2021 AMC 12A Problem 12

Problem 12 of 25IntermediateAlgebra

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?

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Solution

By Vieta’s formulas the six roots sum to

10

(the negative of the

z^5

coefficient) and multiply to

16.

Thus every root is a power

2^{e_i},

with

\sum e_i=4.

Since

2^{e_i}\ge 1+e_i,

the roots have sum at least

6+4=10.

Equality holds only when every

e_i

0

1,

so the roots must be

2,2,2,2,1,1.

So the polynomial is

(z - 1)^2 (z - 2)^4.

Expanding,

\begin{aligned} &(z^2 - 2z + 1) \\ &\quad {}\cdot (z^4 - 8z^3 + 24z^2 - 32z + 16) \\ &= z^6 - 10z^5 + 41z^4 \\ &\quad {}- 88z^3 + 104z^2 \\ &\quad {}- 64z + 16. \end{aligned}

The coefficient of

z^3

B = -88.

Thus, the correct answer is A.