Difference between revisions of "2010 AIME II Problems/Problem 6"
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Let <math>x^2+ax+b</math> and <math>x^2+cx+d</math> be the two quadratics, so that | Let <math>x^2+ax+b</math> and <math>x^2+cx+d</math> be the two quadratics, so that |
Revision as of 10:53, 21 November 2016
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.