1982 AMC 12 Problem 27

Question

Suppose

z=a+bi

is a solution of the polynomial equation

\begin{aligned} c_4z^4+ic_3z^3+c_2z^2\\ {}+ic_1z+c_0=0, \end{aligned}

where

c_0,

c_1,

c_2,

c_3,

c_4,

a,

and

b

are real constants and

i^2=-1.

Which one of the following must also be a solution?

-a-bi

a-bi

-a+bi

b+ai

none of these

Conjugate the entire equation

Compare the conjugated equation at

\bar z

with the original polynomial evaluated at

-\bar z

Conjugating the equation changes each

i

to

-i

and

z

to

\bar z.

Because the odd-powered terms also change sign when the input is negated, this conjugated equation is precisely the original polynomial evaluated at

-\bar z.

Thus

-\bar z=-a+bi

must be a root.

Therefore, the correct answer is C.

Problem 27 in other years