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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The logic behind partial fraction decomposition

In the general case of any function would be interesting but my question is concerning the general case of polynomials with integer powers. I can use the method of partial fractions in the simple case required for an introductory course on…
R R
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Partial Fractions: Why does this shortcut method work?

Suppose I want to resolve $1/{(n(n+1))}$ into a sum of partial fractions. I solve this by letting $1/{(n(n+1))} = {a/n} + {b/(n+1)}$ and then solving for $a$ and $b$, which in this case gives $a=1$ and $b=-1$. But I learnt about a shortcut method.…
Ram Keswani
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Question about partial fractions with irreducible quadratic factors

Given this rational function: $$\frac{-4x^4-2x^3-26x^2-8x-44}{(x+1)(x^2 +3)^2}$$ The decomposition would look like this: $$\frac{A}{x+1} + \frac{Bx+C}{(x^2+3)} + \frac{Dx+E}{(x^2+3)^2}$$ And the final answer would be: $$\frac{-4}{x+1} -…
user1091990
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What is the Mathematical Property that justifies equating coefficients while solving partial fractions?

The McGraw Hill PreCaculus Textbook gives several good examples of solving partial fractions, and they justify all but one step with established mathematical properties. In the 4th step of Example 1, when going from: $$1x + 13 =…
Norman William
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Fast partial-fraction decomposition

I'm studying Laplace transformations for my differential equations class and typically there's a partial fraction decomposition involved, which can be very long and demanding for calculations by hand, if done the standard way. I am aware of some of…
gpo
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Partial fractions decomposition of ${\frac{2x}{(x+2)^2}}$

Express in partial fraction form: $\displaystyle{\frac{2x}{(x+2)^2}}$ I think is $\displaystyle{\frac{2x}{(x+2)^{2}} = \frac{A}{x+2}+\frac{B}{(x+2)^2}}$ However when identifying $A$ and $B$, I'm not sure how to calculate A. E.g. $$2x = A\cdot (x+2)…
Paddington
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Ramanujan's partial fraction decomposition of $\frac{1}{(x^2+a^2)\cdots(x^2+(a+n)^2)}$.

\begin{align*} \frac{1}{(x^2+a^2)\cdots(x^2+(a+n)^2)} &= \frac{2\Gamma(2a)}{\Gamma(n)\Gamma(2a+n)}\left(\frac{a}{x^2+a^2}-\frac{2a}{1!}\frac{n-1}{n+2a}\frac{a+1}{x^2+(a+1)^2}\right. \\ & \qquad +…
Meow
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Not Understanding the Reason Behind This Type of Partial Fraction Decomposition

So I have to teach partial fractions to some undergrads this week and I am looking for someone to help me understand why we solve partial fractions in cases like the following: Suppose we have the following problem (Problem 1): $$\int…
user111707
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Partial fraction decomposition help

In a text that I am reading, they state that the following partial fraction ($r$ fixed) expansion is "readily computed": $$f(z) = \frac{z^r}{(1-z)(1-2z)(1-3z)\cdots (1-rz)} = \frac{1}{r!} \sum_{j=0}^r \binom rj \frac{(-1)^{r-j}}{1-jz}$$ I know how…
angryavian
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partial fraction for complex roots

While solving Laplace transform using Partial fraction expansion. I have confusion in solving partial fraction for complex roots. I have this equation $$ \frac {2s^2+5s+12} {(s^2+2s+10)(s+2)}$$ Please anyone help to tell me to understand the steps…
Shinning Eyes
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Partial fraction expansion question

I have to integrate following expression (but integration is not the problem): $$\frac{x^2+3x-2}{(x-1)(x^2+x+1)^2}$$ It is pretty obvious that: $$\frac{x^2+3x-2}{(x-1)(x^2+x+1)^2}=\frac{A}{x-1} + \frac{Mx+N}{x^2+x+1} + \frac{Px+Q}{(x^2+x+1)^2}$$ The…
King Crimson
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Partial fraction $\frac{x}{(1+x)^2}$

How do you compute $\dfrac{x}{(1+x)^2}$ using partial fractions? The reason I ask is because when I try to solve it I keep getting an impossible $A, B$. $A(1+x) + B(1+x) = x$ $A + Ax + B + Bx = x$ $(A+B)x = 1$ $(A+B) = 0$ However a practice problem…
Billy Thompson
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unresolved partial fraction decomposition

I'm having some trouble doing this partial fraction decomposition: $$\frac{1}{t^3-2t+1}$$ using Ruffini rule i get: $$\frac{1}{t^3-2t+1}= \frac{1}{(t-1)(t^2+t-1)}$$ i would like to decompose the previous result into partial fraction. I did in this…
Duccio Bertieri
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Partial fraction with a constant as numerator

I am trying to express this as partial fraction: $$\frac{1}{(x+1)(x^2+2x+2)}$$ I have a similar exaple that has $5x$ as numerator, it is easy to understand. I do not know what to do with 1 in the numerator, how to solve it?!
Dumbo
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Partial fraction in two variables

I want to find a partial fraction expansion for the following: ($b$ is a constant) $$\frac{1}{(b^2+x^2)(b^2+y^2)}$$ As there are two variables, I am unsure what form the decomposition should be of. I am looking for one where the coefficients of the…
Comp_Warrior
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