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# 2019 AMC 12A Problems/Problem 14

## Problem

For a certain complex number $c$, the polynomial $$P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)$$has exactly 4 distinct roots. What is $|c|$?

$\textbf{(A) } 2 \qquad \textbf{(B) } \sqrt{6} \qquad \textbf{(C) } 2\sqrt{2} \qquad \textbf{(D) } 3 \qquad \textbf{(E) } \sqrt{10}$

## Solution

The polynomial can be factored further broken down into

$P(x) = (x - [1 - i])(x - [1 + i])(x - [2 - 2i])(x - [2 + 2i])(x^2 - cx + 4)$

by using the quadratic formula on the quadratic factors. Since the first four roots are all distinct, the term $(x^2 - cx + 4)$ must be a product of any combination of 2 not necessarily distinct factors from the set: $(x - [1 - i]), (x - [1 + i]), (x - [2 - 2i]),$ and $(x - [2 + 2i])$. We need the two factors to yield a constant term of 4 when multiplied together. One combination that works is $(x - [1 - i])$ and $(x - [2 + 2i])$. When multiplied together, the polynomial is $(x^2 + [-3 + i] + 4)$. Therefore, $c = -3 + i$ and $|c| = \boxed{\textbf{(E)}\sqrt{10}}$.