Problem:
All the roots of polynomial z6−10z5+Az4+Bz3+Cz2+Dz+16 are positive integers. 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=−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
(43)(2⋅2⋅2)+(42)(21)(2⋅2⋅1)+(41)(22)(2⋅1⋅1).
This gives a total of 32+48+8=88. Therefore B=−88.
The problems on this page are the property of the MAA's American Mathematics Competitions