http://planetmath.org/proofthatsylvestersmatrixequalstheresultant
Heres a link to a proof I found concerning the relation between Sylvesters matrix and resultants. Most of it makes sense. I do have one question though. Near the end it says, "To determine the constant of proportionality...", and they conclude that this constant is $1$. I don't understand why though. I would appreciate an explanation.
You compute them directly. For $R'$ you get $\prod_{i=1}^{m}\prod_{j=1}^{n}(1-0)=1$.
For the determinant you need to expand the polynomials to get the matrix. One polynomial is $x^n=x^n+0x^{n-1}+...+0x+0$ the other is $(x-1)^m=\sum_{k=0}^{m}\binom{m}{k}x^k$. Put these numbers in the matrix.
$$\begin{pmatrix}1&0&0&...&0&0\\0&1&0&...&0&0\\\vdots&&&&0&0\\1&\binom{m}{1}&\binom{m}{2}&...&0&0\\\vdots\\0&0&0&...&\binom{m}{1}&1\end{pmatrix}$$
Notice the matrix is diagonal with diagonal consisting of $1$'s.