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Lau 發問於 科學及數學數學 · 5 年前

Reminder Theorum?

Please answer the following question

https://www.flickr.com/photos/124898006@N08/276363...

Thank you very much

1 個解答

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  • 匿名
    5 年前
    最愛解答

    【Question】

    (a)

    Let f(x) = ax⁴ + bx³ + cx² + dx + e.

    (i) If a + c + e = b + d, prove that f(x) is divisible by x + 1.

    (ii) Hence, show that 69487 is divisible by 11.

    (b)

    Let g(x) = 4x⁴ + 9x³ + x² + 9x + 2.

    (i) Find the remainder when g(x) is divided by x + 1.

    (ii) Hence, show that 49192 is divisible by 11.

    【Solution】

    (a) (i)

    When f(x) is divided by x + 1, the remainder is

    f(-1)

    = a - b + c - d + e

    = (a + c + e) - (b + d)

    = 0

    Therefore, f(x) is divisible by x + 1.

    (a) (ii)

    Consider a = 6, b = 9, c = 4, d = 8, e = 7.

    Check that a + c + e = 6 + 4 + 7 = 17 = 9 + 8 = b + d.

    By (a) (i), f(x) is divisible by x + 1.

    For x = 10, that is, f(10) is divisible by 11.

    That is, 6(10000) + 9(1000) + 4(100) + 8(10) + 7(1) = 69487 is divisible by 11.

    (b) (i)

    When g(x) is divided by x + 1, the remainder is

    g(-1)

    = 4 - 9 + 1 - 9 + 2

    = 7 - 18

    = -11

    (b) (ii)

    From (b) (i), it is known that

    g(x) = (x + 1)Q(x) - 11 where Q(x) is a polynomial with integer coefficients.

    Put x = 10,

    g(10) = 11Q(10) - 11 = 11[Q(10) - 1].

    Since g(10) and Q(10) - 1 are integers, g(10) is divisible by 11.

    That is, 4(10000) + 9(1000) + 100 + 9(10) + 2 = 49192 is divisible by 11.

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