Given the weight function $w(t) \equiv 1$, for $t\in [0,1]$ build the Gaussian quadrature formula of order $2$.
The formula in this case has the form
$$ \int_0^1 f(\tau)d\tau \approx b f(c) $$
For this, I find an orthogonal polynomial $1-2t$, whose root is $c:=\frac{1}{2}$. To find $b$, I need to take the interpolated polynomial $$ t-\frac{1}{2} $$
if I'm not mistaken, since
$$p_j(t) = \prod\limits_{k=1, k\ne j}^\nu\frac{t-c_k}{c_j-c_k}$$
where $j=1,...,\nu$. In this case we have just one $c_k$, so it looks like we can just take $t-\frac{1}{2}$.
Then $$ b = \int_0^1 (\tau -\frac{1}{2})d\tau$$ which is just $0$. So something I'm doing is not right. I think I'm erring on the interpolating polynomial side. Would appreciate some help.
The problem is that in your case you are constructing 0th order polynomial which is 1, and not linear.