From Wikipedia:
We then make what Acton calls a 'drastic set of assumptions', that the root we are looking for, say, $x_1$ is a certain distance away from our guess $x$, and all the other roots are clustered together some distance away. If we denote these distances by
I don't understand how such an assumption can be correct. I don't really know what a drastic set of assumptions mean but how can we just write it as $\frac{n-1}b$? They are all different values and we need to expand the expression and do the operations later, isn't it? $x_i$ has actually $n-1$ values.
They are just using the approximation $x-x_i \approx b$ for $i=2,3,\dots,n$, which is basically saying that those roots are relatively clustered together, at least enough that the "true values" $G$ and $H$ aren't modified too much by assuming they're at the same point. The validity of that depends on some things, in particular it depends on $x$ being significantly closer to $x_1$ than to the other roots.
This is structurally necessary for the method since you are basically trying to (perhaps crudely) guess where all the other roots are at once using only three scalar properties of $p$: $p(x_k),p'(x_k),p''(x_k)$.