A Gaussian quadrature formula provides an approximation to the integral

$$\int_{-1}^{1}\left(1-x^{2}\right) f(x) d x \approx \sum_{k=1}^{\nu} b_{k} f\left(c_{k}\right)$$

which is exact for all $f(x)$ that are polynomials of degree $\leqslant(2 \nu-1)$.

Write down explicit expressions for the $b_{k}$ in terms of integrals, and explain why it is necessary that the $c_{k}$ are the zeroes of a (monic) polynomial $p_{\nu}$ of degree $\nu$ that satisfies $\int_{-1}^{1}\left(1-x^{2}\right) p_{\nu}(x) q(x) d x=0$ for any polynomial $q(x)$ of degree less than $\nu .$

The first such polynomials are $p_{0}=1, p_{1}=x, p_{2}=x^{2}-1 / 5, p_{3}=x^{3}-3 x / 7$. Show that the Gaussian quadrature formulae for $\nu=2,3$ are

$$\begin{array}{ll} \nu=2: & \frac{2}{3}\left[f\left(-\frac{1}{\sqrt{5}}\right)+f\left(\frac{1}{\sqrt{5}}\right)\right] \\ \nu=3: & \frac{14}{45}\left[f\left(-\sqrt{\frac{3}{7}}\right)+f\left(\sqrt{\frac{3}{7}}\right)\right]+\frac{32}{45} f(0) \end{array}$$

Verify the result for $\nu=3$ by considering $f(x)=1, x^{2}, x^{4}$.