A closed-loop system has the characteristic function (𝑠2 – 4) (s + 1) + k (s - 1)=0 . Its root locus plot against K is -------Select one:

To plot the root locus of the given closed-loop system, we need to follow these steps:

1. Identify the open-loop transfer function: The characteristic function given in the question represents the open-loop transfer function of the system. In this case, the open-loop transfer function is (𝑠2 – 4) (s + 1) + k (s - 1)=0.

2. Determine the poles and zeros: The poles of the open-loop transfer function are the values of 's' that make the characteristic function equal to zero. In this case, the poles are the roots of the equation (𝑠2 – 4) (s + 1) + k (s - 1)=0.

3. Plot the poles and zeros on the complex plane: Once we have determined the poles and zeros, we can plot them on the complex plane. The poles are usually represented by 'x' marks, and the zeros are represented by 'o' marks.

4. Determine the branches of the root locus: The root locus consists of a set of curves that represent the possible locations of the closed-loop poles as the gain 'k' varies. To determine the branches of the root locus, we need to consider the angles and magnitudes of the transfer function poles and zeros.

5. Sketch the root locus plot: Using the information obtained from steps 3 and 4, we can sketch the root locus plot by connecting the branches of the root locus and indicating the direction in which the poles move as the gain 'k' changes.

By following these steps, we can plot the root locus of the given closed-loop system against the gain 'k'.