# Thread: Divide 1 by a polynomial

1. ## 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 ?

2. Originally Posted by mnova
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.