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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

評分
  • 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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