Vincent Wong 發問於 科學及數學數學 · 1 十年 前

F.4 Maths(Polynomial)

When a polynomial g(x) is divided by x-1, the remainder is 5. When g(x) is divided by 2x+1, the remainder is 2.Find the remainder when g(x) is divided by 2x2-x-1.

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  • 1 十年 前
    最佳解答

    When a polynomial g(x) is divided by x-1, the remainder is 5. When g(x) is divided by 2x+1, the remainder is 2.Find the remainder when g(x) is divided by 2x2-x-1.

    By remainder throrem, when g(x) is divided by x-1, the remainder

    = g(1)

    So g(1) = 5 ............. (1)

    By remainder throrem, when g(x) is divided by 2x+1, the remainder

    = g(-1/2)

    So g(-1/2) = 2 .......... (2)

    As the power of 2x²-x-1 is 2, so the remainder of g(x) divided by 2x²-x-1 is of max. power of 1.

    Let g(x) = (2x²-x-1)Q + R

    Where Q(x) is the quotient and Rx + C is the remainder

    g(x) = (2x²-x-1)Q(x) + Rx + C

    g(x) = (x-1)(2x+1)Q(x) + Rx + C ........ (*)

    Substitute x=1 into (*)

    g(1) = (1-1)(2*1+1)Q + R(1) + C

    g(1) = R + C

    5 = R + C ............. by (1)

    C = 5 - R ............. (3)

    Substitute x=-1/2 into (*)

    g(-1/2) = (-1/2-1)[2*(-1/2)+1]Q + R(-1/2) + C

    g(-1/2) = -R/2 + C

    2 = -R/2 + C ............. by (2)

    2 = -R/2 + (5-R)

    4 = -R + 2(5-R)

    4 = -R + 10 - 2R

    3R = 6

    R = 2

    Substitute R=2 into (3)

    C = 5-2

    C = 3

    So the remainder when g(x) is divided by 2x²-x-1

    = Rx + C

    = 2x + 3

    2006-11-29 18:16:12 補充:

    小小補充:當 g(x) divided by f(x),remainder 會是 f(x) 的 (maximum power of x)-1。所以當 g(x) divided by 2x²-x-1,remainder 可以假設是 Rx + c。加上題目先給予的兩個提示 g(1) 和 g(-1/2) 就可以用兩條方程解兩個 variable。

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