Problem:
All the roots of polynomial z6−10z5+Az4+Bz3+Cz2+Dz+16 are positive integers, possibly repeated. What is the value of B?
Answer Choices:
A. −88
B. −80
C. −64
D. −41
E. −40
Solution:
Because this polynomial has degree 6, there are 6 roots, counting multiplicities. By Vieta's formulas, the sum of the roots is 10 and their product is 16. The only way this can happen is for the roots, listed with repetitions, to be 1,1,2,2,2,2. Thus the polynomial is (z−1)2(z−2)4. By the Binomial Theorem, this polynomial equals
(z2−2z+1)⋅(z4−8z3+24z2−32z+16)
When this product is expanded, the coefficient of z3 is B=−32−48−8=(A)−88​.
OR
Proceed as in the first solution. Then, using Vieta's formulas, observe that −B is the sum of the products of the roots taken 3 at a time, namely
(34​)(2⋅2⋅2)+(24​)(12​)(2⋅2⋅1)+(14​)(22​)(2⋅1⋅1)
This gives a total of 32+48+8=88. Therefore B=(A)−88​.
The problems on this page are the property of the MAA's American Mathematics Competitions