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Math Help - Problem #18: Polynomial

  1. #1
    Forum Admin topsquark's Avatar
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    Problem #18: Polynomial

    Suppose p(x) is a polynomial of degree 7 such that (x - 1)^4 is a factor of p(x) + 1 and (x + 1)^4 is a factor of p(x) - 1. Find p(x).

    (Source: Trinity University Mathematics POTW)

    -Dan
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    Forum Admin topsquark's Avatar
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    Re: Problem #18: Polynomial

    Hint:

    Spoiler:

    p(x) = (x - 1)^4 \cdot Q(x) where Q(x) is a polynomial of degree 3.

    What can you say about the derivative of p(x)?


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    Re: Problem #18: Polynomial

    To get you started, write the formulas for what the two statements mean.
    p(x) + 1 = ...
    P(x) - 1 = ...
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    Re: Problem #18: Polynomial

    Dan,
    Here's a solution assuming the polynomial is over the reals (or the complex field); by the way, good hint.
    Spoiler:
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    Re: Problem #18: Polynomial

    Good job!

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    Re: Problem #18: Polynomial

    Thanks and congratulations to Ishuda and particularly johng for their efforts. No need to post my solution this week as johng's post is virtually identical to mine.

    -Dan
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