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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find $\frac{ax+b}{x+c}$ in partial fractions

$$y=\frac{ax+b}{x+c}$$ find a,b and c given that there are asymptotes at $x=-1$ and $y=-2$ and the curve passes through (3,0) I know that c=1 but I dont know how to find a and b? I thought you expressed y in partial fraction so that you end up with…
maxmitch
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Decomposition of $\frac{10(s+6)}{\left[(s+3)^2+25\right](s^2+25)}$

Determine $\alpha,\beta,\gamma,\delta$ in $$\frac{10(s+6)}{\left[(s+3)^2+25\right](s^2+25)}=\frac{\alpha s+\beta}{(s+3)^2+25}+\frac{\gamma s+\delta}{s^2+25}$$ I have come across the partial fraction and trying to decompose it. I have tried the…
fred
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Are there any shortcuts for computing the coefficients of partial fractions that are not covered by Cover Up Rule?

I am referring to powers of linear factors higher than 1, and all quadratic factors. I was just wondering if there are any obscure techniques to solve the coefficients since in common practice I know there are none.
AndroidV11
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General rule on partial fraction expansion?

I have fractions of the form $$F(p) = \prod_{i=1}^{n} \frac{1}{1+\alpha_i p} $$ and it appears that the partial fraction expansion is $$F(p) = \sum_{i=1}^{n} \frac{1}{(1+\alpha_i p)} \prod_{j=1,j\neq i }^{n} \frac{\alpha_i}{\alpha_i-\alpha_j}$$ at…
Jenny Reininger
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Method to partial fractions.

For Example: $$\frac{ax^2 + bx+c}{(dx+e)(fx^2+g)}\equiv\frac{A}{dx+e}+\frac{Bx+C}{fx^2+g}$$ and $$\frac{ax^4 + bx^3+cx^2+dx+e}{(x+f)(x^2+g)}\equiv Ax+B+\frac{C}{x+f}+\frac{Dx+E}{x^2+g}$$ How do you know how to format the right hand side, in the…
maxmitch
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How to find the partial fraction when the denominator contain repeated root?

I do not know how to find the partial fraction of the following form since the denominator contains a multiplicity of the root with an integer degree $n \geq1$ $$\frac{1}{{\left( {1 + x} \right){{\left( {x + \alpha } \right)}^n}}} = \frac{A}{{\left(…
Thai-Hoc
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Deriving an identity related to partial fractions of large products

I am interested in products of the type \begin{equation} \prod_1^N \frac{x-a_i}{x-b_i} \end{equation} for integer $N$. I would like to prove the following conjecture: \begin{equation} \prod_{i=1}^N \frac{x-a_i}{x-b_i} = 1 + \sum_{i=1}^N…
1010011010
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Cant get partial fraction decomposition

I would like to prove that: $$ L = (\frac{1+z^{-1}}{1+0.5z^{-1}} ) \cdot (\frac{1}{1-z^{-1}}) = \frac{0.166}{z + 0.5} + \frac{1.33}{z - 1} + 1$$ How do I get from left to right? The same problem have other solution that I succeed to get $L =…
arn
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$\frac{23x - 11x^2}{(2x-1)(9-x^2)}$ when resolved into partial fractions is equal to?

$\frac{23x - 11x^2}{(2x-1)(9-x^2)}$ when resolved into partial fractions is equal to? I solved it using $\frac{23x - 11x^2}{(2x-1)(9-x^2)} = \frac{A}{2x-1} + \frac{B}{3-x} + \frac{C}{3+x}$, but this is very long method to solve, as after comparing I…
sawan kumawat
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Theory of Partial Fraction Decomposition

The function in question that I want to decompose is $$\dfrac{8x^3 + 7}{(x+1)(2x+1)^3}$$ I had the idea to to break this down into: $$\dfrac{A}{x+1} + \dfrac{Bx^2 +Cx + D}{(2x+1)^3} + \dfrac{Ex + F}{(2x+1)^2} + \dfrac{G}{2x+1}$$ Well this turns…
Mark
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Partial fraction decomposition when one of the partial fraction coefficients is zero

It is quite some time since I used to do partial fraction (PF) decomposition. I am not able to recall how to PF the following expression? $\frac{9s+9}{(s+1)(s+3)}$. The coefficient of $\frac{1}{s+1}$ comes out to be zero.
Abu Bakar
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Multiple solutions of partial fractions decomposition

I'm learning about series, and there this bit about partial fractions decomposition that I want to ask. Am I right to assume that, there are multiple way to achieve the form $\frac{A}{1 - \alpha_1x}+\frac{B}{1 - \alpha_2x}$, i.e. there might be…
user533068
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Partial fractions with an irreducible quadratic factor in the denominator

So I'm supposed to decompose this expression into partial fractions [Note T_A is a constant in this case] Although I can get the right coefficients, I'm just wondering why it is C and not CT+D - should it not be the latter considering it has a…
Mathlete
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Partial fraction of $\frac{s}{(s^2+2s+2)(s^2-2s+2)}$

I am trying to find the partial fraction of: $$\frac{s}{(s^2+2s+2)(s^2-2s+2)}$$ I started off with: $$\frac{A(s^2-2s+2)}{s^2+2s+2} + \frac{B(s^2+2s+2)}{s^2-2s+2}$$ After that I get the following equations: $A+B = 0$; $-2A+2B =1$ Giving: $A=-B$, and…
Joe Goldiamond
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Partial fraction decomposition trouble

I'm trying to do a partial fraction expansion on $$Y = n \cdot \frac{e^{-pt_{0}}}{(p+n)(p^2+\omega^2)}$$ which gives $A(p^2+\omega^2)+(Bp+C)(p+n) = 1$ which implies $$A = -B$$ $$Bpn=0\rightarrow B=0\rightarrow A=0$$ which is incorrect. Any help…
DS08
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