1962 AMC 12 第 9 题

Question

将

x^9-x

尽可能完全地分解为具有整数系数的多项式和单项式之积，所得因式的个数为：

When

x^9-x

is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:

多于

5

个

more than

5

5

4

3

2

从

x(x^8-1)

开始，反复使用平方差公式

Begin with

x(x^8-1)

and repeatedly use differences of squares

在整数范围内，分解出

x

、

x-1

、

x+1

、

x^2+1

和

x^4+1

后停止

Over the integers, stop after isolating

x,

x-1,

x+1,

x^2+1,

and

x^4+1

在整数范围内分解得

\begin{aligned} x^9-x &=x(x-1)(x+1)\\ &\quad\cdot(x^2+1)(x^4+1) \end{aligned}\text{。}

最后两个非常数因式在整数范围内不可约，所以共有

5

个因式。

所以正确答案是 B。

Factoring over the integers gives

\begin{aligned} x^9-x &=x(x-1)(x+1)\\ &\quad\cdot(x^2+1)(x^4+1). \end{aligned}

The last two nonconstant factors are irreducible over the integers, so there are

5

factors.

Thus, the correct answer is B.

其他年份的第 9 题