2002 AMC 12A Problem 25

Below is the professionally curated solution for Problem 25 of the 2002 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12A solutions, or check the answer key.

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Concepts:polynomial data and graph interpretation

Difficulty rating: 2270

25.

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 $-4 \le x \le 4?$

Solution:

Replacing the nonzero coefficients by their mean keeps the total of the coefficients unchanged, so $P$ and $Q$ have the same coefficient sum. Since $P(1)$ and $Q(1)$ each equal that sum, $P(1) = Q(1).$

Therefore the graphs of $y = P(x)$ and $y = Q(x)$ must cross at $x = 1.$ The only choice showing an intersection at $x = 1$ is graph B. (There, $P(x) = 2x^4 - 3x^2 - 3x - 4$ and $Q(x) = -2x^4 - 2x^2 - 2x - 2.$ )

Thus, the correct answer is B.

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