2010 AIME II Problems/Problem 6

Problem

Find the smallest positive integer $n$ with the property that the polynomial $x^4 - nx + 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

Solution

You can factor the polynomial into two quadratic factors or a linear and a cubic factor.

For two quadratic factors, let $x^2+ax+b$ and $x^2+cx+d$ be the two quadratics, so that

$(x^2 + ax + b )(x^2 + cx + d) = x^4 + (a + c)x^3 + (b + d + ac)x^2 + (ad + bc)x + bd.$

Therefore, again setting coefficients equal, $a + c = 0\Longrightarrow a=-c$ , $b + d + ac = 0\Longrightarrow b+d=a^2$ , $ad + bc = - n$ , and so $bd = 63$ .

Since $b+d=a^2$ , the only possible values for $(b,d)$ are $(1,63)$ and $(7,9)$ . From this we find that the possible values for $n$ are $\pm 8 \cdot 62$ and $\pm 4 \cdot 2$ .

For the case of one linear and one cubic factor, doing a similar expansion and matching of the coefficients gives the smallest $n$ in that case to be $48$ .

Therefore, the answer is $\boxed{008}$ .

2010 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2010 AIME II Problems/Problem 6

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