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Math Help - "simple" polynomial question

  1. #1
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    "simple" polynomial question

    "simple" polynomial question, any help would be appreciated

    Show that (x-1)(x-2) is a factor of

    P(x) = x^n (2^m -1) + x^m(1-2^n) + (2^n - 2^m)

    m and n are positive integers
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  2. #2
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    Well if the expressions x-2 and x-1 are factors of the given polynomial, x=1 and x=2 must be its roots.

    So just substitute x=1 and then x=2 and see if the equation

    x^n(2^m-1) + x^m(1-2^n) + 2^n-2^m = 0 is satisfied.

    so for x=1 , the value of the polynomial is,
    2^m-1+1-2^n+2^n-2^m=0

    Hence the above equation is satisfied. So x-1 is a factor.

    Now, insert x=2 and prove that the value of the polynomial becomes zero.

    so as x=1 and x=2 are the roots of the given polynomial (x-1)(x-2) is a factor.
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  3. #3
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    can you check that again though, cause im pretty sure (x-1)(x-2) is a quadratic
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  4. #4
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    It is. But by proving that x=1 and x=2 are the roots of the given polynomial, it follows that x-1 and x-2 are factors. Hence their product,
    (x-1)(x-2) will also be a factor.
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