2022 AMC 10B 第 21 题

Question

设

P(x)

是一个有理系数多项式。

P(x)

除以

x^2 + x + 1

的余式为

x+2

，

P(x)

除以

x^2+1

的余式为

2x+1

。满足这两个条件且次数最小的多项式唯一。求这个多项式各系数平方和。

Let

P(x)

be a polynomial with rational coefficients such that when

P(x)

is divided by the polynomial

x^2 + x + 1,

the remainder is

x+2,

and when

P(x)

is divided by the polynomial

x^2+1,

the remainder is

2x+1.

There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?

\ 10

\ 13

\ 19

\ 20

\ 23

答案：E

知识点：多项式方程组

难度评级：2150

小提示：

先检验

Q

能否为常数，再尝试一次多项式

First check whether

Q

can be constant, and then try a linear polynomial

大提示：

写出

P=(x^2+x+1)Q+x+2

，再模

x^2+1

化简

Write

P=(x^2+x+1)Q+x+2,

then reduce this expression modulo

x^2+1

解答：

第一个余式条件给出

\begin{aligned}P(x)&=(x^2+x+1)Q(x)\\&\quad+x+2\end{aligned}\text{。}

其中

Q

是某个多项式。模

x^2+1

时有

x^2\equiv-1

，所以

P(x)\equiv xQ(x)+x+2\text{。}

若

Q(x)=c

是常数，则这个余式为

(c+1)x+2

，不可能等于

2x+1

。所以

Q

的次数至少为

1

。

现在令

Q(x)=ax+b

。模

x^2+1

化简可得

P(x)\equiv(b+1)x+(2-a)\text{。}

与

2x+1

比较系数，得到

a=b=1

。这构造出了一个

3

次多项式，而常数情形不成立又说明最低次数为

3

。

因此

\begin{aligned}P(x)&=(x+1)(x^2+x+1)\\&\quad+x+2\\&=x^3+2x^2+3x+3\end{aligned}\text{。}

它的系数平方和为

1^2+2^2+3^2+3^2=23

。

所以正确答案是 E。

The first remainder condition gives

\begin{aligned}P(x)&=(x^2+x+1)Q(x)\\&\quad+x+2.\end{aligned}

for some polynomial

Q.

Modulo

x^2+1,

we have

x^2\equiv-1,

so

P(x)\equiv xQ(x)+x+2.

If

Q(x)=c

is constant, this remainder is

(c+1)x+2,

which cannot equal

2x+1.

Thus

Q

must have degree at least

1.

Now let

Q(x)=ax+b.

Reducing modulo

x^2+1

gives

P(x)\equiv(b+1)x+(2-a).

Matching this with

2x+1

yields

a=b=1.

This constructs a degree-

3

polynomial, and the failed constant case proves that degree

3

is minimal.

Therefore,

\begin{aligned}P(x)&=(x+1)(x^2+x+1)\\&\quad+x+2\\&=x^3+2x^2+3x+3.\end{aligned}

The sum of the squares of its coefficients is

1^2+2^2+3^2+3^2=23.

Thus, the answer is E .

2022 AMC 10B 第 21 题

答案：E

难度评级：2150

小提示：

大提示：

解答：

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