# 唔識做功課55

1. Let f(x)=5x^3+kx^2+115x-25,where k is a constant. It is given that f(x)=(x+5)(ax^2+bx+c) where a,band c are constants.

a題我巳找到a=5,b=24 and c=-5

b. Someone claims that all the roots of the equation f(x)=0 are distinct. Do you agree? Explain your answer.(b題我唔知問我咩,所以唔知点做)

2. Let f(x)=x^2012-5k, where k is a constant. When f(x) is divided by x-1, the remainder is 11.我巳找到k=-2

b. Using the results of (a), find the remainder when 4^2012 is divided by 3.(哩條唔知点計)

### 2 個解答

• 5 年前
最愛解答

1.

(a) [yanyan 學多一次解 (a) 的步驟如下。]

f(x) = 5x³ + kx² + 115x - 25 ≡ (x + 5)(ax² + bx + c)

Comparing coefficient of x³ gives a = 5.

Comparing constance term gives 5c = -25 or c = -5.

Comparing coefficient of x gives c + 5b = 115 or b = 24.

Put x = -5, -625 + 25k - 575 - 25 = 0, k = 49.

(b)

Consider f(x) = 0

(x + 5)(ax² + bx + c) = 0

(x + 5)(5x² + 24x - 5) = 0

(x + 5)(5x - 1)(x + 5) = 0

x = 1/5 or x = -5 (repeated)

Therefore, not all the roots of f(x) = 0 are distinct.

(This is because the three roots are 1/5, -5, -5.)

The claim is not agreed.

2.

(a)

f(x) = x²⁰¹² - 5k

f(1) = 1 - 5k = 11

5k = -10

k = -2

(b)

The meaning of (a) is that for any integer x,

"When f(x) is divided by (x - 1), the remainder is 11."

that is,

"When x²⁰¹² + 10 is divided by (x - 1), the remainder is 11."

Therefore, put x = 4 in the above statement, that becomes

"When 4²⁰¹² + 10 is divided by (4 - 1), the remainder is 11."

That means, when 4²⁰¹² + 10 is divided by 3, the remainder is 11.

You can consider

4²⁰¹² + 10 = 3K + 11 where K is an integer.

Therefore,

4²⁰¹² = 3K + 1.

That means, when 4²⁰¹² is divided by 3, the remainder is 1.

2015-03-22 23:07:59 補充：

"Therefore, not all the roots of f(x) = 0 are distinct."

=

「因此，不是 f(x) = 0 的所有根都相異」

2015-03-22 23:09:31 補充：

當 x²⁰¹² + 10 除以 (x - 1), 餘數是 11.

那麼代 x = 4，即是

當 4²⁰¹² + 10 除以 3, 餘數是 11.

可以寫成 4²⁰¹² + 10 = 3K + 11 其中 K 是整數

即是 4²⁰¹² = 3K + 1

換句話，即是 當 4²⁰¹² 除以 3, 餘數是 1.

2015-03-22 23:09:55 補充：

7 ÷ 3 = 2 ... 1

即是

7 = 3 × 2 + 1

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• 5 年前

Therefore, not all the roots of f(x) = 0 are distinct.哩句中文係咩意思

2015-03-22 22:08:25 補充：

You can consider

4²⁰¹² + 10 = 3K + 11 where K is an integer.

Therefore,

4²⁰¹² = 3K + 1.

哩處我睇極都理解唔到，你可否用中文解比我聽吓

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