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

$a$

$b$

$c$

$-a$

$-b$

Answer: D

Concepts:complex conjugates unit circle Vieta’s formulas

Difficulty rating: 2340

Small Hint:

For a complex number $r$ on the unit circle, compare $\frac{1}{r}$ with $\overline r$

Big Hint:

Real polynomial coefficients make the sum of the roots real

Solution:

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.

1987 AMC 12 Problem 28

Answer: D

Difficulty rating: 2340

Small Hint:

Big Hint:

Solution:

Problem 28 in Other Years