→x^3+1.

c) Find the 17th Legendre coefficient of the function f : [−1, 1] → R : x → sin(x^2).

d) Show by using Rodrigues’ formula that P1 and P2 are orthogonal with respect to the scalar product defined in part (a).