Question

将实系数多项式 $P$ 的所有非零系数都替换成这些系数的平均数，得到多项式 $Q$ 。下列哪一幅可能是在区间 $-4 \le x \le 4$ 上 $y = P(x)$ 和 $y = Q(x)$ 的图像？

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 $-4 \le x \le 4?$

答案：B

知识点：多项式数据与图表解读

难度评级：2270

解答：

将非零系数替换为它们的平均数，会保持系数总和不变，所以 $P$ 和 $Q$ 的系数和相同。由于 $P(1)$ 和 $Q(1)$ 都等于各自的系数和， $P(1) = Q(1)$ 。

因此 $y = P(x)$ 和 $y = Q(x)$ 的图像必须在 $x = 1$ 处相交。唯一显示在 $x = 1$ 处相交的是图 B。（其中 $P(x) = 2x^4 - 3x^2 - 3x - 4$ ， $Q(x) = -2x^4 - 2x^2 - 2x - 2$ 。）

因此，正确答案是 B。

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 第 25 题

答案：B

难度评级：2270

解答：

其他年份的第 25 题