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Thread: Proving a factor of a polynomial

  1. #1
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    Question Proving a factor of a polynomial

    Hi,

    I hope someone can help provide elaboration on the homework question that I'm faced with, and why the solution is what it is.

    The homework question is as follows:
    Determine a general rules to help decide whether x - a and x + a are factors of x^n - a^n and x^n + a^n.

    The homework solution is as follows:
    - For x^n - a^n, if n is even, they're both factors.
    - If n is odd, only x - a is a factor. For x^n + a^n, if n is even, neither is a factor.
    - If n is odd, only x + a is a factor.

    Sincerely,
    Olivia
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  2. #2
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    Re: Proving a factor of a polynomial

    the factor theorem states that

    $(x-a) \text{ divides } p(x) \Rightarrow p(a)=0$

    consider $x^n-a^n=0$

    If $n$ is even, then both $x=\pm a$ satisfy this equation and thus both $(x-a)$ and $(x+a)$ are factors of $p(x)$

    If $n$ is odd, then only $x=a$ satisfies this equation and thus only $(x-a)$ is a factor of $p(x)$

    Apply this same reasoning to $x^n + a^n$ to obtain the other two results.

    I leave that to you.
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  3. #3
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    Re: Proving a factor of a polynomial

    Thank you That makes total sense now.
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