Partial Fraction Decomposition when denominator can't be further factored

In summary, the fraction is not able to be decomposed into partial fractions since the numerator and denominator have the same order. However, it can be integrated by using a common trick of rewriting it as a difference of fractions.
  • #1
tmt1
234
0
I have this fraction

$$x^2 / (x^2 + 9)$$

I'm not sure how to approach this problem since the denominator can't be further factored. What is the right approach for this type of problem?
 
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  • #2
In order to do partial fractions we must have an expression that has a lower order in the numerator than in the denominator. In your case they are the same. So you would be looking for a form like \(\displaystyle \frac{Ax + B}{x^2 + 9}\).

I would suggest looking at it like this:
\(\displaystyle \frac{x^2}{x^2 + 9} = \frac{x^2 + 9 - 9}{x^2 + 9} = 1 - \frac{9}{x^2 + 9}\)
which is now in the desired form.

This is a fairly common trick.

-Dan
 
  • #3
tmt said:
I have this fraction: $$x^2 / (x^2 + 9)$$

I'm not sure how to approach this problem since the denominator can't be further factored.
What is the right approach for this type of problem?

it depends on what you intend to do with it.

If you are trying to decompose it into Partial Fractrions, nothing can be done.

If you are trying to integrate it:

. . [tex]\int\frac{x^2}{x^2+9}dx \;=\;\int\frac{x^2+9 - 9}{x^2+9} dx[/tex]

. . [tex]=\;\int\left(\frac{x^2+9}{x^2+9} - \frac{9}{x^2+9}\right)dx \;=\;\int\left(1 - \frac{9}{x^2+9}\right) dx \;\cdots\;\text{etc.}[/tex]
 

FAQ: Partial Fraction Decomposition when denominator can't be further factored

1. What is partial fraction decomposition?

Partial fraction decomposition is a method used in algebra to break down a rational function into simpler fractions. It involves finding the individual fractions that make up the original function.

2. When is partial fraction decomposition used?

Partial fraction decomposition is typically used when the denominator of a rational function cannot be factored any further. It is also used to simplify complex rational functions and make them easier to integrate or differentiate.

3. How do you perform partial fraction decomposition?

To perform partial fraction decomposition, you first factor the denominator of the rational function into its irreducible factors. Then, you equate the original function to the sum of the individual fractions with each irreducible factor as the denominator. Finally, you solve for the unknown coefficients in each fraction.

4. What is the purpose of partial fraction decomposition?

The purpose of partial fraction decomposition is to simplify complex rational functions and make them easier to work with. It also allows for easier integration and differentiation of rational functions.

5. Are there any restrictions to using partial fraction decomposition?

Yes, there are a few restrictions when using partial fraction decomposition. The denominator of the original rational function must be a polynomial, and the degree of the numerator must be less than the degree of the denominator. Additionally, if there are repeated factors in the denominator, the decomposition will be more complicated.

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