2021 AMC 12A Spring Problem 12

Below is the professionally curated solution for Problem 12 of the 2021 AMC 12A Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Spring solutions, or check the answer key.

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Concepts:Vieta’s Formulas polynomial

Difficulty rating: 1710

12.

All the roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, possibly repeated. What is the value of $B?$

$-88$

$-80$

$-64$

$-41$

$-40$

Solution:

By Vieta's formulas the six roots sum to $10$ (the negative of the $z^5$ coefficient) and multiply to $16.$ Six positive integers with sum $10$ and product $16$ must be $2, 2, 2, 2, 1, 1.$

So the polynomial is $(z - 1)^2 (z - 2)^4.$ Expanding, $(z^2 - 2z + 1)(z^4 - 8z^3 + 24z^2 - 32z + 16) = z^6 - 10z^5 + 41z^4 - 88z^3 + 104z^2 - 64z + 16.$ The coefficient of $z^3$ is $B = -88.$

Thus, the correct answer is A.

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