唔識做功課55

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

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