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




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

where f and g are polynomials, is its expression as











{\displaystyle {\frac {f(x)}{g(x)}}=p(x)+\sum _{j}{\frac {f_{j}(x)}{g_{j}(x)}}}
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
  1. Physics Slayer

    A doubt in Partial fraction decomposition

    Say you want to find the following Integrals $$\int \frac{1}{(x-1)(x+2)} (dx)$$ $$\int \frac{1}{(x-1)(x^2 + 2)} (dx)$$ The easiest way to solve them will be by using partial fraction decomposition on both the given functions. Decomposing the first function, $$\frac{1}{(x-1)(x+2)} =...
  2. A

    Partial fraction decomposition with Laplace transformation in ODE

    Hello! Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
  3. opus

    Partial Fraction Decomposition

    Homework Statement Find the partial fraction decomposition for: ##\frac{1}{\left(x^2-1\right)^2}## Homework EquationsThe Attempt at a Solution Please see my attached images. I think the image shows my thought process better and it would take me well over an hour to type all that out! Im...
  4. opus

    B Partial Fraction Decomposition - "Telescoping sum"

    There is a problem in a PreCalculus book that I'm going over that states: Express the sum ##\frac{1}{2⋅3}+\frac{1}{3⋅4}+\frac{1}{4⋅5}+...+\frac{1}{2019⋅2020}## as a fraction of whole numbers in lowest terms. It goes on to state that each term in the sum is of the form...
  5. Q

    Stuck on this integral (using partial fraction decomposition)

    Homework Statement \int\frac{x^2}{\sqrt{x^2+4}}dx Homework Equations n/a The Attempt at a Solution Letting x=2tan\theta and dx=2sec^2\theta d\theta \int\frac{x^2}{\sqrt{x^2+4}}dx=\int\frac{4tan^2\theta}{\sqrt{4+4tan^2\theta}}2sec^2\theta d\theta=\int\frac{8tan^2\theta...
  6. Mr Davis 97

    Partial fraction decomposition with cos() in the numerator

    Homework Statement See below Homework EquationsThe Attempt at a Solution I am looking at a particular integral, and to get started, my text gives the indication that one should use partial fraction decomposition with ##\displaystyle \frac{\cos (ax)}{b^2 - x^2}##. Specifically, it says "then...
  7. Mr Davis 97

    Partial fraction decomposition

    Homework Statement Find the partial fraction decomposition of ##\displaystyle \frac{1}{x^4 + 2x^2 \cosh (2 \alpha) + 1}## Homework EquationsThe Attempt at a Solution Using the identity ##\displaystyle \cosh (2 \alpha) = \frac{e^{2 \alpha} + e^{- 2\alpha}}{2}##, we can get the fraction to the...
  8. karush

    MHB partial fraction decomposition

    $\tiny{}$ $\textsf{Use the method of partial fraction decomposition}\\$ \begin{align*} \displaystyle I_{44} &=\int \frac{4x^3+6x^2+128x}{x^5+32x^3+256x}dx\\ &=4\int \frac{1}{x^2+16} \, dx +6\int \frac{x}{(x^2+16)^2} \, dx +64\int \frac{1}{(x^2+16)^2} \, dx \end{align*}$\textsf{so far...
  9. Ryaners

    Finding sum of infinite series

    [Please excuse the screengrabs of the fomulae - I'll get around to learning TeX someday!] 1. Homework Statement Find the sum of this series (answer included - not the one I'm getting) The Attempt at a Solution So I'm trying to sum this series as a telescoping sum. I decomposed the fraction...
  10. karush

    MHB Partial fraction decomposition

    $\tiny{206.8.5,42}\\$ $\textsf{partial fraction decomostion}\\$ \begin{align} \displaystyle && I_{42}&=\int\frac{3x^2+x-18}{x^3+9x}\, dx& &(1)&\\ && \frac{3x^2+x-18}{x^3+9x} &=\frac{Ax+B}{x^2+9} +\frac{C}{x} & &(2)& \end{align} $\textit{just seeing if this is set up ok before finding values} $
  11. Cosmophile

    I Partial Fraction Decomposition With Quadratic Term

    Hey, all! I'm learning partial fraction decomposition from Serge Lang's "A First Course in Calculus." In it, he gives the following example: \int\frac{x+1}{(x-1)^2(x-2)}dx He then decomposes this into the following sum: \frac{x+1}{(x-1)^2(x-2)} =...
  12. defaultusername

    Breakdown of a Logistic Equation

    Homework Statement I feel so stuck. Given the Logistic Equation: $$\frac{dP}{dt}=kP(1-\frac{P}{A})$$ a.). Find the equilibrium solutions by setting $$\frac{dP}{dt}=0$$ and solving for P. b.). The equation is separable. Separate it and write the separated form of the equation. c.). Use partial...
  13. T

    MHB Partial Fraction Decomposition when denominator can't be further factored

    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?
  14. D

    Partial fraction decomposition exercise 2

    Homework Statement Hello! Here is my second post on the subject partial fraction decomposition. The subject looks pretty easy to learn, but when I try exercises, I do not get to the correct answer. Please, take a look at the exercise below and help me to see my mistakes. Homework Equations...
  15. D

    Partial fraction decomposition using matrix

    Homework Statement Hello! I am doing a chapter on partial fraction decomposition, and it seems I do not understand it correctly. Here is the exercise doing which I get wrong answers. Please, take a look at the way I proceed and, please, let me know what is wrong in my understanding. Homework...
  16. L

    MHB Partial fraction decomposition

    Hello everybody! I have to decompose to simple fractions the following function: V(z)=\frac{z^2-4z+4}{(z-3)(z-1)^2}. I know I can see the function as: V(z)=\frac{A}{z-3}+\frac{B}{(z-1)^2}+\frac{C}{z-1}, and that the terms A, B, C can be calculated respectively as the residues in 3 (single pole)...
  17. yuming

    Telescopic sum issues, cant get Sk

    1. Homework Statement Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges Homework Equations 3/(1*2*3) + 3,/(2*3*4) + 3/(3*4*5) +...+ 3/n(n+1)(n+2) The Attempt at a Solution the first try, i tried using partial fraction which equals...
  18. O

    Definite integral involving partial fractions

    Homework Statement Homework Equations trigonometric identities The Attempt at a Solution I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = 1-u2/1+u2 dθ = 2/(1+u2) du = 1/2 sec2(θ/2) dθ I simplified a bit and changed the bounds to get 2du/(5u2 + 1)(1 + u2)2...
  19. A

    What's the logic behind partial fraction decomposition?

    Ok so I took partial fraction decomposition in Calc II, and now I'm taking it again in Differential Equations course. The problem is that I don't really understand what I'm doing. I understand the procedure when having simple real roots, for example 2x+1/(x+1)(x+2), it becomes A/(x+1) + B/(x+2)...
  20. J

    Very long Taylor expansion/partial fraction decomposition

    Homework Statement I want to express the following expression in its Taylor expansion about x = 0: $$ F(x) = \frac{x^{15}}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)} $$ The Attempt at a Solution First I tried to rewrite the function in partial fractions (its been quite a while since I've last...
  21. G

    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. G

    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. StrangeCharm

    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. D

    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. C

    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. E

    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. S

    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. M

    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. S

    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. binbagsss

    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. icesalmon

    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. S

    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. jdawg

    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. S

    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. S

    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. jdawg

    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. J

    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. MarkFL

    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. J

    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

    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. S

    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. N

    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. MarkFL

    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. MarkFL

    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. O

    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. T

    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. T

    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. R

    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. T

    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. D

    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