2021 AMC 12A Fall 第 15 题

Question

回忆：复数

w = a + bi

（其中

a

和

b

为实数，

i = \sqrt{-1}

）的共轭是复数

\overline{w} = a - bi

。对任意复数

z

，令

f(z) = 4i\overline{z}

。多项式

P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1

有四个复根：

z_1

、

z_2

、

z_3

和

z_4

。设

\begin{aligned} &Q(z) = z^4 + Az^3 + Bz^2 \\ &\quad {}+ Cz + D \end{aligned}

是以

f(z_1)

、

f(z_2)

、

f(z_3)

和

f(z_4)

为根的多项式，其中系数

A

、

B

、

C

和

D

都是复数。

B + D

是多少？

Recall that the conjugate of the complex number

w = a + bi,

where

a

and

b

are real numbers and

i = \sqrt{-1},

is the complex number

\overline{w} = a - bi.

For any complex number

z,

let

f(z) = 4i\overline{z}.

The polynomial

P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1

has four complex roots:

z_1,

z_2,

z_3,

and

z_4.

Let

\begin{aligned} &Q(z) = z^4 + Az^3 + Bz^2 \\ &\quad {}+ Cz + D \end{aligned}

be the polynomial whose roots are

f(z_1),

f(z_2),

f(z_3),

and

f(z_4),

where the coefficients

A,

B,

C,

and

D

are complex numbers. What is

B + D?

-304

-208

12i

208

304

答案：D

知识点：韦达定理复数

难度评级：1800

小提示：

\overline{z_j}

的基本对称式是

z_j

的对应对称式的共轭；

P

的系数全为实数，所以这些对称和也是实数

The elementary symmetric functions of

\overline{z_j}

are the conjugates of those of

z_j;

all of

P

’s coefficients are real, so those symmetric sums are real

大提示：

B = (4i)^2 \sum_{i\lt j}\overline{z_i}\,\overline{z_j}

，且

D = (4i)^4 \prod \overline{z_j}

B = (4i)^2 \sum_{i\lt j}\overline{z_i}\,\overline{z_j}

and

D = (4i)^4 \prod \overline{z_j}

解答：

对

P

，使用韦达定理，

\sum_{i\lt j} z_iz_j = 3

，且

\prod z_j = 1

，二者都是实数，所以它们的共轭仍为

3

和

1

。

Q

的根为

4i\overline{z_j}

。于是

B

是两两乘积之和：

B = (4i)^2 \sum_{i\lt j}\overline{z_i}\,\overline{z_j}

= -16 \cdot 3 = -48

。且

D = (4i)^4 \prod \overline{z_j} = 256 \cdot 1 = 256

。

所以

B + D = -48 + 256 = 208

。

所以正确答案是 D。

By Vieta on

P,

\sum_{i\lt j} z_iz_j = 3

and

\prod z_j = 1,

both real, so their conjugates are also

3

and

1.

The roots of

Q

are

4i\overline{z_j}.

Then

B

is the sum of products of pairs:

B = (4i)^2 \sum_{i\lt j}\overline{z_i}\,\overline{z_j}

= -16 \cdot 3 = -48.

And

D = (4i)^4 \prod \overline{z_j} = 256 \cdot 1 = 256.

So

B + D = -48 + 256 = 208.

Thus, the correct answer is D.

2021 AMC 12A Fall 第 15 题

答案：D

难度评级：1800

小提示：

大提示：

解答：

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