Problem:
The zeros of the function are integers. What is the sum of the possible values of ?
Answer Choices:
A.
B.
C.
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E.
Solution:
The zeros of are integers and their sum is , so is an integer. If is an integer zero, then or
So must be an integer, and the possible values of are 6,4 , , and -2 . The possible values of are , and -1 , all of which yield integer zeros of , and their sum is .
OR
As above, must be an integer. The function has zeros at
These values are integers only if for some integer . Solving for in terms of gives , so must be a perfect square. The only integer solutions for are 0 and \textbackslash pm 3 , from which it follows that the values of are , and -1 , all of which yield integer values of . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions