For how high a degree is the following formula exact? $$\int_{-1}^1f(x) dx = \frac{5}{9}f(-\sqrt{\frac{3}{5}})+\frac{8}{9}f(0)+\frac{5}{9}f(\sqrt{\frac{3}{5}})$$
Verify your answer by continuing the method of undetermined coefficients until an equation is not satisfied.
Now I'm stuck, I dont really understand Gaussian quadrature and can some one help me use the method of undetermined coefficients STEP BY STEP to solve the problem? This isn't homework it was just apart of my review sheet
My mindset believes that I just need to derive formula by using integrals from $-1$ to $1$ of $1,x^2,x^3,x^4$ then doing the same thing till $x^5$. But I am not sure how to do this, the book only gives one example and doesnt show you how to go on from there.

The problem is symmetric, so taking odd functions for $f$ gives zero on both sides.
Using $f(x)=x^6$ one gets
$$
\int_{-1}^1 x^6\,dx
=\frac27
$$
On the other hand
$$
f(\pm\sqrt{\tfrac35})=(\tfrac35)^3=\frac{27}{125}
$$
so that the right side gives
$$
2·\frac59·\frac{27}{125}=\frac{6}{25}=\frac27·\frac{21}{25}
$$
which is close, but not exact.
