1966 AMC 12 第 30 题

Question

若

x^4+ax^2+bx+c=0

的三个根为

1

、

2

和

3

，则

a+c

的值为：

If three of the roots of

x^4+ax^2+bx+c=0

are

1,

2,

and

3,

then the value of

a+c

is:

35

24

-12

-61

-63

缺少

x^3

项意味着四个根之和为零

The missing

x^3

coefficient makes the sum of all four roots zero

利用第四个根求两两乘积之和

a

与四根之积

c

Use the fourth root to compute the pairwise sum

a

and product

c

因为根之和为零，所以第四个根是

-6

。两两乘积之和为

\begin{aligned} a&=2+3-6+6\\ &\quad-12-18=-25 \end{aligned}\text{，}

而四根之积为

c=(1)(2)(3)(-6)=-36

。因此

a+c=-61

。

因此，正确答案是 D。

The fourth root is

-6

because the root sum is zero. The sum of pairwise products is

\begin{aligned} a&=2+3-6+6\\ &\quad-12-18=-25, \end{aligned}

while the product is

c=(1)(2)(3)(-6)=-36.

Thus

a+c=-61.

Therefore, the correct answer is D.

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