2010 AIME II Problems/Problem 6
Problem
Find the smallest positive integer with the property that the polynomial
can be written as a product of two nonconstant polynomials with integer coefficients.
Solution
You can always factor a polynomial into quadratic factors.
Let and
be the two quadratics, so that
Therefore, again setting coefficients equal, ,
,
, and so
.
Since , the only possible values for
are
and
. From this we find that the possible values for
are
and
. Therefore, the answer is
.
See also
2010 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.