I am in a first semester numerical analysis course and we are going over numerical integration and more specifically quadrature forms. So far we have gone over standard quadrature as well as Gaussian quadrature. The problem is:
Show that the quadrature of the form
$\displaystyle \int_a^b f(x)dx \approx c_1f(x_1)+c_2f(x_2)+c_3f(x_3)+c_4f(x_4)$
cannot be exact for all polynomials of degree 4.
I seem to be having an issue with the fact that through Gaussian quadrature exact accuracy is achieved for polynomials of $2k-1$ where $k$ denotes the number of nodes. In this case we are using 4 nodes so we can accurately integrate a polynomial of degree 7. I don't know how to prove this because of this. As well I've tested it for standard polynomials when using Gaussian quadrature (i.e. $x^4$) and it held.