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.
You can factor the polynomial into two quadratic factors or a linear and a cubic factor.
For two 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 .
For the case of one linear and one cubic factor, doing a similar expansion and matching of the coefficients gives the smallest in that case to be .
Therefore, the answer is .
Let . From this, we get that and . Plugging this back into the equation, we get . Expanding gives us . Therefore . Simplifying gets us . Since and must be integers, we can use guess and check for values of because must be a factor of . Note that cannot be negative because would be imaginary. After guessing and checking, we find that the possible values of are and . We have that . Plugging in our values for and , we get that the smallest positive integer value can be is . -Heavytoothpaste
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