Problem:
Suppose z=a+bi is a solution of the polynomial equation
c4​z4+ic3​z3+c2​z2+ic1​z+c0​=0
where c0​,c1​,c2​,c3​, a and b are real constants and i2=−1. Which one of the following must also be a solution?
Answer Choices:
A. −a−bi
B. a−bi
C. −a+bi
D. b+ai
E. none of these
Solution:
One cannot simply use the theorem that solutions come in conjugate pairs, because that theorem applies to polynomials with real coefficients only. However, one can use the technique for proving that theorem to work this problem too. Namely, conjugate both sides of the original equation
That is, −zˉ=−a+bi is also a solution of the original equation. (One may check by example that neither −a−bi nor a−bi need be a solution. For instance, consider the equation 0=z4+iz and the solution a+bi=21​(3​−i). Neither 21​(−3​+i) nor 21​(3​+i) is a solution.