Mechanical Integration

Age 16 to 18
Challenge Level

There is a theory of polynomials which enables you to find the integral of a polynomial by simply evaluating it at some special points and adding certain multiples $\Lambda_1, \Lambda_2,\ldots$ of these values. For example, for all polynomials $q$ of degree less than six, the special points are $-\sqrt{3/5}$, $0$ and $+\sqrt{3/5}$, and $$\int_{-1}^1 q(x)dx = \Lambda_1q(-\sqrt{3/5}) + \Lambda_2q(0) + \Lambda_3q(+\sqrt{3/5}).\quad (1)$$ Find the multiples $\Lambda_1,\ \Lambda_2$ and $\Lambda_3$ by considering the three polynomials $q(x) = 1$, $q(x) = x$ and $q(x) = x^2$.

With these values of $\Lambda_1,\ \Lambda_2$ and $\Lambda_3$ show that the mechanical integration given by equation (1), which uses the values of the polynomial at the three special points, gives the value of the integral of ALL quadratic polynomials.

Now go on to show that the same formula gives the integral of ALL cubic, quartic and quintic polynomials.

Does the formula (1) hold for $q(x)=x^6$?