2023 WSMO Team Round Problems/Problem 6

Problem

A quartic real polynomial $f(x)$ satisfying $f(3+2i) = 0$ has 3 distinct roots. If the sum of the three roots is $12,$ find their product.

Solution

From the conjugate root theorem, since $3+2i$ is a root, $3-2i$ must also be a root. Since there are only three distinct roots, let $x$ be the value of the remaining two roots. We have $x+(3+2i)+(3-2i) = 12\implies x+6 = 12\implies x = 6.$ So, the product of the four roots is $(3+2i)(3-2i)(6)(6) = (3^2+2^2)(6)(6) = \boxed{468}.$

~pinkpig

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2023 WSMO Team Round Problems/Problem 6

Problem

Solution