Partial Fractions: Reducing x^5, Example Help Needed

[zanazzi78]

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

I know i`m supossed to reduce the numerator, but i`m 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 i`m lost.

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

Cheers

 

[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,

 

[Architeuthis Dux]

Hi there,

I'm glad you're seeking help with this problem. Partial fractions can be tricky to understand at first, but once you get the hang of it, it becomes much easier.

First, let's review what partial fractions are. They are a method of breaking down a complex fraction into smaller, simpler fractions. This is done by finding the unique factors in the denominator and expressing the original fraction as a sum of fractions with those factors in the denominator.

In this case, we have a numerator of x^5 and a denominator of x^3 - x. The first step is to factor the denominator. In this case, x^3 - x can be written as x(x^2 - 1). Now, we can express our original fraction as:

x^5 / x(x^2 - 1)

Next, we can break this down further by factoring x^2 - 1. This gives us (x - 1)(x + 1). So now, our fraction can be written as:

x^5 / x(x - 1)(x + 1)

To reduce the numerator, we can use the exponent rule for division, which states that when dividing two terms with the same base, we can subtract the exponents. In this case, we can rewrite x^5 as x^3 * x^2. So now our fraction is:

x^3 * x^2 / x(x - 1)(x + 1)

We can now cancel out one of the x's in the numerator and denominator, leaving us with:

x^2 / (x - 1)(x + 1)

This is the final expression in reduced form.

I hope this example helps clarify the process for you. It's important to remember to always factor the denominator first and then work on reducing the numerator. If you have any further questions or need more clarification, please don't hesitate to ask. Happy problem-solving!