# What is Partial fractions: Definition and 297 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
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x
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g
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,

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

where f and g are polynomials, is its expression as

f
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g
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=
p
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+

j

f

j

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g

j

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{\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.

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1. ### Partial fractions with complex linear terms

I am interested specifically in solving this problem by factoring the quadratic term into complex linear factors. $$s^2+4=0$$ $$\implies s=\pm 2i$$ $$\frac{5s+6}{(s-2i)(s+2i)(s-2)}=\frac{A}{s-2i}+\frac{B}{s+2i}+\frac{C}{s-2}$$ We can solve for ##C## using the cover-up method with ##s=2## to...
2. ### I Using Residues (Complex Analysis) to compute partial fractions

Dear Everybody, I am wondering how to compute the partial fraction decomposition of the following rational function: ##f(z)=\frac{z+2}{(z+1)^2(z^2+1)}.## I understand how to do the simple poles of the function and how it is related to the decomposition's constants, i.e...
3. ### Solve the given problem that involves integration

For part (a), Using partial fractions (repeated factor), i have... ##7e^x -8 = A(e^x-2)+B## ##A=7## ##-2A+B=-8, ⇒B=6## $$\int {\frac{7e^x-8}{(e^x-2)^2}}dx=\int \left[{\frac{7}{e^x-2}}+{\frac{6}{(e^x-2)^2}}\right]dx$$ ##u=e^x-2## ##du=e^x dx## ##dx=\dfrac{du}{e^x}## ... also ##u=e^x-2##...

33. ### 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...
34. ### Integral with partial fractions

Homework Statement I'm currently in Calculus 3, and the professor gave us a "retro assignment" which is basically a bunch of tough integrals from Calculus 2. I think my process here is valid, but when I check my answer on Wolfram, they're getting a slightly different final answer...
35. ### Partial fractions integral

Homework Statement integral(0>1) of (x^2+x)/(x^2+x+1)dx Homework Equations Factor denominator, and set numerator with A,B,C, etc. multiply both sides by the common denominator. The Attempt at a Solution Since the denominator won't factor at all I don't really know where to start, I could...
36. ### Partial Fractions: Why Does (x+1)2(2x+1) Need 3 Terms?

Why, when a fraction has repeated linear terms in its denominator e.g. (11x2+14x+5)/[(x+1)2(2x+1)] does it have to be split into three partial fractions, A/(x+1) + B/(x+1)2 + C/(2x+1)? When my first saw this example, my initial reaction was to split it into A/(x+1)2 +B/(2x+1), but after working...
37. ### Inverse laplace transform without partial fractions

Homework Statement take inverse laplace of: 6/[s^4(s-2)^2] Homework Equations 6/[s^4(s-2)^2] The Attempt at a Solution I used partial fractions. I was wondering if It could be manipulated to where I could use the laplace table?
38. ### Partial Fractions - irreducibility question

1. x^2-x+1 Is this factorable? My initial thinking is NO. However, I can complete the square and it becomes (x-1/2)^2-3/4, but this doesn't seem to help me. Would this be considered factorable? 2. Turn 1/x^2-x+1 into partial fractions Clearly, after I answer #1 correctly, #2 will be more...
39. ### How to find the partial fractions for this expression?

Homework Statement Find the partial fractions for this expression. (((n+1)*(sqrt(n)) - n*(sqrt(n+1))) / (n*(n+1))) The Attempt at a Solution The final answer is 1/sqrt(n) - 1/(sqrt(n+1)) My work: A/n - B/(n+1) = n*sqrt(n+1) - (n+1)*(sqrt(n)) I am subbing in n = -1 and n = 0 to solve for...
40. ### Partial Fractions in Differential Equations

I'm a little rusty with partial fractions, and I can't seem to find my error once I get up to that point. Homework Statement dy/dx = (y^2 - 1) / x Homework Equations The Attempt at a Solution Cross-mutliply x dy = (y^2 - 1) dx Divide by the appropriate terms dy / (y^2...
41. ### MHB Tricky Partial Fractions Question

Hey guys, Here is another pair of questions that I'm doubting at the moment: I used partial fractions for A and got (Bx+C)/x^2 + Ax/(x-1)^2 + Dx(x-1) which led me to compute A=1, B=0, C= -1, and D=0, which already sounds off. Do you guys have any suggestions? Also, for 5b, I calculated B=...
42. ### Partial Fractions - Integration

Homework Statement Evaluate the integral. (Remember to use ln |u| where appropriate. Use C for the constant of integration.) \int \frac {5x^2 - 20x +45}{(2x+1)(x-2)^2}\, dx Homework Equations 5x^2 - 20x +45 = 5 (x^2 -4x +9) The Attempt at a Solution I'm able to come up with an...
43. ### MHB Understanding Partial Fraction Decomposition in Integrals

First the example problem. This is an integral of the whole thing (3x^3+24x^2+56x-5) / (x^2+8x+17)^2 The answer comes out to be 3/2 ln(x^2+8x+17) - (49/2 tan^-1(x+4)) - (25x+105 / 2(x^2+8x+17) + C I would show all the steps but I'm still not sure on how to use the format tools, so that...
44. ### MHB Quick Integral (U-substitution and partial fractions) Questions

Hey guys, I'd really appreciate it if I could get some quick help for this problem set I'm working on. For question one, I just did a quick u substitution for x^4 and managed to get x^4 * sin(x^4)+cos(x^4) + C. For part b, I used integration by parts and took ln(4t) as u and the rest as...
45. ### Calculating Partial Fractions find A, B and C

Homework Statement For the equation shown below: x2+2x+3 / (x2+9)(X-3) = Ax+B/(x2+9) + C/(x-3) Find A, B and C Homework Equations The Attempt at a Solution C = 1 B = 2 A = ? Find C which = 1 by putting x=3 and working out x2+2x+3/(x2+9), then multiply out equation...
46. ### Partial Fractions - 3 Unknowns

Hello, i've come across a partial fractions problem that I don't know how to solve - Usually, the denominator of the fraction I need to split up into two separate fractions is a quadratic, but in this instance it's a cubic. Specifically, the problem I'm having is that two of the factors to...
47. ### Integration by Partial Fractions

Homework Statement Find the indefinite integral of the below, using partial fractions. \frac{4x^2+6x-1}{(x+3)(2x^2-1)} Homework Equations ?The Attempt at a Solution First I want to say there is probably a much easier and quicker way to get around certain things I have done but I have just...
48. ### Using trig substitution or partial fractions?

When would you use trig substitution vs. partial fractions? I know partial fractions is when you have a polynomial over a polynomial, but some of the problems in the trig substitution section in my book had polynomial over polynomial and used trig substitution?
49. ### How to Solve Partial Fractions Expansion?

Homework Statement Find the partial fractions expansion in the following form, G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2} Homework Equations The Attempt at a Solution I expanded things out and found the following, 1 = A(s^{2} + 4)...
50. ### Multiplying Partial Fractions: Understanding the Rules

Homework Statement Homework Equations After looking through this on Wiki, I'm a little confused as to how these partial fractions are multiplied out. Is there a rule or something for this? With simpler partials I can do it but this one is something else! The Attempt at a Solution