3.4 余式定理 Remainder Theorem - 练习题 - 独中课程 | 线上补习

高中一 | 高级数学

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1. Calculate the remainder when $5(x +3)^3 + 2(x +4)^2$ is divided by $x + 2$ .
试计算 $5(x +3)^3 + 2(x +4)^2$ 除以 $x + 2$ 所得余数。

2. Given that $x^3 - 2x^2 - 3x - 11$ and $x^3 - x^2 - 9$ have the same remainder when divided by $x + a$ , find the possible values of a.
已知 $x^3 - 2x^2 - 3x - 11$ 和 $x^3 - x^2 - 9$ 除以 $x + a$ 所得余数相等，求a 的可能值。

3. The expression $x^3 + 4ax^2 - 8x + b$ leaves a remainder of $19$ and $-8$ when divided by $x + 2$ and $x - 1$ respectively. Calculate the value of a and b.
当式子 $x^3 + 4ax^2 - 8x + b$ 除以 $x + 2$ 和 $x - 1$ 所得余数分别为 $19$ 和 $-8$ . 试计算a 和 b 的值。
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]f (–2) = 19, f (1) = –8, 解联立[/bg_collapse]

4. The expression $ax^2 + bx - 1$ leaves a remainder of k when divided by $x + 2$ and a remainder of $3k + 5$ when divided by $x - 3$ . Show that $a = 3b - 1$ .
式子 $ax^2 + bx - 1$ 当除以 $x + 2$ 时所得余数为 k 和除以 $x - 3$ 时所得余数为 $3k + 5$ . 试证 $a = 3b - 1$ .
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]f (–2) = k, f (3) = 3k + 5，用代入消元法消掉k。[/bg_collapse]

5. A polynomial $f (x)$ has a remainder 15 when divided by $(x - 1)$ , and 9 when divided by $(x + 2)$ . Find the remainder when $f (x)$ is divided by $(x^2 + x - 2)$ .
一多项式 $f (x)$ 当除以 $(x - 1)$ 时所得余数为15, 及当除以 $(x + 2)$ 时所得余数为 9. 求当 $f (x)$ 除以 $(x^2 + x - 2)$ 时所得余式.
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设 $f (x) = (x^2 + x - 2)Q(x) + ax + b$ , 则 $f (1) = 15$ , $f (-2) = 9$ 后解联立得a, b。余式为 $ax + b$ [/bg_collapse]

6. If deg $f (x) = 4$ , and f (x) is divided by $(x - 1)^3$ , $x - 2$ , $x + 2$ , the remainders are $3$ , $6$ and $138$ respectively. Find $f (x)$ .
若 $f (x)$ 为四次多项式，且当 $f (x)$ 除以 $(x - 1)^3$ , $x - 2$ , $x + 2$ , 所得余数分别为 $3$ , $6$ 和 $138$ . 求 $f (x)$ .
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设 $f (x) = (x - 1)^3(ax + b) + 3$ , 则 $f (2) = 6$ , $f (-2) = 138$ 后解联立得 $a$ , $b$ . 把 $a$ , $b$ 代入 $f (x)$ [/bg_collapse]

7. $f (x) = x^3 + kx^2 + lx + m$ where $k, l, m$ are constants. If satisfy the following conditions:
$f (x) = x^3 + kx^2 + lx + m$ 其中 $k, l, m$ 为常数。若其满足以下各条件：

(i) when $f (x)$ divided by $x^2 + 3x + 2$ , obtain the remainder $5x - 3$ ,
当 $f (x)$ 除以 $x^2 + 3x + 2$ , 所得余式为 $5x - 3$ ,

(ii) when $f (x)$ divided by $x - 1$ , we obtain the remainder $-4$ .
当 $f (x)$ 除以 $x - 1$ , 所得余数为 $-4$ .
Find the values of $k$ , $l$ and $m$ .
求 $k$ , $l$ 和 $m$ 的值.
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]从 (i) 可得 $f (x) = (x^2 + 3x + 2)(x + a) + 5x - 3$ . 从 (ii) 可得 $f (1) = -4$ . 代入 $f (x)$ 可得 $a$ ，将 $f (x)$ 展开可得 $k, l$ 和 $m$ 的值[/bg_collapse]

8. When the polynomial $f (x)$ is divided by $x^2 + 3x-2$ , $2x+1$ , the remainder is $x + 3$ and $-1$ respectively. Find the remainder when $f (x)$ is divided by $(x^2 + 3x - 2)(2x+ 1)$ .
当多项式 $f (x)$ 除以 $x^2 + 3x-2$ , $2x+1$ , 所得余式分别为 $x + 3$ 和 $-1$ . 求当 $f (x)$ 除以 $(x^2 + 3x - 2)(2x+ 1)$ 所得余式.
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设 $f (x) = (x^2 + 3x - 2)(2x+ 1)Q(x) + k(x^2 + 3x - 2) + x + 3$ , $f (-frac{1}{2}) = -1$ 可得 $k$ 。余式为 $k(x^2 + 3x - 2) + x + 3$ [/bg_collapse]

Answer 答案：

(1) 13

(2) 1, 2

(3) 1, –5

(5) 2x + 13

(6) $(x - 1)^3(2x - 1) + 3$

(7) 1, 1, –7

(8) $\frac{1}{13}\left(14x^2+55x-11\right)$