Question

If \( Z = x + iy \) is a complex number, then the number of distinct solutions of the equation \[ z^3 + \bar{z} = 0 \] is:

• A. \( 1 \)
• B. \( 3 \)
• C. \( \text{Infinite} \)
• D. \( 5 \)

Hint:

For polynomial equations involving complex numbers, consider symmetry properties and the fundamental theorem of algebra.

Answer 1

The Correct Option is D

Step 1: Express \( \bar{z} \) in Terms of \( z \) Since \( z = x + iy \), we rewrite: \[ z^3 + \bar{z} = 0 \Rightarrow z^3 = -\bar{z} \] Step 2: Find Distinct Solutions Solving using complex number properties, we obtain 5 distinct roots. Thus, the correct answer is 5.