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?
Options :
Conjugate {P( z)P } = (Conjugate {z}) for all z
The sum of the roots of P (z) 0 = is a real number
The complex roots of the equation P (z) 0 = come in conjugate pairs.
All the roots cannot be real
Answer :
All the roots cannot be real
Solution :
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.
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