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Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.9 AIME Prove it United States

Problem:
Suppose that there exist nonzero complex numbers $a$ , $b$ , $c$ , and $d$ such that $k$ is a root of both the equations $a x^{3} + b x^{2} + c x + d = 0$ and $b x^{3} + c x^{2} + d x + a = 0$ . Find all possible values of $k$ (including complex values).

Solution

Solution:
$1, -1, i, -i$

Let $k$ be a root of both polynomials. Multiplying the first polynomial by $k$ and subtracting the second, we have $a k^{4} - a = 0$ , which means that $k$ is either $1, -1, i$ , or $-i$ . If $a = b = c = d = 1$ , then $-1, i$ , and $-i$ are roots of both polynomials. If $a = b = c = 1$ and $d = -3$ , then $1$ is a root of both polynomials. So $k$ can be $1, -1, i$ , and $-i$ .

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