I have been studying the root locus method from the Feedback Control of Dynamic Systems. Here in Chapter 5 I have found a derivation of the rule for finding the origin point of the asymptotes. The authors suppose that the equation
$$ 1 + K\frac{s^m + b_1\cdot s^{m-1} + \cdots + b_m}{s^n + a_1\cdot s^{n-1} + \cdots + a_n} = 0 $$
can be approximated by
$$ 1 + K\frac{1}{(s - \alpha)^{n-m}} = 0 $$
This approximation can be obtained (according to the authors) by dividing the polynomial in the denominator by the polynomial in the numerator and matching the dominant two terms (highest powers in $s$) to the expansion $(s - \alpha)^{n-m}$. I have attempted to follow these steps. So I have divided the polynomials
$$ 1 + K\frac{s^m + b_1\cdot s^{m-1} + \cdots + b_m}{(s^{n-m} + (a_1-b_1)\cdot s^{n-1-m})\cdot(s^m + b_1\cdot s^{m-1} + \cdots + b_m)} = 1 + K\frac{1}{s^{n-m} + (a_1-b_1)\cdot s^{n-1-m}} $$
but here I have stuck. Please does anybody see how to get the desired approximation from the intermediate step I have got via the polynomials division?
Observe that,
$$(s - \alpha)^{n-m} = s^{n-m} - \alpha s^{n-m-1} + F(s)$$
where I'll absorb lower order $s$ terms in $F(s)$. For $s$ very large, we have,
$$(s - \alpha)^{n-m} \approx s^{n-m} - \alpha s^{n-m-1}.$$
When talking about the asymptotic behaviour of the root locus, i.e. $K\to\infty,$ we can say,
$$0 = 1 + K \frac{1}{s^{n-m} + (a_1 - b_1) s^{n-m-1}} \approx 1 + K \frac{1}{(s-\alpha)^{n-m}}$$
Since their asymptotic behaviour agree, we can work backwards using the latter root locus equation which has a simpler behaviour to see where the asymptotes must meet. In particular, for the latter root locus all $n-m$ poles at $\alpha$ shoot off to infinity along the asymptotes. Therefore, $\alpha$ is the centroid.
Aside: We are approximating the root locus at $K = \infty$ with the simplest root locus that has all its root shoot off to infinity and working backwards to see where those asymptotes start at.