Numerical Analysis Gaussian-quadrature

Question

How do you show that the 4 point Gaussian quadrature rule is exact for all polynomials of degree less than or equal to 7?

Answer 1

It is done by consistency requirement.

We require the exactness of $$ \int _{-1}^1 f(x)dx= c_1f(x_1)+ c_2f(x_2)+c_3f(x_3)+ c_4f(x_4)$$

for polynomials of degree up to $7$.

Let $$f(x)= 1,x,x^2,x^3,x^4,x^5,x^6,x^7$$ into the above equation and solve for $$\{c_1,c_2,c_3, c_4,x_1,x_2,x_3,x_4\}$$

Note that you have $8$ equations and $8$ unknowns.

As you know there are better ways to calculate the nodes and weights than solving systems, but that is how it started.