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Math Help - Divide 1 by a polynomial

  1. #1
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    Divide 1 by a polynomial

    I know how to use synthetic division to divide a polynomial by one of lower degree, but how does one divide a poly. by one of higher degree?

    In particular, a text I am studying says
    1 divided by x-1 =1 + x + x^2 + x^3 + ...
    iff -1 < x < 1,
    but I don't know how to find this out myself.

    How do you divide 1 by x-1 ?
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  2. #2
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    Quote Originally Posted by mnova View Post
    I know how to use synthetic division to divide a polynomial by one of lower degree, but how does one divide a poly. by one of higher degree?

    In particular, a text I am studying says
    1 divided by x-1 =1 + x + x^2 + x^3 + ...
    iff -1 < x < 1,
    but I don't know how to find this out myself.

    How do you divide 1 by x-1 ?
    This is done using the geometric series formula:

     \displaystyle \sum_{k=0}^{\infty} ar^k = \frac{a}{1-r}

    In your case r = x, a = 1

     \displaystyle  \frac{1}{1-x} = \sum_{k=0}^{\infty} x^k = x^0 + x^1 + x^2 + x^3 + x^4 ...

    As you can see, the result goes on to infinity. There is no method of doing this in the same way that you can divide using synthetic division/long division.
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