Question

Let $p(z) = z^3 + (1 + j)z^2 + (2 + j)z + 3$, where $z$ is a complex number. Which one of the following is true?

• A. $\text{conjugate}\{p(z)\} = p(\text{conjugate}\{z\})$ for all $z$
• B. The sum of the roots of $p(z)=0$ is a real number
• C. The complex roots of $p(z)=0$ come in conjugate pairs
• D. All the roots cannot be real

Hint:

Polynomials with non-real coefficients do not guarantee conjugate-pair roots; only real-coefficient polynomials do.

Answer 1

The Correct Option is D

The polynomial is \[ p(z) = z^3 + (1 + j)z^2 + (2 + j)z + 3 \]
A polynomial has real roots or complex conjugate pairs only if all coefficients are real. But here the coefficients include complex terms: - $(1 + j)$
- $(2 + j)$
Thus, the polynomial has non-real coefficients, so: - Complex roots do not necessarily occur in conjugate pairs → option (C) is false.
- Conjugation symmetry $\overline{p(z)} = p(\overline{z})$ holds only when coefficients are real, so (A) is false.
- The sum of the roots = $-(1 + j)$, which is not real, so (B) is false.
Since coefficients are complex, the roots cannot all be real. This makes (D) true.
Final Answer: All roots cannot be real