Question

Which one of the following options correctly describes the locations of the roots of the equation s⁴ + s² + 1 = 0 on the complex plane?

A. Four left half plane (LHP) roots
B. One right half plane (RHP) root, one LHP root and two roots on the imaginary axis
C. Two RHP roots and two LHP roots
D. All four roots are on the imaginary axis

Answer 1

The equation s⁴ + s² + 1 = 0 has two complex conjugate roots, one right half plane (RHP) root, one left half plane (LHP) root, and two roots on the imaginary axis. The correct answer is B. One right half plane (RHP) root, one LHP root and two roots on the imaginary axis.

Step-by-step explanation:

The equation s⁴ + s² + 1 = 0 is a quartic equation, which means it has degree 4. To determine the locations of its roots on the complex plane, we can use the discriminant.

The discriminant is given by Δ = b² - 4ac, where a = 1, b = 0, and c = 1 in this equation. Plugging in these values, we get Δ = 0 - 4(1)(1) = -4.

Since the discriminant is negative, the equation has two complex conjugate roots. Therefore, the correct option is B. One right half plane (RHP) root, one LHP root and two roots on the imaginary axis.