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Math Help - complicated factoring problem

  1. #1
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    complicated factoring problem

    how do you factor this polynomial?

    x^(2k+1) - y^(2k+1)

    my teacher's answer is:
    [x^2 (x^(2k-1) + y^(2k-1))] + [(y^2 - x^2)y^(2k-1)]

    what is the reasoning involved to factor something as complicated as this?
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  2. #2
    Member Abu-Khalil's Avatar
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    x^{2k+1}-y^{2k+1}=x^{2k+1}+x^2y^{2k-1}-x^2y^{2k-1}-y^{2k+1}=x^2\left(x^{2k-1}-y^{2k-1}\right)-\left(x^2-y^2\right)y^{2k-1}

    It's easily to see in this way when you are using induction to prove divisibility.
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  3. #3
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    Hello, oblixps!

    How do you factor this polynomial?

    . . x^{2k+1} - y^{2k+1}

    my teacher's answer is: . \bigg[x^2 \left(x^{2k-1} + y^{2k-1}\right)\bigg] \:{\color{red}+}\: \bigg[\left(y^2 - x^2\right)y^{2k-1}\bigg]
    Wrong! . . . That is not factored!

    The difference of two odd powers can always be factored.

    x^{2k+1} - y^{2k+1} \;=\;\bigg(x-y\bigg)\,\bigg(x^{2k} + x^{2k-1}y + x^{2k-2}y^2 + x^{2k-3}y^3 + \hdots + x^2y^{2k-2} + xy^{2k-1} + y^{2k} \bigg)

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