All Questions
Tagged with partial-fractions algebra-precalculus
157
questions
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30
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Partial fractions with a repeated factor [duplicate]
I am looking to find a derivation, or a intuitive explanation as for why a partial fraction with a repeated factor needs to include a factor in the expansion for each power possible. How does one ...
- 855
2
votes
2
answers
245
views
How can i do the following partial decomposition?
I need to prove that:
$$
\frac{1}{(x-a)(x-b)} = \frac{1}{(b-a)(x-b)}- \frac{1}{(b-a)(x-a)},
$$
and I must note that I need to go from the left expression to the right (because of the exercise).
So, I ...
-2
votes
2
answers
109
views
I found an interesting question but I keep getting stuck in a loop. [closed]
Find all values of A, B, C and C such that:
$$
\frac{x-1}{(x-1)(x-2)(x-2)} = \frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{(x-2)^2}
$$
I keep getting into a loop in which:
$$
x - 1 = Ax^2 - 4Ax + 4A + Bx^2 ...
- 15
2
votes
3
answers
83
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Converting a proper fraction into partial fraction
For solving integration-related questions, a rational proper fraction of the form $\frac{px^{2}+qx+r}{(x-a)(x^{2}+bx+c)}$ is decomposed into the sum of the expressions,
$$\frac{A}{x-a} + \frac{Bx+C}{x^...
- 535
1
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2
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62
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Contour integral over function $P(x)/Q(x)$: $P(x) = 1$ and $Q(x)$ can be broken into linear factors
a. Let $z_1,z_2,...,z_n$ be distinct complex numbers $(n \geq 2)$. Show that in the partial fractions decomposition
\begin{equation}
\frac{1}{(z-z_1)(z-z_2)\cdots(z-z_n)} = \frac{A_1}{z-z_1}+\frac{A_2}...
- 142
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3
answers
190
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Partial Fractions Decomposition-Unsure Which Method To Use When
So I was working on this problem and could not use the cover up method to solve it. I was getting the wrong answer.
Find B.
$$\frac{1}{s^2(s^2+4)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+4}$$
$$s=0: ...
- 53
3
votes
2
answers
224
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Usual method of partial fractions decomposition over the reals seems to fail.
I assumed that it would be straightforward to find the partial fraction decomposition over the reals of the rational function $$f(x) = \frac{1}{(x^2 +1)^2}.$$ However, when I try what I thought would ...
- 188
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180
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Question on partial fractions; why numerator has to be one lower degree than denominator?
When decomposing into a partial fraction, why does the highest degree of the numerator have to be one lower than the numerator?
For example:
$\frac{x}{x^3-1} = \frac{a}{x-1} + \frac{bx+c}{x^2+x+1}$
...
- 1,190
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1
answer
230
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Express in partial fractions and expand the terms using binomial expansion up to $x^3$ [closed]
$$
\frac{2}{(1-x)\left(1+x^{2}\right)}
$$
This is then split into partial fractions
$$
\frac{A}{1-x}+\frac{B x+C}{1+x^{2}}
$$
Computing this i had gotten
\begin{equation}
2=A\left(1+x^{2}\right)+(B x+...
- 161
1
vote
1
answer
124
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Which integrating technique should I use?
Just some context: In the mathematical course, I have undertaken this year, I've just learnt how to integrate using partial fractions, substitution(not trig though, just a variable) and integrating ...
- 1,190
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2
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106
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Solution to $\int\frac{\ln(1+x^2)}{x^2}dx$
Q: $\int\frac{\ln(1+x^2)}{x^2}dx$
Here is my entire working:
So, overall, I started with the reverse product rule, then onto reverse chain rule and then tried to partial fraction, however, I still ...
- 1,190
2
votes
4
answers
85
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Finding the partial fractions decomposition of $\frac{9}{(1+2x)(2-x)^2} $
So this is basically my textbook work for my class, where we are practicing algebra with partial fractions.
I understand the basics of decomposition, but I do not understand how to do it when then the ...
- 85
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5
answers
97
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How to decompose $\frac{1}{(1 + x)(1 - x)^2}$ into partial fractions
Good Day.
I was trying to decompose $$\frac{1}{(1 + x)(1 - x)^2}$$ into partial fractions.
$$\frac{1}{(1 + x)(1 - x)^2} = \frac{A}{1 + x} + \frac{B}{(1 - x)^2}$$
$$1 = A(1 - x)^ 2 + B(1 + x)$$
...
- 1,858
2
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0
answers
299
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Are there applications of partial fraction decomposition ( of a rational function) outside integration problems?
I've been recently acquainted with a well known technique called " partial fraction decomposition" which allows, for example to express $\frac {x} {x^2-1}$ as $\frac {1}{2(x+1)} + \frac {1} ...
- 1,796
1
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1
answer
393
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Skepticism concerning Heaviside's "Cover-up Method" for partial fraction decomposition
I was reading this paper from MIT and it introduces Heaviside’s Cover-up Method for partial fraction decomposition. In that paper in Example $1$ it solves a problem using that method and just when ...
user964242
1
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1
answer
95
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Find the partial fraction for: $\frac{1}{(x^2+1)^2(x-1)}$
I want to find the partial fraction of: $$\frac{1}{(x^2+1)^2(x-1)}$$
Although I want this expressed through the format of finding the complex number that goes into $x$ so that I can get this equation ...
- 361
1
vote
2
answers
87
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Find the partial fraction: $\frac{x^2}{(x-1)^2(x-2)^2(x-3)}$
I'm trying to find the partial fraction for the following $\frac{x^2}{(x-1)^2(x-2)^2(x-3)}$ by the process of long-division, here's what I have tried:
Leaving $(x-3)$ as it is in the denominator and ...
- 361
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2
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90
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Partial Fraction Decomposition of $\frac{z+4}{(z+1+2i)(z+1-2i)}$
I have the fraction $\frac{z+4}{(z+1+2i)(z+1-2i)}$. I want to partial decompose this fraction, but I am not seeing how to do it. I know that the answer is a=$\frac{1}{2}$+$\frac{3i}{4}$ and b=$\frac{1}...
- 49
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votes
3
answers
52
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Clarification on partial fraction expansion
I would like to use the cover up method for the following equation.
$$\frac{1}{x^2(x+1.79)}$$
and it breaks down into
$$\frac{A}{x}\quad\frac{B}{x^2}\quad\frac{C}{(x + 1.79)}$$
I realize that you ...
- 23
1
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3
answers
284
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Why does this partial fraction decomposition work, even with division by $0$? [duplicate]
Here is how partial fraction decomposition works. First, take a fraction like $\frac{1}{n(n+1)}$. You can express this fraction as the sum $\frac{A}n + \frac{B}{n+1}$ for some constants $A$ and $B$. ...
- 2,685
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3
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82
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Compute $\sum_{n=1}^\infty (\frac34)^n \frac{7n+32}{n(n+2)}$
Question: Compute $$\sum_{n=1}^\infty \left(\frac34\right)^n \frac{7n+32}{n(n+2)}$$
I first did the partial fraction decomposition into:
$$\sum_{n=1}^\infty \left(\frac34\right)^n \frac{16}n - \sum_{...
- 157
1
vote
5
answers
75
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how to solve this into partial fractions
I'm having a bit of a hard time putting this into partial fractions:
$$\frac{10}{x^2+2x+1+\pi^2}.$$
I know that the roots of the denominator are $-1 \pm i\pi$, but I dont know how to proceed on puting ...
- 17
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2
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179
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Finding nth derivative of $\frac{1}{x^4+4}$
I am supposed to find the nth order derivative of:
$$\frac{1}{x^4+4}$$
I tried to resolve into partial fractions. But it didn't work out for me.
Edit- where I am stuck
$$\frac{1}{x^4+4}=\frac{1}{(x-1+...
- 1,239
1
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1
answer
64
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Evaluation of a telescoping sum
I have come to a problem in a book on elementary mathematics that I don't understand the solution to. The problem has two parts :
a.) Factorize the expression $x^{4} + x^{2} + 1$
b.) Compute the ...
- 595
1
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1
answer
53
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Decomposing fractions
I am not sure how these two terms are equal from this wiki:
$$
I(X,Y) = KL(p(x,y) || p(x)p(y)) = \sum_{x,y}p(x,y) \log[\frac{p(x,y)}{p(x)q(y)}] \\
I(X,Y) = \sum_{x,y}p(x,y) \log[\frac{p(x,y)}{p(y)}...
- 736
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1
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131
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Partial Fractions With Repeated Quadratics
I'm told that given a function $f(x)=\frac{P(x)}{Q(x)}$, if $\deg(P)>\deg(Q)$ then $f$ is improper, which makes sense when I think of real numbers like $5/2$. And in this case we would have to do ...
- 2,072
4
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1
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651
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Partial fractions zero coefficients
In partial fractions, there can be terms in the coefficients in the partial form that turn out to be zero. One case of this is when the variable is "linear" in a power of $x$, for example, ...
- 701
3
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2
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93
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Binomial Expansion Of $\frac{24}{(x-4)(x+3)}$
Can somebody help me expand $\frac{24}{(x-4)(x+3)}$ by splitting it in partial fractions first and then using the general binomial theorem?
This is what I've done so far:
$$\frac{24}{(x-4)(x+3)}$$
$$=\...
- 327
2
votes
1
answer
597
views
Why do some partial fractions have x or a variable in the numerator and others don't?
Why do rational expressions like $\left(\frac{1}{(x-2)^3}\right)$ do not have x in the numerator of the partial fraction but a rational expression like $\left(\frac{1}{(x^2+2x+3)^2}\right)$ does have ...
0
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0
answers
52
views
Partial Fractions $\frac{2}{s^2+4} = \frac{As+B}{s^2+4}$
I have this relatively simple partial fraction
$$\frac{2}{s^2+4} = \frac{As+B}{s^2+4}$$
I multiply each side by $(s^2+4)$ and all that remains is $2 = As + B$. Then can I match the coefficients up ...
- 2,409
1
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1
answer
81
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Arbitrarily long decomposition into partial fractions.
$$\int \frac{dx}{x(x+p)(x+2p)(x+3p)...(x+(n-1)p)}= ?$$
I'm trying to solve this integral, and as I usually do in these cases, I break the expression into partial fractions, but I find this case ...
1
vote
1
answer
209
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Reasoning behind the resolution of partial fractions when denominator is the product of linear factors where some of them are repeating
The following text is from Mathematics for Class XII by Dr. R.D.Sharma, chapter "Indefinite Integrals", topic "Integration of Rational Algebraic Functions by using Partial Fractions&...
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Partial fraction decomposition of $\frac{1}{x^a(x+c)^b}$
We have the partial fraction decomposition
$$\frac{1}{x^a(x+c)^b}=\frac{d_a}{x^a}+\frac{d_{a-1}}{x^{a-1}}+...+\frac{d_{1}}{x}+\frac{e_b}{(x+c)^b}+\frac{e_{b-1}}{(x+c)^{b-1}}+...+\frac{e_{1}}{x+c},$$
...
- 351
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2
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Integration of rational of polynomials
I want to evaluate the indefinite integral for:
$$
\int\frac{x^3+3x−2}{x^2-3x+2}dx,\quad \text{for } x>2
$$
I did long division and factoring, simplifying it to
$$
\int x+3\,dx + \int\frac{10x-8}{(...
- 115
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1
answer
53
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How does this imply $x = z^2$?
I read that given the equation:
$$
x + \frac{1}{x} = z^2 + \frac{1}{z^2}
$$
, we can imply that $x=z^2$.
But that $x=z^2$ is not obvious to me... I know we can match the variables on the left hand ...
- 735
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votes
1
answer
94
views
Partial fractions Integration - Distributing Coefficients
Given the following Integral
$\int \frac{2x^3+ 2x^2+ 2x+ 1}{x^2 (x^2+1)}$
I would expand my fractions like the following
$\frac{A}{x} + \frac{B}{x^2}+\frac{Cx+D}{x^2+1}$
When I look at the ...
- 3
2
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4
answers
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How to get the value of $A + B ?$
I have this statement:
If $\frac{x+6}{x^2-x-6} = \frac{A}{x-3} + \frac{B}{x+2}$, what is the value of $A+B$ ?
My attempt was:
$\frac{x+6}{(x-3)(x+2)} = \frac{A(x+2) + B(x-3)}{(x-3)(x+2)}$
$x+6=(...
- 3,173
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1
answer
200
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Resolve into partial fractions $(x^2 + 3x - 5)/[(2x - 7)(x^2 + 3)^2]$
Resolve into partial fractions $\frac{x^2 + 3x - 5}{(2x - 7)(x^2 + 3)^3}$
The question has to do with the denominator being one linear and a repeated quadratic factor. Although, I am familiar with ...
1
vote
3
answers
253
views
Integrating quadratics in denominator
I'm following a book on Calculus that introduces partial fraction expansion. They discuss common outcomes of the partial fraction expansion, for example that we are left with an integral of the form:
...
- 2,493
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votes
3
answers
51
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Partial fraction expansion inquiry
How can I expand $\frac{a + 5}{(a^2-1)(a+2)}$ so that the sum of partial fractions is equal to $\frac{1}{a-1} - \frac{2}{a+1} + \frac{1}{a+2}$ ?
Thanks in advance!
- 631
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votes
2
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60
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Equations with 3 unknowns [closed]
I have this equation/problem:
Find the value of A and B
if 2/(x−5)(x+3) = A /(x−5) + B/(x+3)
How can I solve/approach this?
Thank you for your advice.
Regards
Lisa
- 43
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votes
6
answers
222
views
Why partial fraction decomposition of $\frac{1}{s^2(s+2)}$ is $\frac{A}{s}+\frac{B}{s^2}+\frac{C}{(s+2)}$?
Can someone please explain why: $$\frac{1}{s^2(s+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{(s+2)}$$
And not:$$\frac{1}{s^2(s+2)}=\frac{A}{s^2}+\frac{B}{(s+2)}$$
I'm a bit confused where the extra s ...
- 533
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votes
1
answer
41
views
How would I apply partial fraction expansion to this expression?
$$\displaystyle\frac{1}{X\bigg(1-\dfrac{X}{Y}\bigg)\bigg(\dfrac{X}{Z}-1\bigg)}$$
I want it in the form $$\frac{A}{X} + \frac{B}{(1-\dfrac{X}{Y})}+\frac{C}{(\dfrac{X}{Z}-1)}$$where I am required to ...
0
votes
1
answer
912
views
Confusion about "picking values of $x$", partial fraction decomposition
One of the methods of decomposing a fraction and finding the constants, $A, B, C,$ etc., is to "pick values of $x$". For example, to find $A$ and $B$ of $${{3x}\over(x-1)(x+2)} = {A\over(x-1)} + {B\...
- 945
0
votes
0
answers
70
views
Mistake in the computation via partial fractions
This is a computation made in Titchmash's Introduction to Zeta functions. I was trying to reverse the computation. However, I kept missing factors.
Consider $\frac{1- xyz^2}{(1-z)(1-xz)(1-yz)(1-xyz)}...
- 8,550
0
votes
1
answer
82
views
About a statement of partial fraction in an answer
I'm reading this answer of The logic behind partial fraction decomposition, I think my question is too basic and not directly related to the answer so I don't comment there. I don't understand why:
...
- 229
2
votes
4
answers
79
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Show $\frac{1}{1-\frac{x}{3-x}+\frac{x}{4-x}}$ is equivalent to $1+\frac{1}{2}\left(\frac{1}{2-x}-\frac{3}{6-x}\right)$ for $\lvert x\rvert < 1$
I have been asked to show that $$\frac{1}{1-\frac{x}{3-x}+\frac{x}{4-x}}$$ is equivalent to writing $$1+\frac{1}{2}\left(\frac{1}{2-x}-\frac{3}{6-x}\right)$$
From here I just tried to work out the ...
- 29
0
votes
3
answers
48
views
How do we make $A(x+1)(x^2+4x+5)+B(x^2...$ to be equal to $2x^2$??
$$\frac{2x^2}{(x+1)^2(x^2+4x+5)}$$
$$2x^2=A(x+1)(x^2+4x+5)+B(x^2+4x+5)+(Cx+D)(x+1)^2$$
We can get $B=1$ if we put $-1$ for $x$.
But I don't know how can we solve for $A$, $C$ and $D$ since we can't ...
1
vote
3
answers
121
views
Partial Fraction problem solution deviates from the Rule
Question:
Compute $\displaystyle \int\frac{x^2+1}{(x^2+2)(x+1)} \, dx$
My Approach:
As per my knowledge this integral can be divided in partial Fraction of form $\dfrac{Ax+B}{x^2+px+q}$ and then do ...
user517784
0
votes
1
answer
48
views
Find the Partial Fraction Decomposition
$\frac{2x^5+3x^4-3x^3-2x^2+x}{2x^2+5x+2}$, I am not sure as to where to start with this one; I have already done the factoring process of the denominator but not sure how to continue the algebraic ...
