Difference between revisions of "2010 AIME II Problems/Problem 6"
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− | There are two ways for a monic fourth degree polynomial to be factored into two non-constant polynomials with real coefficients: into a | + | There are two ways for a monic fourth degree polynomial to be factored into two non-constant polynomials with real coefficients: into a cubic and a linear equation, or 2 [[quadratic]]s. |
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Revision as of 01:27, 11 January 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
There are two ways for a monic fourth degree polynomial to be factored into two non-constant polynomials with real coefficients: into a cubic and a linear equation, or 2 quadratics.
- Case 1: The factors are cubic and linear.
Let be the linear root, where is a root of the given quartic, and let be the cubic.
By the Rational Root Theorem, then , or . Observe that
Setting coefficients equal, we have , , and , and .
It follows that , , or , which reach minimum when , where .
- Case 2: The factors are quadratics.
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 |
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