Question

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 $-4 \le x \le 4?$

Answer: B

Concepts:polynomial data and graph interpretation

Difficulty rating: 2270

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.

2002 AMC 12A Problem 25

Answer: B

Difficulty rating: 2270

Solution:

Problem 25 in Other Years