中四MATH - Ans.fyi 參考答案

中四MATH

2012-02-03 12:11 am

1. if a polynomial f(x) is divisible by x - 1 , then f(x - 1) is divisible by ?

2. 1 - (1/x^2 - 1)/(1 - 1/x) = ?

3. if A and B are the roots of the equation x^2 - x - 3 = 0,find the quadratic equation whose roots are 1/A and 1/B.

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回答 (1)

用戶2053

2012-02-03 12:41 am

✔ 最佳答案

1)f(x) is divisible by x - 1 , by remainder theorem ,
f(1) = 0For
f(x -1) = 0
f(x -1) = f(1)
x - 1 = 1
x = 2So f(x - 1) is divisible by x - 2 = 0.
2)1 - (1/x² - 1)/(1 - 1/x)= 1 - (1/x - 1)(1/x + 1) / (1 - 1/x)= 1 + (1/x + 1)= 2 + 1/x
or (2x + 1)/x
3)Since 1 / (1/A) = A , 1 / (1/B) = B , The required equation is(1/x)² - 1/x - 3 = 0
1 - x - 3x² = 0
3x² + x - 1 = 0

2012-02-02 16:42:21 補充：
1) So f(x - 1) is divisible by x - 2 , (not x - 2 = 0)

2012-02-02 16:45:45 補充：
3)

Method 2 :

A+B = 1 , AB = - 3

So

1/A + 1/B = (A+B)/(AB) = 1/3

and 1/A * 1/B = 1/(AB) = - 1/3

The required equation is

x² - (1/A + 1/B)x + 1/A * 1/B = 0
x² - (1/3)x + 1/3 = 0
3x² + x - 1 = 0

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