HMMT February

Algebra Difficulty 5.0 AIME Find the answer

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).

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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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