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