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