Difference between revisions of "2010 AIME II Problems/Problem 6"

Revision as of 10:53, 21 November 2016

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 always factor a polynomial into 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 $\pm 4 \cdot 2$ . Therefore, the answer is $\boxed{008}$ .

@@ Line 3: / Line 3: @@
 ==Solution==
-<br/>
 You can always factor a polynomial into quadratic factors.

2010 AIME II (Problems • Answer Key • Resources)
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Difference between revisions of "2010 AIME II Problems/Problem 6"

Revision as of 10:53, 21 November 2016

Problem

Solution

See also