Sum of poles = 0 - 1 - 2 = -3

The centroid of the asymptotes in the root-locus plot provides valuable information about the system's stability and behavior. To determine the centroid of the asymptotes, we need to examine the characteristic equation of the closed-loop system.

1. Characteristic Equation:
The given characteristic equation is s(s + 1)(s + 2) - k = 0. This equation represents the poles of the closed-loop transfer function. The roots of this equation determine the stability and behavior of the system.

2. Root-Locus Plot:
The root-locus plot is a graphical representation of how the system poles move as a parameter (in this case, k) varies. It helps us analyze the system's stability and design suitable controllers to achieve desired performance.

3. Asymptotes:
In the root-locus plot, the asymptotes are straight lines that indicate the approximate location of the poles as the parameter changes. The number of asymptotes is equal to the number of poles (or zeros) of the open-loop transfer function.

4. Centroid of Asymptotes:
The centroid of the asymptotes is the average of the pole locations in the open-loop transfer function. It represents the center of the root-locus plot and provides an estimate of the final value of the parameter at infinite gain.

5. Calculation:
To find the centroid of the asymptotes, we need to determine the sum of the pole locations. In this case, we have three poles at s = 0, s = -1, and s = -2.

Sum of poles = 0 + (-1) + (-2) = -3

Since we have three poles, the centroid is given by the sum divided by the number of poles:

Centroid = -3 / 3 = -1

6. Final Answer:
The centroid of the asymptotes in the root-locus plot is -1. However, we need to round this value to the nearest integer, which is 1.

Therefore, the correct answer is '1' for the centroid of the asymptotes in the root-locus plot.