Problem:
All the roots of polynomial are positive integers, possibly repeated. What is the value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because this polynomial has degree , there are roots, counting multiplicities. By Vieta's formulas, the sum of the roots is and their product is . The only way this can happen is for the roots, listed with repetitions, to be . Thus the polynomial is . By the Binomial Theorem, this polynomial equals
When this product is expanded, the coefficient of is .
Proceed as in the first solution. Then, using Vieta's formulas, observe that is the sum of the products of the roots taken at a time, namely
This gives a total of . Therefore .
The problems on this page are the property of the MAA's American Mathematics Competitions