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, trivial mistake

I'm doing a trivial mistake for sure, but I struggling to find it... I have: $$ \frac{160}{4s^2+4.8s+4} = \frac{160}{(s+0.6-0.8i)(s+0.6+0.8i)} = \frac{K_1}{s+0.6-0.8i} +\frac{K_1^*}{s+0.6+0.8i} $$$$ K_1= -100i $$ but if I first multiply by…
Miguel Horta
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Why is this true with partial fractions?

Suppose you have a fraction like $\frac{x^2+2x}{x(x-2)^2}$. You can rewrite that as $$\frac{x^2+2x}{x(x-2)^2}=\frac{A}{x}+\frac{B}{x-2}+\frac{C}{(x-2)^2}.$$ Why is it that you must put the linear version and then the quadratic too? Why isn't it just…
King Squirrel
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Show that f(x)=1-x+5x^2

$6$. Let $$f(x) =\frac{9x^2 + 4}{(2x + 1)(x − 2)^2}$$ (i) Express $f(x)$ in partial fractions. (ii) Show that, when $x$ is sufficiently small for $x^3$ and higher powers to be neglected, $$f(x) = 1 − x + 5x^2.$$ The answer for the first part is…
user133028
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Partial Fraction Decomposition equating coefficiants

$$\frac{x^2 + x + 1}{(2 x + 1) (x^2 + 1)}$$ I'm having issues with coming with up with the coefficients for this....my conclusion is $1=A+2b \\ 1=2c+b \\ 1=a+c$ am i on the right track? and I'm a little stumped on how to solve the unknowns.
vic
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Partial Fraction Decomposition with x and y in numerator and denominator

I am trying to find the partial fraction decomposition of $$\dfrac{x^2+y^2}{x^2-y^2}$$ I got it down to $x^2+y^2 = (A+B)x+(A-B)y$, but I cannot simplify it down anymore. Is this the wrong approach or am I not seeing something?
Gary H
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Extra Square in Partial Fraction

So I understand why this is true: $$ \frac{x}{(x+2)(x+1)} = \frac{A}{x+2} + \frac{B}{x+1} $$ But there's a special rule in partial fraction that I just couldn't get it. When you have a term that is squared, I must add another fraction with the term…
Derek 朕會功夫
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partial fraction i can't figure it out

$$X(s)=\frac{4s+1}{s(2s^2+2s+1)}$$ Please somebody show me how to factor this out step by step so I can take the inverse laplace using tables. I already have the answer: $1+(3e^{0.5t})\sin0.5t-(e^{0.5t})\cos0.5t$ The partial fraction is difficult…
user122415
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Partial Fraction Question, Quite Basic

Express $$\frac{2x}{(x^2 + 1)(x + 1)^2} = \frac{A_1 x + A_2}{(x^2+1)} + \frac{B}{(x+1)^2} + \frac{C}{x+1}$$ in partial fractions. I know I have to decompose it into three fractions with numerators $(x^2 + 1), (x + 1)$ and $(x + 1)^2$.
Vladimir Nabokov
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Basic Questions Concerning Partial Fractions

Express $\displaystyle \frac x{(x^2+1)(1+x)}$ in partial fractions, I'm having a little difficulty with this. Can you tell me what the two numerators should be please.
Vladimir Nabokov
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Partial fraction of 1 over (x^2+1)^2

Its been years since I solved PF. Now I am having hard time solving this partial fraction $$ F = \frac{1}{\left( x^2+1\right)^2} $$ I proceeded with(Is this right ?) $$ \frac{1}{\left( x^2+1\right)^2} = \frac{A}{\left( x+\iota\right)} +…
moki
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Polynomials and Partial Fractions Decomposition

In a bid to check my understanding I have this case: Decompose: $ \dfrac{x^4-8}{x^2+2x}$ Here I see it is an improper factions- degree of $x$ is higher on the numerator than on denominator. Using long division to divide, I get $$x^2-2x+4+…
Sylvester
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Can someone help me with partial fraction expansion, please?

$$\frac{1}{(1-x^a)(1-x^b)}=\frac{A}{(1-x)^2}+\frac{B}{(1-x)}+\sum_{r^a=1}^{ ‎ }\frac{C_r}{(1-x/r)}+\sum_{t^b=1}^{ ‎ }\frac{D_t}{(1-x/t)}$$ $(t,r\neq1; A, B, C, D$ are real numbers) How did the author know that he need to separate the case r,t=1? I…
ChemistryLearner
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Algebraic manipulation to turn a fraction into "half a difference" of fractions

I'm reading the resolution of a solved exercise about Telescoping Series in a "Finite Math" course I'm attending. The resolution has a step that is not explained because it's left as an exercise to the reader: $$ \cdots = \frac{1}{(2k + 3)(2(k + 1)…
ricmarques
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Integrate reciprocal of polynomial (coefficient of partial fraction decomposition)

I am working on the integral of the reciprocal of polynomial such that $\int{\frac{x^{q-1}dx}{x^{p}-ax^{q}+1}}$ where $p$ and $q$ are coprime integers. I tried to solve it by partial fraction decomposition as…
Summer
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How do I solve a partial fractions question when $x$ can’t equal any solutions?

How do I solve $A(i)$ and $A(ii)$ when $x$ can't equal $-2/5$ or $1/2$? The only method I’m aware of for partial fractions requires these values to be subbed in and I’m confused. Sorry if the image is obstructed by my scruffy writing. If it’s…
Lorcan Kearney
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