Problem:
Suppose z=a+bi is a solution of the polynomial equation
c4βz4+ic3βz3+c2βz2+ic1βz+c0β=0
where c0β,c1β,c2β,c3β, a and b are real constants and i2=β1. Which one of the following must also be a solution?
Answer Choices:
A. βaβbi
B. aβbi
C. βa+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, βzΛ=βa+bi is also a solution of the original equation. (One may check by example that neither βaβbi nor aβbi need be a solution. For instance, consider the equation 0=z4+iz and the solution a+bi=21β(3ββi). Neither 21β(β3β+i) nor 21β(3β+i) is a solution.