1983 AMC 12 第 6 题

Question

将

x^5

、

x+\frac1x

和

1+\frac2x+\frac3{x^3}

相乘，所得多项式的次数为

When

x^5,

x+\frac1x

and

1+\frac2x+\frac3{x^3}

are multiplied, the product is a polynomial of degree

2

3

6

7

8

从每个因式中各取一项，追踪所得项的最大指数

Track the largest exponent produced by choosing one term from each factor

最高次项来自

x^5\cdot x\cdot1

The leading term comes from

x^5\cdot x\cdot1

最大指数来自

x^5\cdot x\cdot1=x^6

。其余各项的指数都更小，而最小的可能指数为

5-1-3=1

，所以乘积确实是一个多项式。它的次数为

6

。

所以正确答案是 C。

The greatest exponent comes from

x^5\cdot x\cdot1=x^6.

Every other term has smaller exponent, and the smallest possible exponent is

5-1-3=1,

so the product is indeed a polynomial. Its degree is

6.

Therefore, the correct answer is C.

其他年份的第 6 题