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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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Finding the Partial Fractions Using Coefficients

(3-2x)/(x+1)(1-x)^2 For the above question when I find the partial fractions using the elimination method I got the following answer. (3-2x)/(x+1)(1-x)^2 = 5/4(x+1) - 5/4(1-x) + 1/2(1-x)^2 But When I tried to find the partial fractions using the…
Pegasus008
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How to do the PFE of a function whose polynomial does not easily expand?

I want to do a PFE of $$ y(x) = \frac{1}{ x^2 + \sqrt{2}x +1} $$ when I try to expand the polynomial I end up with $$ y(x) = \frac{1}{(x + \sqrt{\frac{1}{2}})( x + \sqrt{\frac{1}{2}}) + \frac{1}{2}} $$ I would know how to proceed if the…
neolith
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Partial fractions when numerator polynomial has greater degree than denominator

Consider the 'The Big example' as shown in this site, in it we are tasked to split: $$ \frac{x^2 +15}{(x^2+3)(x+3)^2} $$ They split it as: $$ \frac{A_1}{x+3} + \frac{A_2}{(x+3)^2} + \frac{Bx+C}{(x^2+3)}$$ Suppose, the degree of the numerator…
tryst with freedom
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partial fraction expansion with As+B and irreducible quadratic

My question is at the bottom. In an electrical engineering text on Laplace methods for solving electrical circuits a section on discussing p.f.e. the author says that when the denominator has an irreducible quadratic (does not factorize without…
relayman357
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Partial fractions with irreducible denominators above degree 2

In all courses and textbooks I have seen covering partial fractions, they only cover cases where the denominator is reducible to linear or quadratic factors. However, I can not see any reason why having an irreducible factor of degree greater than 2…
Nile Waldal
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Why do we need a $Bx$ term for partial fractions with irreducible quadratic factors?

Why is there Bx+c term when we try to split partial fraction with irreducible quadratic? Eg: $$\frac{1}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}$$ I think that splitting partial fraction is intuition when we directly put it as $\frac{A}{linear…
tryst with freedom
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Knowing the zero coefficients in partial fractions from the beginning

While solving partial fractions and getting the coefficients, can we know from the beginning that some coefficients will be zero? How? For example $$\frac{1}{S^2(S^2+4)}=\frac{A}{S^2}+\frac{B}{S}+\frac{CS+D}{S^2+4}$$ If we compute the coefficients,…
MCS
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Partial fraction decomposition corresponding numerator question

Consider the two decompositions: $\frac{x^2+1}{(x+2)(x-1)(x^2+x+1)}= \frac{A}{x+2} + \frac{B}{x-1} + \frac{Cx +D}{x^2+x+1}$, and $\frac{x^2+1}{(x-1)^3(x^2+1)^2}= \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{(x-1)^3} + \frac{Dx +E}{x^2+1}+ \frac{Fx…
Lopey Tall
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Polynomial division of $\frac{x^5+1}{x^4 + x^3 + x^2}$

Task: Do polynomial division for $$\frac{x^5+1}{x^4 + x^3 + x^2}$$ The question is really about finding all primitive functions via partial fractions, but I know how to do all the other steps without a problem once I have it in a form that I can use…
MathInferno
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Can anyone explain this problem on Partial fraction

(CAUTION- PLEASE DO NOT TAKE MY QUESTIONS VERY SERIOUSLY. I received a ban from asking questions,I don't know what to say really,I am just a student trying to learn,not a professional mathematician,so of course the questions could have been,well…
Rishabh Narula
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When doing partial fraction decomposition, how do I know when to put a constant to the numerator?

I am trying to find the partial fraction decomposition of $ (x^4+2)/(x^5+4x^3)$, which happens to be $(A/x) + (B/x^2) + C/(x^3) + ((Dx+E)/(x^2+6)$. My question is, why is there a " $+E$ " in the final term instead of being just $D$? How do I know…
Niko H
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What is the partial fraction of $\frac{x}{((x)^2+1)^2}$

I was trying to find the partial fraction of $$\frac{x}{(x^2+1)^2}$$ By the method of assuming $$\frac{x}{(x^2+1)^2}=\frac{(Ax+B)}{(x^2+1)} + \frac{(Cx+D)}{(x^2+1)^2} $$ But, my values for $A, B$ and $D$ are coming $0$. i.e. $$A=B=D=0$$ and…
Naved THE Sheikh
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Why doesn’t the equation "break" when we put a value $x$ while solving partial fractions?

Let’s suppose we want to decompose $\frac {9}{9-x^2}$ by partial fractions. I was taught that we proceed by writing $$ \frac{9}{9-x^2} = \frac{a}{3+x} + \frac{b}{3-x}, $$ where $a$ and $b$ are unknown constants. Clearing the fractions, we…
Vasu090
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How to separate an equation into partial fractions?

I am looking at a math question that has simplified this: into this: Can somebody explain the process for how this simplification was made? i.e. how does the denominator get broken down to those two terms?
Startec
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Partial Fraction: Already irreducible?

I have this partial fraction: $${3x+7}\over{(x-4)^2+25}$$ As far as I can tell, I do not think this can be decomposed. Is that a correct assumption? Sorry for the very short question, there isn't much work I could show, I think. Thank you!
JustHeavy
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