# 2002 AMC 12A Problems/Problem 25

## 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 $-4\leq x \leq 4$?

## Solution

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 between the two graphs at $x = 1$ is $(B)$. (The polynomials in the graph are $P(x) = 2x^4-3x^2-3x-4$ and $Q(x) = -2x^4-2x^2-2x-2$.)

## Solution 2

We know every coefficient is equal, so we get $ax^n + ... + ax + a = 0$ which equals $x^n + ... + x + 1 = 0$. We see apparently that x cannot be positive, for it would yield a number greater than zero for $Q(x)$. We look at the zeros of the answer choices. A, C, D, and E have a positive zero, which eliminates them. B is the answer.

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