Question

The number of zeros of the polynomial $P(s) = s^3 + 2s^2 + 5s + 80$ in the right-half plane is _____.

Hint:

Routh-Hurwitz criterion is a systematic method to find the number of roots of a polynomial with positive real parts (right-half plane). Count the sign changes in the first column of the Routh array.

Answer 1

Correct Answer: 2

Step 1: Write the polynomial.
\[ P(s) = s^3 + 2s^2 + 5s + 80 \] Step 2: Use Routh-Hurwitz criterion.
Construct the Routh array: \[ \begin{array}{c|cc} s^3 & 1 & 5
s^2 & 2 & 80
s^1 & \frac{(2 . 5 - 1 . 80)}{2} = \frac{10 - 80}{2} = -35 & 0
s^0 & 80 &
\end{array} \] Step 3: Check sign changes in the first column.
First column: $[1, 2, -35, 80]$.
Number of sign changes = 2.
Step 4: Interpret.
Thus, there are 2 roots in the right-half plane.