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Difference between revisions of "2004 AIME I Problems/Problem 7"

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== Solution ==
 
== Solution ==
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Let our [[polynomial]] be <math>P(x)</math>.  It is clear that the [[coefficient]] of <math>x</math> in <math>P(x)</math> is <math>-1 + 2 - 3 + \ldots + 14 - 15 = -8</math>, so <math>P(x) = 1 -8x + Cx^2 + Q(x)</math> where <math>Q(x)</math> is some polynomial [[divisibility | divisible]] by <math>x^3</math>.  Then <math>P(-x) = 1 + 8x + Cx^2 + Q(-x)</math> and so <math>P(x)\cdot P(-x) = 1 + (2C - 64)x^2 + R(x)</math>, where <math>R(x)</math> is some polynomial divisible by <math>x^3</math>.  However, we also know <math>P(x)\cdot P(-x) = (1 - x)(1 + x)(1 +2x)(1 - 2x) \cdots (1 - 15x)(1 + 15x) = (1 - x^2)(1 - 4x^2)\cdots(1 - 225x^2) = 1 - (1 + 4 + \ldots + 225)x^2 + R(x)</math>.
+
Let our [[polynomial]] be <math>P(x)</math>.  It is clear that the [[coefficient]] of <math>x</math> in <math>P(x)</math> is <math>-1 + 2 - 3 + \ldots + 14 - 15 = -8</math>, so <math>P(x) = 1 -8x + Cx^2 + Q(x)</math> where <math>Q(x)</math> is some polynomial [[divisibility | divisible]] by <math>x^3</math>.  Then <math>P(-x) = 1 + 8x + Cx^2 + Q(-x)</math> and so <math>P(x)\cdot P(-x) = 1 + (2C - 64)x^2 + R(x)</math>, where <math>R(x)</math> is some polynomial divisible by <math>x^3</math>.  However, we also know <math>P(x)\cdot P(-x) = (1 - x)(1 + x)(1 +2x)(1 - 2x) \cdots (1 - 15x)(1 + 15x)</math> <math>= (1 - x^2)(1 - 4x^2)\cdots(1 - 225x^2)</math> <math>= 1 - (1 + 4 + \ldots + 225)x^2 + R(x)</math>.
 
Equating coefficients, we have <math>2C - 64 = -(1 + 4 + \ldots + 225) = -1240</math>, so <math>-2C = 1176</math> and <math>|C| = 588</math>.
 
Equating coefficients, we have <math>2C - 64 = -(1 + 4 + \ldots + 225) = -1240</math>, so <math>-2C = 1176</math> and <math>|C| = 588</math>.
  

Revision as of 20:20, 27 December 2007

Problem

Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$

Solution

Let our polynomial be $P(x)$. It is clear that the coefficient of $x$ in $P(x)$ is $-1 + 2 - 3 + \ldots + 14 - 15 = -8$, so $P(x) = 1 -8x + Cx^2 + Q(x)$ where $Q(x)$ is some polynomial divisible by $x^3$. Then $P(-x) = 1 + 8x + Cx^2 + Q(-x)$ and so $P(x)\cdot P(-x) = 1 + (2C - 64)x^2 + R(x)$, where $R(x)$ is some polynomial divisible by $x^3$. However, we also know $P(x)\cdot P(-x) = (1 - x)(1 + x)(1 +2x)(1 - 2x) \cdots (1 - 15x)(1 + 15x)$ $= (1 - x^2)(1 - 4x^2)\cdots(1 - 225x^2)$ $= 1 - (1 + 4 + \ldots + 225)x^2 + R(x)$. Equating coefficients, we have $2C - 64 = -(1 + 4 + \ldots + 225) = -1240$, so $-2C = 1176$ and $|C| = 588$.

See also

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