Course Content
第四章:部分分式 Partial Fraction
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第六章:角的形成及单位 Angles and Measurements
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第十三章: 方程组 Simultaneous Equations
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第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming
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高中一 | 高级数学
About Lesson
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 - 11x^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 + 2x - 1所得余数分别为 19-8. 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).
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, 6138. 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,
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.
f (x) 除以 x - 1, 所得余数为 -4.
Find the values of k, l and m.
k, lm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, lm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 + 3-1. 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)

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