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Sixth Degree Polynomial With Four Distinct Roots

Source: 2019 AMC 12A #14

February 8, 2019
AMCAMC 12polynomialcomplex numbers2019 AMC 12Aalgebra2019 AMC

Problem Statement

For a certain complex number cc, the polynomial P(x)=(x2−2x+2)(x2−cx+4)(x2−4x+8) P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8) has exactly 4 distinct roots. What is ∣c∣|c|?
<spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>6<spanclass=′latex−bold′>(C)</span>22<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>10<span class='latex-bold'>(A) </span> 2 \qquad <span class='latex-bold'>(B) </span> \sqrt{6} \qquad <span class='latex-bold'>(C) </span> 2\sqrt{2} \qquad <span class='latex-bold'>(D) </span> 3 \qquad <span class='latex-bold'>(E) </span> \sqrt{10}
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