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Problem of the Day #110: Because The Other Problem was Too Easy July 7, 2011

Posted by Seungln in : potd , trackback

Alex is given a 6th degree monic polynomial $V(x)$ with integer coefficients. Unfortunately, he does not know any of the coefficients (except for the first one, obviously, which is $1$), but he has some clues about the polynomial.

1. $V(x)$ crosses the x-axis four times, but touches it five times.

2. All of the roots of $V(x)$ are integers.

3. The roots of $V(x)$, when added up, sum up to $10$, and multiply up to $4620$.

4. Three numbers, $-74$, $-224$ and $1578$, are products of the coefficients and the exponent of $x$ in three of the term in $V(x)$ (but not necessarily in that order).

Alex was almost done figuring out the answer, but he got stuck in the middle of it. When SeungIn tells him that one of the terms of $V(x)$ is $-3764 x$, however, he realizes that he can now solve the problem.

Help Alex figure out $V(3)$.



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