2021 AMC 10A Problem 14

Problem 14 of 25IntermediateAlgebra

All the roots of the polynomial

\begin{aligned} &z^6-10z^5+Az^4+Bz^3\\ &\quad+Cz^2+Dz+16 \end{aligned}

are positive integers, possibly repeated. What is the value of

B?

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Solution

By Vieta’s formulas, the six roots have sum

10

and product

16.

Because the product is a power of

2,

every positive integer root is a power of

2.

Distributing the four factors of

2

among six roots gives the least possible sum when four roots are

2

and two roots are

1

; that sum is already

10.

Hence the roots are

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

The coefficient

B

is the negative of the sum of all products of three roots. Choosing zero, one, or two of the two roots equal to

1

gives

\begin{aligned} B&=-\left(\binom43 2^3+2\binom42 2^2\right.\\ &\qquad\left.+\binom41 2\right)\\ &=-88. \end{aligned}

Thus, A is the correct answer.