AMC12 2002 A

AMC12 2002 A · Q25

AMC12 2002 A · Q25. It mainly tests Polynomials, Functions basics.

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 $-4\le x\le 4$?

将一个实系数多项式 $P$ 的所有非零系数都替换为它们的平均值，从而得到多项式 $Q$。下列哪一个可能是区间 $-4\le x\le 4$ 上 $y=P(x)$ 与 $y=Q(x)$ 的图像？

(A)

(B)

(C)

(D)

(E)

Answer

Correct choice: (B)

正确答案：（B）

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 $Q(x) = -2x^4-2x^2-2x-2$.)

$P$ 与 $Q$ 的系数之和相等，因此 $P(1) = Q(1)$。唯一在 $x = 1$ 处两图像相交的选项是 (B)。（图中的多项式为 $P(x) = 2x^4-3x^2-3x-4$ 和 $Q(x) = -2x^4-2x^2-2x-2$。）

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