# Partial Fraction

1. Oct 18, 2005

### zanazzi78

Ive been asked to complete the following expresion
$$\frac{x^5}{x^3 - x}$$

I know im supossed to reduce the numerator, but im a little stuck getting started.

The problem i have is how do you reduce $$x^5$$?

is it simply $$x^3(x^2)$$? but then how do you get rid of the $$x^3$$

AAHH im lost.

I obviously don`t want you to give me the answer so an example of your choice would be greatly appreciated.

Cheers

Last edited: Oct 18, 2005
2. Oct 18, 2005

### VietDao29

You can first divide both denominator and numerator by 'x'. They have 'x' in common. So:
$$\frac{x ^ 5}{x ^ 3 - x} = \frac{x ^ 4}{x ^ 2 - 1}$$
You can use 'polynomial long division' to reduce the degree of the numerator. You can click here for more information.
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Or you can also do it a little bit differently:
You notice that x4 = (x2)(x2). But the denominator is x2 - 1, so:
x4 = x2(x2 - 1) + x2.
Now:
$$\frac{x ^ 4}{x ^ 2 - 1} = \frac{x ^ 2(x ^ 2 - 1) + x ^ 2}{x ^ 2 - 1} = x ^ 2 + \frac{x ^ 2}{x ^ 2 - 1}$$
Now just do the same for $$\frac{x ^ 2}{x ^ 2 - 1}$$:
$$\frac{x ^ 2}{x ^ 2 - 1} = \frac{x ^ 2 + ... - ...}{x ^ 2 - 1} = ...$$
Viet Dao,

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