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 of derivative over polynomial

We know about standard partial fraction decomposition that says if $f$ and $g$ are two non-zero polynomials over a field $K$ with $g = \displaystyle\prod_{i=1}^k p_i^{n_i}$ being a product of irreducible polynomials, then we have unique polynomials…
Jeff
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Why Partial Fractions Decomposition works

Consider this partial fraction. $$\dfrac{x-25}{x^2+5x-24}=\dfrac{A}{x-3}+\dfrac{B}{x+8}$$ Multiply both sides by the quadratic $x^2+5x-24$. $$x-25=A(x+8)+B(x-3)$$ From here, I've seen many textbooks set $x=3$, solve for $A$, then set $x=-8$ and…
Anirudh Yamunan Govindarajan
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Partial Fraction decomposition when denominator is in $x^2 + a$ form

Why isn't the expansion of $$ \frac{s^3 - 2s^2 + 16s - 2}{(s^2+1)(s^2+16)} $$ in the form of $$ \frac{As+B}{s^2+1} + \frac{Cs+D}{s^2+16} $$ since (as I recall) the denominator is of square power and you should decompose it (in the numerator) until…
40pro
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Write down the partial fraction of $\frac{3x^2+1}{(x+1)\left(x-5\right)^2}$.

One such decomposition is: $$ \dfrac{A}{x+1} + \dfrac{B}{x-5} + \frac{C}{\left(x-5\right)^2}. $$ Why can't we decompose like below? $$ \dfrac{A}{x+1} + \dfrac{B}{(x-5)^2} $$
Haaarish
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Tricky partial fractions

Im trying to do partial fraction but cant seem to get it right, and the examiner have not showed how he did the partial fraction, just the answer. I want to partial fraction; $$\frac{1}{((s+\frac{1}{2})^2+\frac{3}{4})(s+\frac{1}{2})}$$ My solution…
uoiu
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Partial Fractions

I here would like to clear my doubt on the question below: $$\frac{1}{x(x-1)(x-2)}\;,$$ that is, we want to bring it into the form: $$\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-2}\;,$$ in which the unknown parameters are $A,B$, and $C$. Multiplying these…
Sugi
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Two partial fraction approaches, one is wrong, the other is right, why?

I want to do a partial fraction on \begin{equation} \frac{z}{(z-4)(z+\frac{1}{2})} \end{equation} Method one, which apparently is…
Luthier415Hz
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Where am I going wrong with this proof for partial fraction decomposition?

We have to prove that $ \frac{F}{G} $ can be written as $\frac{F_1}{G_1}+ \frac{F_2}{G_2}$ if $G=G_1G_2$ and $G_1,G_2$ are co-prime polynomials. This was the proof given in my textbook: $$There \space exists \space polynomials \space C,D \space…
Tatai
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A partial fraction decomposition of an inverse of a generic polynomial of an arbitrary order.

Let $d \ge 2 $ be an integer. Then let $\vec{n}:=\left(n_\xi \right)_{\xi=1}^d $ be integers (to be termed exponents) each of which is bigger or equal to one and then let $\left( a_\xi \right)_{\xi=1}^d $ and $\left( b_\xi \right)_{\xi=1}^d $ be…
Przemo
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factor $1/(x\pm y)$ as $f(x)g(y)$

Is it possible to rewrite $$\frac{1}{x\pm y}$$ as multiplication of an expression containing only $x$ and another containing only $y$ ($x$ and $y$ are real independent variables), i.e. $$\frac{1}{x\pm y}\stackrel{?}{=}f(x)g(y)$$ If it is possible,…
Masa
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I need help with a stupid webassign problem. It's about partial fraction

The question is following: Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficients. $$\frac{1}{x^2+x^4}$$ I know how to solve this. but the problem is I…
OneMoreGamble
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In partial fraction decomposition, do the terms in the denominator have to be irreducible polynomials?

Somewhere I read that in PFD the polynomials in the denominator should be irreducible. For curiosity I tried it with a reducible polynomial and got an answer out of it. I think I might have not understood what they meant. This the the fraction and…
Aaron de Windt
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partial fraction expansion...

I tried partial fraction expansion in this way, but it's just too cumbersome to solve in a test. Is there another way?
NK Yu
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Partial Fractions Decomposition of $\frac{25s}{(s^2+16)(s-3)(s+3)}$

So this is the problem.. $$ \frac{25s}{(s^2+16)(s-3)(s+3)} $$ So what I did was... $$ \frac {25s}{(s^2+16)(s-3)(s+3)} = \frac {A}{s^2+16}+\frac {B}{s-3}+\frac{C}{s+3} $$ then... $$\begin{align} 25s &= A(s-3)(s+3)+B(s^2+16)(s+3)+C(s^2+16)(s-3) \\ 25s…
William Purificacion MoteQ
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Prove that$\frac{x^{2^{k-1}}}{\left(1-x^{2^{k}}\right)}$= $\frac{1}{1-x^{2^{k-1}}}-\frac{1}{1-x^{2^{k}}}$

QuestionProve that$\frac{x^{2^{k-1}}}{\left(1-x^{2^{k}}\right)}$= $\frac{1}{1-x^{2^{k-1}}}-\frac{1}{1-x^{2^{k}}}$ My Approach R.H.S…
Mohan Sharma
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