Resultant of two single-variable polynomials via long division 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?
 A: From Wikipedia:
Invariance under change of polynomials
If $a$ and $b$ are nonzero constants (that is they are independent of the indeterminate $x$), and $A,B$ are polynomials of degree $d,e$ respectively, then
$$\operatorname{res}(aA,bB) =a^eb^d\operatorname{res}(A,B).$$
If $d\ge e$, $a$ is a constant and $b_0$ is the leading coefficient of $B$, and if $C$ is a polynomial of degree at most $d-e$, then 
$$b_0^{d-e}\operatorname{res}(aA-CB,B) =a^e\operatorname{res}(A,B).$$
These properties imply that in the Euclidean algorithm for polynomials, the resultant of two successive remainders differs from the resultant of the initial polynomials by a factor, which is easy to compute. Moreover, the constant $a$ in above second formula may be chosen in order that the successive remainders have their coefficients in the ring of coefficients of input polynomials. This is the starting idea of the subresultant-pseudo-remainder-sequence algorithm for computing the greatest common divisor and the resultant of two polynomials. This algorithm works for polynomials over the integers or, more generally, over an integral domain, without any other division than exact divisions (that is without involving fractions). It involves $O(de)$ arithmetic operations, while the computation of the determinant of the Sylvester matrix with standard algorithms require $O((d+e)^3)$ arithmetic operations.
