Question

The number of roots of $s^3 + 5s^2 + 7s + 3 = 0$ in the right half of the s-plane is _______.

• A. zero
• B. one
• C. two
• D. three

Hint:

No sign change in the Routh table’s first column means all roots lie in the left half-plane.

Answer 1

The Correct Option is A

Step 1: Routh-Hurwitz Stability Criterion.
To determine number of RHP roots, construct the Routh table:

Given polynomial: \( s^3 + 5s^2 + 7s + 3 \) \[ \begin{array}{c|cc} s^3 & 1 & 7 \\ s^2 & 5 & 3 \\ s^1 & \frac{(5 \times 7 - 1 \times 3)}{5} = \frac{32}{5} & 0 \\ s^0 & 3 & - \end{array} \] Step 2: Check sign changes in first column.
First column: 1, 5, \( \frac{32}{5} \), 3 → all positive, no sign change

Conclusion: \( \boxed{0} \) roots lie in right-half s-plane.