# 2011 AIME I Problems/Problem 15

## Problem

For some integer , the polynomial has the three integer roots , , and . Find .

## Solution

With Vieta's formula, we know that , and .

since any one being zero will make the the other 2 .

. WLOG, let .

Then if , then and if , .

We know that , have the same sign. So . ( and )

Also, maximize when if we fixed . Hence, .

So .

so .

Now we have limited a to .

Let's us analyze .

Here is a table:

We can tell we don't need to bother with ,

, So won't work. ,

, , which is too small to get

, is not divisible by or or , we can clearly tell that is too much

Hence, , . , .

Answer: