2000 AMC 12 第 22 题

Question

下图显示了由四次多项式

P(x) = x^4 + ax^3 + bx^2 + cx + d

定义的曲线的一部分。下列哪一项最小？

The graph below shows a portion of the curve defined by the quartic polynomial

P(x) = x^4 + ax^3 + bx^2 + cx + d.

Which of the following is the smallest?

P(-1)

P

的零点的乘积

The product of the zeros of

P

P

的非实零点的乘积

The product of the non-real zeros of

P

P

的系数之和

The sum of the coefficients of

P

P

的实零点之和

The sum of the real zeros of

P

答案：C

知识点：多项式韦达定理复数

难度评级：2030

小提示：

图像显示恰有两个实零点，且都为正数，所以

P

有两个非实（共轭）零点。

The graph shows exactly two real zeros, both positive, so

P

has two non-real (conjugate) zeros

大提示：

所有零点的乘积等于

d

，即

y

-截距；再除以实零点的乘积。

The product of all zeros equals

d,

the

y

-intercept; divide it by the product of the real zeros

解答：

图像与

x

-轴恰好相交两次，且两次都在正数处，所以

P

有两个实零点和两个非实（复共轭）零点。

从图像读出：系数之和为

P(1) \gt 3

；

P(-1) \gt 4

；实零点之和大于

4.5

；所有零点的乘积为

d

，即

y

-截距，小于

6

。

设实零点的乘积为

R

，则

R \gt 4.5

。非实零点的乘积为

\dfrac dR

，它小于

\dfrac{6}{4.5} \lt 2

。

这比列出的其他每一个量都小。

因此，正确答案是 C。

The graph crosses the

x

-axis exactly twice, both times at positive values, so

P

has two real zeros and two non-real (complex conjugate) zeros.

Reading off the graph: the sum of the coefficients is

P(1) \gt 3;

P(-1) \gt 4;

the sum of the real zeros is greater than

4.5;

and the product of all zeros is

d,

the

y

-intercept, which is less than

6.

Let

R

be the product of the real zeros, so

R \gt 4.5.

The product of the non-real zeros is

\dfrac dR,

which is less than

\dfrac{6}{4.5} \lt 2.

This is smaller than every other listed quantity.

Thus, the correct answer is C.

2000 AMC 12 第 22 题

答案：C

难度评级：2030

小提示：

大提示：

解答：

其他年份的第 22 题