Chapter 9.8, Problem 39E

To determine

To explain why the sum of the imaginary parts of the sum of two complex conjugates is $0$ , the imaginary part of the sum of all the roots must be $0$ .

Explanation of Solution

Given:

The given complex conjugate.

Calculation:

The roots are the vertices of a regular polygon.

Since one of the roots must be a positive real number, a vertex of the polygon lies on the positive real axis and the polygon is symmetric about the real axis.

This means the non-real complex roots occur in conjugate pairs.

Since the imaginary part of the sum of two complex conjugates is $0$ , the imaginary part of the sum of all the roots must be $0$ .