2019 AMC 12A Problem 14

Below is the professionally curated solution for Problem 14 of the 2019 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 12A solutions, or check the answer key.

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Concepts:complex number polynomial Vieta’s Formulas

Difficulty rating: 1690

14.

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 $|c|?$

$2$

$\sqrt{6}$

$2\sqrt{2}$

$3$

$\sqrt{10}$

Solution:

The factors $x^2 - 2x + 2$ and $x^2 - 4x + 8$ have roots $1 \pm i$ and $2 \pm 2i,$ which are $4$ distinct values.

For $P$ to have exactly $4$ distinct roots, the roots of $x^2 - cx + 4$ must lie among these. Their product must equal $4,$ and the only such pair is one root from each factor, for example $(1 + i)(2 - 2i) = 4.$

Then $c = (1 + i) + (2 - 2i) = 3 - i,$ so $|c| = \sqrt{3^2 + 1^2} = \sqrt{10}.$

Thus, the correct answer is E.

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