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Math Help - polynomial : x^n - a^n divided by x + a , x^n + a^n divided by x+a

  1. #1
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    polynomial : x^n - a^n divided by x + a , x^n + a^n divided by x+a

    from my text book it writes: [see pic]
    i don't know how could it prove (2)and(3). why doesn't it show to proving?? i know factor thm but i just don't understand how are they derived from factor theorem. many thanks!
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    (2) : As n is a positive even integer, it can be written as n=2k for k\in\mathbb{N}.

    f(x)=x^n-a^n=x^{2k}-a^{2k}=(x^2)^k-a^{2k}

    Hence, f(-a)=((-a)^2)^k-a^{2k}=(a^2)^k-a^{2k}=a^{2k}-a^{2k}=0. As -a=\frac{-a}{1}, x^n-a^n can be factored by 1\cdot x-(-a)=x+a thanks to the factor theorem.

    Does it help ?
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