Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [partial-fractions]

Rewriting rational function in the form of partial fractions is often useful when calculating integrals.

Rewriting rational function in the form of partial fractions is often useful when calculating integrals. The possibility of decomposing a rational function into a sum of simplified fractions is guaranteed by the fundamental theorem of algebra.

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Partial fraction decomposition with a 3rd degree numerator

I have the following function to decompose using PFD: $H(x) = \frac{x^3 + 4x^2 - 11x - 48}{x^3 + 6x^2 + 3x - 10}$ The poles are $1$, $-2$ and $-5$ so I tried to do it this way: finding $a$, $b$ and $c$ such as: $H(x) = \frac{a}{x-1} + \frac{b}{x+2}…
Damian Taylor
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Repeated linear factors in partial fractions

I have a question about the following partial fraction: $$\frac{x^4+2x^3+6x^2+20x+6}{x^3+2x^2+x}$$ After long division you get: $$x+\frac{5x^2+20x+6}{x^3+2x^2+x}$$ So the factored form of the denominator…
Jinzu
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Partial Fraction Decomposition of A/[x(x-a)^m]

I'm having difficulty understanding why / how to go about proving the fact that a partial fraction decomposition of $\frac{A}{x(x-a)^m}=\frac{A_1}{x}+\frac{A_2}{(x-a)^1}+\frac{A_3}{(x-a)^2}+ ... +\frac{A_m}{(x-a)^m}$ . Why is it not just something…
Chung Ren Khoo
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Partial Fractions Decomposition $\frac{6x^2 - 29x - 29}{(x+1)(x-3)^2}$ explanation repeated factors

I am trying to solve the fraction $$\frac{6x^2 - 29x - 29}{(x+1)(x-3)^2}$$ into partial fractions. Now, I thought it could be solved into the following $$\frac{6x^2 - 29x - 29}{(x+1)(x-3)^2} = \frac{A}{x+1} + \frac{B}{(x-3)^2}$$ but this is…
Evan Francis
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Repeated Linear Factors in denominator of fraction e.g. $\frac{2x^2 + 2x + 18}{x(x-3)^2}$

I have the following fraction below and I must find the partial fraction decomposition. $$\frac{2x^2 + 2x + 18}{x(x-3)^2}$$ Now, I thought I could simplify this into the following... $$\frac{2x^2 + 2x + 18}{x(x-3)^2} = \frac{A}{x} +…
vik1245
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Where does the x come from when you decompose non-linear denominators?

I have done >4h of research with different text books and I'm still stucked. I also have looked at enter link description here But still I don't get it, where does this x at Bx+C come from?: $$ \frac{3}{(x+1)(x²+4)} =…
J.Doe
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How to solve the partial fraction decomposition $\frac{x^3+5x^2+3x+6}{2x^2+3x}$.

I have the following integral: $$\int\frac{x^3+5x^2+3x+6}{2x^2+3x}dx$$ I'm trying to use partial fraction decomposition but I'm getting stuck at the following formula: $$\int\frac{(x+6)(1+5x+x^2)}{x(2x+3)}-\frac{x+27}{2x+3}dx$$ I can't necessarily…
user56834
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Partial Fraction with Unknown Constants

I'm having trouble doing the partial fraction decomposition here due to the unknown constants. I need to break down $$x(s)=\frac{F_0\omega}{(s^2+\omega^2)(s^2-\omega_0^2)}$$ where $F_0, \omega,$ and $\omega_0$ are all constants.
H. Rumo
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Decomposition into partial fractions of an inverse of a generic polynomial with three distinct roots.

Let $d \ge 2$ be an integer. Let $\left\{ m_j \right\}_{j=1}^d$ be strictly positive integers and $\left\{ b_j \right\}_{j=1}^d$ be parameters. Define the following quantity: \begin{equation} {\mathfrak F}_d(x) := \frac{1}{\prod\limits_{j=1}^d…
Przemo
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Partial fraction decomposition of function

How can I decompose the following function into partial fractions in order to integrate it? $$\frac{1}{v(\ln(v) - 2)}$$
Shoaib Ashraf
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How can I decompose this 6th-degree polynomial fraction into partial fractions?

How can I use partial-fraction decomposition for this fraction? $$f(x)\equiv\dfrac{-x^{5}+2x^{4}-3x^{3}+4x^{2}-5x+6}{7(x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1)}.$$
k2532184
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partial fraction decomposition: product in denominator

A fraction $\dfrac{a}{bc}$ can be split into $\dfrac{x}{b} + \dfrac{y}{c}$ by solving for $x$ and $y$ from $a = by + cx$. Now, then a term $\dfrac{a}{cd}$ should be able to be split like this $$\frac{x}{bc} + \frac{y}{d}$$ But apparently that is not…
user3578468
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Simple partial fraction expansion

I would like to ask for help solving a simple Partial fraction example: (It contains complex roots) $\frac{1}{(x^3 - 3x^2 + 4x - 2)}$ According to my book we can factor it as a product of the real roots and quadratic…
Charlie
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Why partial fraction decomposition requires a proper fraction on input?

Is this only a desirable condition to make the job easier or a strict constraint of the method itself? ("Proper fraction" = a fraction where numerator is smaller then denominator)
ribitskiyb
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Integration of Partial Fraction Expansion

Hi This is my first time posting a question on this website. Thank you advance for helping me out here. My question is Suppose the density of $X$ is $$f(x) = \frac{Kx^2}{(1 + x)^5}$$ when $x > 0$. Find the constant $K$ and the density of $Y =…
King Kong
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