2010 AIME II Problems/Problem 6
Find the smallest positive integer with the property that the polynomial can be written as a product of two nonconstant polynomials with integer coefficients.
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 .
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