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Math Help - pattern for factoring

  1. #1
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    pattern for factoring

    I need some help continuing the pattern for limits here is what I have so far:

    a^3-b^3=(a-b)(a^2+ab+b^2)
    a^4-b^4=(a-b)(a+b)(a^2+b^2)
    a^5-b^5=?
    a^6-b^6=?
    a^7-b^7=?
    a^8-b^8=?
    a^9-b^9=?
    a^{10}-b^{10}=?

    Thanks!
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  2. #2
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    Quote Originally Posted by qbkr21 View Post
    I need some help continuing the pattern for limits here is what I have so far:


    Thanks!
    In general for n\geq 2 we have,
    x^n-y^n=(x-y)(x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1})
    Now there are two cool ways of writing that long expression,
    \sum_{k=1}^{n-1} x^ky^{n-1-k}
    Another way is,
    \sum_{\begin{array}{c}i+j=n-1\\i,j\geq 0 \end{array}}x^iy^j

    The second one means that it is the sum of all possibilities of getting an exponent sum of n-1.
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