Question

Let p(z) = z³ + (1 + j) z² + (2 + j) z + 3, where z is a complex number.

Which one of the following is true?  

1. All the roots cannot be real
2. conjugate {p(z)} = p(conjugate {z}) for all z
3. The sum of the roots of p(z) = 0 is a real number
4. The complex roots of the equation p(z) = 0 come in conjugate pairs

Answer 1

Correct Answer - Option 1 : All the roots cannot be real

Concept:

The general form of a cubic equation is ax³ + bx² + cx + d = 0.

Where a, b, c, and d are constants and a ≠ 0.

Let the roots be p, q, and r

• The sum of the roots (p + q + r) = - b/a
• The product of the roots (pqr) = - d/a
• The sum of the product of any two roots (pq + qr + rp) = c/a

 

Calculation:

Given p(z) = z3 + (1 + j) z2 + (2 + j) z + 3

Sum of the roots (p + q + r) = - (1 + j)

Product of the roots (pqr) = - 3

Sum of the roots is complex, so all the roots cannot be real.