# What is Partial fraction decomposition: Definition and 79 Discussions

In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz.In symbols, the partial fraction decomposition of a rational fraction of the form

f
(
x
)

g
(
x
)

,

{\displaystyle \textstyle {\frac {f(x)}{g(x)}},}

where f and g are polynomials, is its expression as

f
(
x
)

g
(
x
)

=
p
(
x
)
+

j

f

j

(
x
)

g

j

(
x
)

{\displaystyle {\frac {f(x)}{g(x)}}=p(x)+\sum _{j}{\frac {f_{j}(x)}{g_{j}(x)}}}
where
p(x) is a polynomial, and, for each j,
the denominator gj (x) is a power of an irreducible polynomial (that is not factorable into polynomials of positive degrees), and
the numerator fj (x) is a polynomial of a smaller degree than the degree of this irreducible polynomial.
When explicit computation is involved, a coarser decomposition is often preferred, which consists of replacing "irreducible polynomial" by "square-free polynomial" in the description of the outcome. This allows replacing polynomial factorization by the much easier to compute square-free factorization. This is sufficient for most applications, and avoids introducing irrational coefficients when the coefficients of the input polynomials are integers or rational numbers.

View More On Wikipedia.org

21. ### Partial fraction decomposition

Homework Statement What is the partial fraction decomposition in ##\mathbb{R}[X]## of ##F = \frac{1}{X^{2n} - 1 } ##, ##n\ge 1##. Homework EquationsThe Attempt at a Solution Is this correct ? ## F = \frac{1}{2n}(\frac{1}{X-1} - \frac{1}{X+1} + 2 \sum_{k = 1}^{n-1} \frac{ \cos...
22. ### MHB Partial fraction decomposition (x-3)/(x^2+4x+3)

(x-3)/(x^2+4x+3) After i factor the denominator what do i do next to find A and B? =(x-3)/(x+3)(x+1) =A/(x+3)+B/(x+1)
23. ### Integration by Partial Fractions Help

Homework Statement ∫ [x^(3)+4] / [x^(2)+4] dx Homework Equations N/A The Attempt at a Solution I know that the fraction is improper, so I used long division to rewrite it as x+(-4x+4)/[x^(2)+4]. Given the form S(x)+R(x)/Q(x), Q(x) is a distinct irreducible quadratic factor [x^(2)+4]. I used...
24. ### MHB Partial fraction decomposition

please help decompose$\frac{4x^2y}{(x^2-2xy+2y^2)(x^2+2xy+2y^2)}$ I've used the cases I know for this problem but to no avail. please help me.
25. ### Partial Fraction Decomposition

Homework Statement To find the decomposition of a polynomial with a repeated factor in the denominator, you should separate them into (x+a)^1 + ... + (x+a)^n. But, my question is why? For example, why should you decompose it in the following way: \frac{x+2}{(x+1)(x+3)^2} = \frac{A}{x+1}...
26. ### Partial fraction decomposition

Alright, I am here again with another question... When I have a rational function, let's say (x+4)/(x-2)(x-3) I rewrite it like A/(x-2) + B(x-3) and then solve it for A & B. But when we have for e.g (x^2 + 3x + 2)/(x(x^2 +1 )) the book tells me to rewrite it like: A/x + (Bx + C)/(x^2 + 1)...
27. ### MHB Find the partial fraction decomposition for the rational function.

Find the partial fraction decomposition for the rational function. \frac{-4x^2 - 8x - 19}{(x^2 + 2)(x-9)} I'm not sure what to do.
28. ### Integration with Partial Fraction Decomposition

Homework Statement \int \frac{-2x + 4}{(x-1)^{(2)}(x^{(2)}+1)}Homework Equations The Attempt at a Solution I've done the problem a couple times but the answers keep coming out differently so I'm assuming I am messing up the setup. This is what I have for the first part of the setup: -2x +...
29. ### Partial Fraction Decomposition problem

Homework Statement Evaluate ∫((secx)^2)/[((tanx)^2)+(3tanx)+2] Homework Equations Partial fraction decomposition The Attempt at a Solution So here's what I did: But this is incorrect. It says the correct answer is -2lnabs(\frac{1}{2tanx+3}+\sqrt{4(tanx+3/2)^{2}-1}), which was...
30. ### Laurent Series & Partial Fraction Decomposition.

Okay so the partial fraction decomposition theorem is that if f(z) is a rational function, f(z)=sum of the principal parts of a laurent expansion of f(z) about each root. I'm working through an example in my book, I am fine to follow it. (method 1 below) But instinctively , I would have...
31. ### Partial Fraction Decomposition

Homework Statement use partial fraction decomposition to re-write 1/(s2(s2+4) The Attempt at a Solution I thought it would break down into (A/s) + (B/s2) + ((cx+d)/(s2+4) but it doesn't.
32. ### MHB Partial Fraction Decomposition

When I'm evaluating a problem like \int \frac{2x^2 + 8x + 9}{(x^2 + 2x + 5)(x + 2)} = \frac{Ax + B}{x^2 + 2x + 5} + \frac{C}{x+2} I understand how to get the C part, that's simple. But what is a Good trick to know that I need to have Ax + B over the x^2 + 2x + 5 denominator? Is there a way I...
33. ### Partial Fraction Decomposition with Integration

Homework Statement ∫(2x3-4x-8)/(x2-x)(x2+4) dx Homework Equations The Attempt at a Solution ∫(2x3-4x-8)/x(x-1)(x2+4) dx Next I left off the integral sign so I could do the partial fractions: 2x3-4x-8=(A/x)+(B/(x-1))+((Cx+D)/(x2+4))...
34. ### MHB Partial Fraction Decomposition Evaluation

Ok I'm stuck I have \int \frac{x^2 - 5x + 16}{(2x + 1)(x - 2)^2} \, dx and I got to this part: x^2 - 5x + 16 = A(x - 2)^2 + B(x - 2)(x + \frac{1}{2}) + c(x + \frac{1}{2})So do i need to distribute all of these and factor out or is there a simpler way? I found a solution where they are just...
35. ### MHB Partial Fraction Decomposition

Quick question... I know that if the numerator is greater than the denominator I need to divide out by long division BUT If the numerator is equal to the denominator (the exponent is what I'm talking about to be specific) then, do I need to do anything? Because I'm stuck on this problem \int...
36. ### Partial Fraction Decomposition

Homework Statement (t4+9)/(t4+9t2) Homework Equations The Attempt at a Solution I'm not completely sure if I'm using the correct method to solve this. Since the degrees of the numerator and denominator are the same, wouldn't you divide the denominator into the numerator? Here is...
37. ### MHB Partial fraction decomposition

Q3.) Express as partial fractions. a) \frac{3x+4}{x^2+3x+2} b) \frac{5x^2+5x+8}{(x+2)\left(x^2+2 \right)} c) \frac{x^2+15x+21}{(x+2)^2(x-3)}
38. ### MHB Integration by Partial Fraction Decomposition - Yahoo Answers

Here is the question: I have posted a link there so the OP can view my work.
39. ### Partial Fraction Decomposition

My professor asks us to solve the integral of: [x/(x^4 + 1)]dx This expression is not factorable; what should I do? She is asking us to solve specifically using PFD, not u-substitution.
40. ### I don't understand partial fraction decomposition

if there is something like (x^2+3x+6) in the denominator for one of the terms in a partial fraction problem, why do we put Ax+B instead of just A? and if the denominator is (x^2+3x+6)^2, why do we do {(Ax+B)/(x^2+3x+6)}+{(Cx+D)/(x^2+3x+6)^2}? i was always just told to memorize it, but why do we...
41. ### Partial Fraction Decomposition

Hello I am stuck on an ODE involving substitution. I have done the correct substitutions, but have become stuck on decomposing the fraction. i have the following ∫(1/x)dx + ∫(u+1)/(u^2+1)du = 0 Im stuck on breaking the u down into a partial decomposition. Could anyone offer some advice on...
42. ### Partial fraction decomposition: One quick question

Homework Statement Give the partial fraction decomposition of 1/z4+z2 Homework Equations The Attempt at a Solution My question is about the final answer. The book gives the answer to be 1/z2+ 1/2i(z+i)- 1/2i(z-i). For my answer I keep getting a negative for both of the 1/2i...
43. ### MHB DW123's question at Yahoo Answers regarding partial fraction decomposition

Here is the question: Here is a link to the question: Decompose the equation into two simpler fractions? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
44. ### MHB Partial Fraction Decomposition Help - Calculus BC

Here is the question: Here is a link to the question: Help with Calculus BC: partial fractions!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
45. ### Octave partial fraction decomposition help

So I'm trying to figure out how to decompose the following using octave: 85000/[(s^2+250^2)(0.2s^2+40s+10000)] I tried using the residue command but I think that only works if the polynomials have real roots, which these don't. When I do use residue I get the following: b =...
46. ### Partial fraction decomposition

I have been having trouble of late with partial fraction decomposition. Not so much the maths, but the intuition behind it. What I mean by this, but a question in front of me, I now what procedure to follow to get the answer, but I don't get why you follow the said produced. I will give an...
47. ### Partial fraction decomposition with complex function

As part of a project I have been working on I fin it necessary to manipulate the following expression. e^(icx)/(x^2 + a^2)^2 for a,c > 0 I would like to decomp it into the form A/(x^2 + a^2) + B/(x^2 + a^2) = e^(icx)/(x^2 + a^2)^2 but I am having trouble getting a usable outcome.
48. ### Partial fraction decomposition

Homework Statement \frac{2e^3}{((s^2)-6s+9)*s^3} you can factorize the denominator into s,s,s,(s-3),(s-3) that gives you 5 residuals. the first 3 should all be the same value but that's apparently not correct, so where am I going wrong?
49. ### Help With Partial Fraction Decomposition

Homework Statement I'm supposed to decompose 1 / x(x2 + 1)2 Also, we haven't learned matrices yet so I can't use that technique to solve it. Homework Equations None. The Attempt at a Solution 1 / x(x2 + 1)2 = A/x + (Bx + C) / (x2 + 1) + (Dx + E) / (x2 + 1)2 I multiplied...
50. ### Help Factorial Partial Fraction Decomposition

Homework Statement Show that n/(n+1)!=(1/n)-(1/(n+1)!) I am totally lost on the algebraic steps taken to come to this conclusion. It is for an Infinite series. Thanks