1987 AMC 12 Problem 28

Question

Let

a,

b,

c,

d

be real numbers. Suppose that all the roots of

z^4+az^3+bz^2+cz+d=0

are complex numbers lying on a circle in the complex plane centered at

0+0i

and having radius

1.

The sum of the reciprocals of the roots is necessarily

a

b

c

-a

-b

For a complex number

r

on the unit circle, compare

\frac{1}{r}

with

\overline r

Real polynomial coefficients make the sum of the roots real

If

|r|=1,

then

\frac{1}{r}=\overline r.

Therefore the sum of the reciprocals is the conjugate of the sum of the roots. By Vieta’s formulas, the sum of the roots is

-a,

which is real. Its conjugate is still

-a.

Thus the correct answer is D.

Problem 28 in other years