I need to calculate the resultant of $Q=X^{10}+X^9 + \cdots + 1$ and $P= X^3+X^2+1$ by hand, and I already know it should be $23$. I'm obviously not gonna take the naive way via the coefficient matrix. As I hint I was told to divide $Q$ by $P$ via long division, but I cannot find how this should help:
We get $Q = PS+R$ where $S = X^7+X^5-X^4+2X^3-2X^2+4X-5$ and $R=8X^2-3X+6$.
We also know that there are two polynomials $U,V$ with $\deg U < \deg Q$ and $\deg V < \deg P$ such that
$$ \operatorname{Res}(P,Q) = UP+VQ = P(U+VS)+VR $$
Here we can see that if there is a common zero of $P$ and $Q$, it must also be a zero of $R$ and we could repeat the procedure with $P,R$ to show that there is no common zero.
I don't know how to go any further from here. Can anyone tell me how to proceed?