Difference between revisions of "2002 AMC 12A Problems/Problem 25"

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The nonzero coefficients of a polynomial <math>P</math> with real coefficients are all replaced by their mean to form a polynomial <math>Q</math>. Which of the following could be a graph of <math>y = P(x)</math> and <math>y = Q(x)</math> over the interval <math>-4\leq x \leq 4</math>?
 
The nonzero coefficients of a polynomial <math>P</math> with real coefficients are all replaced by their mean to form a polynomial <math>Q</math>. Which of the following could be a graph of <math>y = P(x)</math> and <math>y = Q(x)</math> over the interval <math>-4\leq x \leq 4</math>?
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[[File:2002AMC12A25.png]]
  
 
==Solution==
 
==Solution==

Revision as of 20:35, 26 January 2011

Problem

The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?

2002AMC12A25.png

Solution

(B) The sum of the coefficients of $P$ and of $Q$ will be equal, so $P(1) = Q(1)$. The only answer choice with an intersection at $x = 1$ is at (B). (The polynomials in the graph are $P(x) = 2x^4-3x^2-3x-4$ and $Q(x) = -2x^4-2x^2-2x-2$.)

See Also

2002 AMC 12A (ProblemsAnswer KeyResources)
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