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Math Help - Challenging question on geometric progression?

  1. #1
    Super Member fardeen_gen's Avatar
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    Challenging question on geometric progression?

    For what values of n is the polynomial 1 + x^2 + x^4 + ... + x^(2n -2) divisible by 1 + x + x^2 + ... + x^(n - 1)?

    ^ - Raised to
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  2. #2
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    Quote Originally Posted by fardeen_gen View Post
    For what values of n is the polynomial 1 + x^2 + x^4 + ... + x^(2n -2) divisible by 1 + x + x^2 + ... + x^(n - 1)?

    ^ - Raised to
    p_1(x) = 1 + x^2 + x^4 + ... + x^{2n -2} = 1 + x^2 + (x^2)^2 + \, .... \, + (x^2)^{n-1} = \frac{1 - x^{2n}}{1 - x^2}.

    p_2(x) = 1 + x + x^2 + ... + x^{n-1} = \frac{1 - x^{n}}{1 - x}.

    In both cases the formula for the sum of a geometric series has been used: 1 + r + r^2 + \, ... \, + r^m = \frac{1 - r^{m+1}}{1 - r}. Note that r = x^2 and m = n-1 in p_1(x).

    \frac{p_1}{p_2} = \frac{(1 - x^{2n}) (1 - x)}{(1 - x^n) (1 - x^2)} = \frac{1+x^n}{1 + x}.

    And x + 1 is a factor of p(x) = x^n + 1 if p(-1) = 0 ......
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