1980 AMC 12 第 24 题

Question

对某个实数

r

，多项式

8x^3-4x^2-42x+45

可被

(x-r)^2

整除。下列哪个数最接近

r

？

For some real number

r,

the polynomial

8x^3-4x^2-42x+45

is divisible by

(x-r)^2.

Which of the following numbers is closest to

r?

1.22

1.32

1.42

1.52

1.62

重根会以一次因式的平方出现

A repeated root appears as a squared linear factor

把三次多项式完全因式分解，再把重根与所列小数比较

Factor the cubic completely and then compare the repeated root with the listed decimals

该多项式可分解为

\begin{aligned} &8x^3-4x^2-42x+45\\ &\qquad=(2x-3)^2(2x+5) \end{aligned}\text{。}

因此重根为

r=\frac{3}{2}=1.5

，所列数中最接近它的是

1.52

。

因此，正确答案是 D。

The polynomial factors as

\begin{aligned} &8x^3-4x^2-42x+45\\ &\qquad=(2x-3)^2(2x+5). \end{aligned}

Thus the repeated root is

r=\frac{3}{2}=1.5,

and the nearest listed number is

1.52.

Therefore, the correct answer is D.

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