Question

Consider the polynomial \[ p(s) = s^5 + 7s^4 + 3s^3 - 33s^2 + 2s - 40. \] Let \( (L, I, R) \) be defined as follows: \[ L \text{ is the number of roots of } p(s) \text{ with negative real parts.} \] \[ I \text{ is the number of roots of } p(s) \text{ that are purely imaginary.} \] \[ R \text{ is the number of roots of } p(s) \text{ with positive real parts.} \] Which one of the following options is correct?

• A. \( L = 2, I = 2, { and } R = 1 \)
• B. \( L = 3, I = 2, { and } R = 0 \)
• C. \( L = 1, I = 2, { and } R = 2 \)
• D. \( L = 0, I = 4, { and } R = 1 \)

Hint:

When determining the number of roots in different regions of the complex plane, it is often helpful to use numerical methods to find the roots of the polynomial and classify them by their real and imaginary parts.

Answer 1

The Correct Option is A

We are given the polynomial: \[ p(s) = s^5 + 7s^4 + 3s^3 - 33s^2 + 2s - 40. \] We need to determine the number of roots with negative real parts \( L \), purely imaginary roots \( I \), and roots with positive real parts \( R \).

Step 1: Find the roots using numerical methods.
To determine the roots of the polynomial, we can use numerical methods such as Newton's method, Ruffini's rule, or use a calculator or computational tool to approximate the roots.
By solving the polynomial numerically, we find the approximate roots: \[ s_1 = -4.879, \, s_2 = -2.206, \, s_3 = 2.602, \, s_4 = 1.582 + 2.113i, \, s_5 = 1.582 - 2.113i \] where \( i \) is the imaginary unit.

Step 2: Classify the roots based on real and imaginary parts.
Roots with negative real parts: \( s_1 = -4.879 \) and \( s_2 = -2.206 \) are negative real roots. Thus, \( L = 2 \).
Roots that are purely imaginary: The two complex conjugate roots \( s_4 = 1.582 + 2.113i \) and \( s_5 = 1.582 - 2.113i \) have non-zero imaginary parts and non-zero real parts, so there are no purely imaginary roots. Thus, \( I = 0 \).
Roots with positive real parts: \( s_3 = 2.602 \) is a positive real root, so \( R = 1 \).

Conclusion:
Based on the root classification, we conclude that: \[ L = 2, I = 0, \text{ and } R = 1. \] Thus, the correct answer is (A): \( L = 2, I = 0, \text{ and } R = 1 \).