唔識做功課76

1.
f(x) = x³ - 8x + 8, solve the equation f(x) = 0.

Try f(2) = 2³ - 8(2) + 8 = 8 - 16 + 8 = 0.
Therefore, 2 is a root of f(x) = 0 or (x - 2) is a factor of f(x).

By long division or factorization, consider
f(x)
= x³ - 8x + 8
= x³ - 2x² + 2x² - 4x - 4x + 8
= x²(x - 2) + 2x(x - 2) - 4(x - 2)
= (x - 2)(x² + 2x - 4)

Then, for f(x) = 0
(x - 2)(x² + 2x - 4) = 0
x - 2 = 0 or x² + 2x - 4 = 0
x = 2 or x² + 2x + 1 = 5
x = 2 or (x + 1)² = 5
x = 2 or x = -1 ± √5

2.
(a)
Solve the equation x² + (1 - b)x - b = 0 where b is a constant.

x² + (1 - b)x - b = 0
(x - b)(x + 1) = 0
x = b or x = -1

(b)
Using the result of (a), solve the equation
(log y)² + (log 5)(log y) - log 2 = 0

Let x = log y and note that (log 5) + (log 2) = log 10 = 1, so put b = log 2, then, 1 - b = log 5.

The equation
(log y)² + (log 5)(log y) - log 2 = 0

becomes
x² + (1 - b)x - b = 0

By (a),
x = b or x = -1

That is,
log y = log 2 or log y = -1
y = 2 or y = 10⁻¹
y = 2 or y = 1/10

(c)
Using the result of (b), solve the simultaneous equation:
{a = 5b
{a^(log b) = 2

{a = 5b
{a^(log b) = 2

{log a = log(5b)
{log[a^(log b)] = log 2

{log a = log 5 + log b
{(log a)(log b) = log 2

(log 5 + log b)(log b) = log 2

(log b)² + (log 5)(log b) = log 2

(log b)² + (log 5)(log b) - log 2 = 0

By (b),
b = 2 or b = 1/10

Recall that a = 5b,
when b = 2, a = 5(2) = 10;
when b = 1/10, a = 5(1/10) = 1/2.

Therefore,
{a = 10
{b = 2
or
{a = 1/2
{b = 1/10

3.
Suppose log 2 = a, log 3 = b and log 7 = c.
Express the following numbers in terms of a, b and c. log(sin(π/3))

log(sin(π/3))
= log(√3 / 2)
= log(√3) - log (2)
= (1/2)log(3) - log (2)
= (1/2)b - a
= b/2 - a


2015-07-20 16:46:36 補充：
你正常應該用長除法 (long division)，但我在這裏很難可以打出長除法，所以我自己用代數式子自己慢慢寫出 x - 2 這個因式 (factor) 的因式分解 (factorization)。

如果你看書本或者回憶學校的教學，你肯定是學長除法的。

即是知道 x - 2 是 x³ - 8x + 8 的 factor 之後，你會做
　　　＿＿＿＿＿＿＿＿＿＿＿＿＿
ｘ－２）ｘ³ ＋ ０ｘ² － ８ｘ ＋ ８

　　　　　　．．．

2015-07-20 19:10:51 補充：
回意見 003：

其實呢題對於一個無讀 M2 的同學是很怪的。
但學學下無妨。
其實在數學上 π 即是 180°。
（原來角度的單位除了度 degree (°) 之外，還有其他，叫 radian 弧度）

因此，π/3 其實是 180°/3 = 60°

sin(π/3) = sin(60°) = √3 / 2

明白嗎？
ヽ(❀ฺ◕ฺ‿ฺ◕ฺ)ノ

2015-07-20 21:20:53 補充：
無錯，邊位都好 老師的方法使用了 恆等式 (identity) 也可以。

總之長除法也好，恆等式也好，回答區的「長除式因式分解」也好，只要可以正確地 factorize 就好了。

x³ - 8x + 8
＝x³ - 8 - 8x + 16
＝(x³ - 8) - 8(x - 2)
＝(x - 2)(x² + 2x + 4) - 8(x - 2)
＝(x - 2)(x² + 2x - 4)

由 x³ - 8x + 8轉成x³ - 2x² + 2x² - 4x - 4x + 8係咪要自己諗,或是有一啲特定嘅方法可以記?

2015-07-20 18:45:21 補充：
我本暑期作業因包含中五課程，所以我唔知点解log(sin(π/3))會= log(√3 / 2)，我睇返中四啲書冇試過log數中有sin和π嘅出現,你再教我吓吖...thx

2015-07-20 18:55:30 補充：
第一條我明白返晒喇!thx

2015-07-20 19:35:47 補充：
hehe...明白晒喇!