Partial Fractions: Deciding Formula Without Memorization

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In summary, the conversation discusses the technique of partial fractions and how to determine which formula to use when decomposing a fraction into its partial fractions. The only prime polynomials over R are linear or quadratic, and the numerators are either constant or linear. When f^n is a factor of the denominator, where f is prime, we need to allow for all fractions whose greatest common denominator is f^r. The conversation ends with the realization that partial fractions are not as mystifying as they may seem.
  • #1
pivoxa15
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I have never remebered the different partial fraction forumlas. Is there a way to decide which partial fraction formula to use just by looking at the expression?
 
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  • #2
Partial fraction formulae?

I thought you just did partial fractions... what's the context? Normally you just split the fraction after factoring the denominator into a sum of fractions with denominators equal to the different factored parts
 
  • #3
over R, the only prime polynomials are linear or quadratic. the numerators are either constant or linear respectively.

if f^n is a factor of the denominator, where f is prime, we need to allow for all fractions whose greatest common denom is f^r, so we have

to allow a/f , b/f^2, c/f^3, ..., d/f^n, for appropriate numerators.

e.g. to decom,pose h/[x^2(x^2+x+1)^3] we set it equal to

a/x + b/x^2 + (cx+d)/(x^2+x+1) +(ex+f)/(x^2+x+1)^2 + (gx+h)/(x^2+x+1)^3.where a,b,c,d,e,f,g,h, are constants.
 
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  • #4
mathwonk said:
over R, the only prime polynomials are linear or quadratic. the numerators are either constant or linear respectively.

if f^n is a factor of the denominator, where f is prime, we need to allow for all fractions whose greatest common denom is f^r, so we have

to allow a/f , b/f^2, c/f^3, ..., d/f^n, for appropriate numerators.

e.g. to decom,pose h/[x^2(x^2+x+1)^3] we set it equal to

a/x + b/x^2 + (cx+d)/(x^2+x+1) +(ex+f)/(x^2+x+1)^2 + (gx+h)/(x^2+x+1)^3.


where a,b,c,d,e,f,g,h, are constants.

Is that all there is to partial fractions? It isn't as mystifying as it appears.
 
  • #5
pivoxa15 said:
Is that all there is to partial fractions?
Yes it is! :smile:
It isn't as mystifying as it appears.
No, it isn't! :smile:
 

Related to Partial Fractions: Deciding Formula Without Memorization

1. What are partial fractions?

Partial fractions are a method used to split a complicated rational function into simpler fractions. This allows for easier integration and manipulation of the function.

2. Why is it important to learn how to decide the formula for partial fractions without memorization?

Memorizing formulas can be time-consuming and limit understanding of the concept. By learning how to decide the formula without memorization, one can have a deeper understanding of the method and apply it to a wider range of problems.

3. How do you decide the formula for partial fractions without memorization?

The key is to factor the denominator of the rational function and then write it as a sum of simpler fractions with unknown coefficients. These coefficients can then be solved for using algebraic manipulation and the given equation.

4. Can you give an example of using partial fractions to solve a problem?

Sure, let's say we have the rational function (3x+2)/(x^2+4x+3). We can factor the denominator to (x+1)(x+3) and then write the rational function as (A/(x+1))+(B/(x+3)). From there, we can use algebra to solve for A and B and rewrite the function as (1/(x+1))+(2/(x+3)), making it easier to integrate or manipulate in other ways.

5. Are there any common mistakes when using partial fractions?

One common mistake is forgetting to include all the possible terms in the partial fraction decomposition. Another mistake is not checking for repeated roots in the denominator, which would require a different approach. It's important to carefully factor the denominator and check the final decomposition to avoid these errors.

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