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.

1254 questions
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Mistake in partial fraction

Can anyone spot my mistake? $$\frac{1-\frac{1}{2}z^{-1}}{1+\frac{3}{4}z^{-1}+\frac{1}{8}z^{-2}}$$ Set $x = z^{-1}$ $$\frac{1-\frac{1}{2}x}{1+\frac{3}{4}x+\frac{1}{8}x^{2}}$$ Multiply by 8/8 and factorise…
Rory McEwan
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Problem with a partial fraction decomposition

I don't know why, but for some reason I cannot solve the following partial fraction decomposition no matter how much I try. $$\frac{1}{(v-1)^2(v+1)^2}$$ When decomposing that to $\frac{1}{(v-1)^2(v+1)^2} = \frac{A_1}{v-1} + \frac{A_2}{(v-1)^2} +…
pavus
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steps to solve a simple partial fractions

I want to solve for the following $$\frac{1}{1+t^4} $$ I start by following $$\frac{1}{1+t^4} = \frac{1}{(1 +t^2)^2 - (\sqrt{2}t)^2} = \frac{1}{(1+t^2- \sqrt{2}t)(1+t^2+\sqrt{2}t)} $$ $$\frac{1}{(1+t^2- \sqrt{2}t)(1+t^2+\sqrt{2}t)} = \frac{A}{1+t^2…
SJa
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What is the form of partial fraction decomposition when the exponent is inside the factor?

I have a partial fraction problem where I need to decompose $$\frac{1}{(-2x-x^4)}$$ which becomes $$\frac{1}{(x)(-x^3-2)}.$$ I'm used to dealing with partial fractions where the factor $(x-2)$ is raised to the third power, like $(x-2)^3$, but what…
Pro Q
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partial fraction decomposition special case

I would like to have the following partial fraction decomposed : $$\frac{2r+1}{r^2{(r+1)}^2}$$ Since the denominator does not contain any constant the approch is non-trivial to me. Any help would be highly appreciated. Thank you in advance.
mysterium
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Partial fractions with 2 squared terms

Attempting the following partial fraction equation and was wondering how to approach the $s^2$ outside the brackets: $$\frac{1}{s^2(s^2+2s+10)}$$
Matlab rookie
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Binomial expansion involving partial fractions

Sorry I do not know how to use the formatting will try my best. Q. Find the binomial expansion up to $x^2$ of: $$\frac{3+2x^2}{(2x+1)(x-3)^2}$$ For the partial fraction I get: $$\frac{2}{7}\frac{1}{2x+1} + \frac{6}{7}\frac{1}{x-3} +…
boi Shift
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Are these two partial fraction the same?

I tried solving a pretty standard partial fraction equation. $$ \frac{5-x}{2x^2+x-1} $$ Which becomes: $$ \frac{A}{x-0.5} + \frac{B}{x+1} $$ Solving the partial fraction: $$ 5-x = A(x+1)+B(x-0.5) $$ $$ 5-x = Ax+A+Bx-0.5B $$ $$ 5 = A -0.5B $$ $$ -x =…
Carrein
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Partial Fractions with multiple variables

I have been trying this problem for a 1 week now but for some reason can not get my head around how to even approach this problem. $$ \frac{\omega K}{(sT+1)(s^{2}+ \omega^{2})} $$ I first tried using the most straight forward way of…
user285454
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Confusion with how partial fractions work

I have trouble with the logic behind Partial Fractions, which I will elaborate about here. I have two problems with it. Firstly, why, if I have a polynomial like $$f(x)= \frac{3}{(x+4)^2(x+3)}$$ That when I decompose it I get something…
sangstar
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Partial fraction decomposition - denominator to the power of 3

How to get correct partial fraction decomposition of this expression? $$\frac{3x+3y-z}{(x+y+z)^3}$$
Vid
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Why is: $\frac{x^2}{(x^2-1)(x^2+1)} = \frac{1}{2}\left(\frac{1}{x^2-1}\right)+\frac{1}{2}\left(\frac{1}{x^2+1}\right)$

I am trying to understand the following. $\dfrac{x^2}{(x^2-1)(x^2+1)} = \dfrac{1}{2}\left(\dfrac{1}{x^2-1}\right)+\dfrac{1}{2}\left(\dfrac{1}{x^2+1}\right)$ If I start with the right side I can easily get to the left side of the equation but not…
Sylvester Stallone
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How to find the partial fractions of the given irreducible equation?

$$H(z) = \frac{(z^2 + \frac 13 z)(z^2 -\frac 78 z)}{(z^2 - 2z +2)(z^2 -\frac 34z + \frac 18)}.$$ How can i find the partial fractions of the given function above? I know the equation in the denominator is irreducible. Can anyone solve this equation?
Ross
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how to do partial fraction decomposition on this equation

I have a fractional function $\frac{1}{x(1+x^{n-1})}$. using PFD: $\frac{A}{(1+x^{n-1})}+ \frac{B}{x}$, that means $Ax+(1+x^{n-1})B=1$. For this to hold, we need $A=0, B=0, B=1$, which is of course impossible. Does that mean that this fraction…
user56834
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Partial fraction decomposition of a parameter-dependent rational function

Let $n$ and $m$ be strictly positive integers such that $n< 2 m$. By using mathematical induction in $n$ we have derived the following equality identity: \begin{equation} \frac{x^n}{\left(1-(1+a) x+x^2\right)^m}= \sum\limits_{\xi=0}^{\lfloor…
Przemo
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