2002 AMC 12A 第 25 题

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

小提示：

P

的系数和等于

Q

的系数和。

The sum of the coefficients of

P

equals the sum of the coefficients of

Q

大提示：

代入

x = 1

得

P(1) = Q(1)

，所以两条图像必须在

x = 1

处相交。

Evaluating at

x = 1

gives

P(1) = Q(1),

so the two graphs must intersect at

x = 1

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

小提示：

大提示：

解答：

其他年份的第 25 题