All Questions
Tagged with partial-fractions algebra-precalculus
157
questions
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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
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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
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109
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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:
$$
x - 1 = Ax^2 - 4Ax + 4A + Bx^2 ...
- 15
2
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3
answers
83
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Converting a proper fraction into partial fraction
For solving integration-related questions, a rational proper fraction of the form $\frac{px^{2}+qx+r}{(x-a)(x^{2}+bx+c)}$ is decomposed into the sum of the expressions,
$$\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
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191
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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.
$$\frac{1}{s^2(s^2+4)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+4}$$
$$s=0: ...
- 53
3
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2
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224
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Usual method of partial fractions decomposition over the reals seems to fail.
I assumed that it would be straightforward to find the partial fraction decomposition over the reals of the rational function $$f(x) = \frac{1}{(x^2 +1)^2}.$$ However, when I try what I thought would ...
- 188
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180
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Question on partial fractions; why numerator has to be one lower degree than denominator?
When decomposing into a partial fraction, why does the highest degree of the numerator have to be one lower than the numerator?
For example:
$\frac{x}{x^3-1} = \frac{a}{x-1} + \frac{bx+c}{x^2+x+1}$
...
- 1,190
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1
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230
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Express in partial fractions and expand the terms using binomial expansion up to $x^3$ [closed]
$$
\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}}
$$
Computing this i had gotten
\begin{equation}
2=A\left(1+x^{2}\right)+(B x+...
- 161
1
vote
1
answer
124
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Which integrating technique should I use?
Just some context: In the mathematical course, I have undertaken this year, I've just learnt how to integrate using partial fractions, substitution(not trig though, just a variable) and integrating ...
- 1,190
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2
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106
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Solution to $\int\frac{\ln(1+x^2)}{x^2}dx$
Q: $\int\frac{\ln(1+x^2)}{x^2}dx$
Here is my entire working:
So, overall, I started with the reverse product rule, then onto reverse chain rule and then tried to partial fraction, however, I still ...
- 1,190
2
votes
4
answers
85
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Finding the partial fractions decomposition of $\frac{9}{(1+2x)(2-x)^2} $
So this is basically my textbook work for my class, where we are practicing algebra with partial fractions.
I understand the basics of decomposition, but I do not understand how to do it when then the ...
- 85
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5
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97
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How to decompose $\frac{1}{(1 + x)(1 - x)^2}$ into partial fractions
Good Day.
I was trying to decompose $$\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)$$
...
- 1,858
2
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0
answers
299
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Are there applications of partial fraction decomposition ( of a rational function) outside integration problems?
I've been recently acquainted with a well known technique called " partial fraction decomposition" which allows, for example to express $\frac {x} {x^2-1}$ as $\frac {1}{2(x+1)} + \frac {1} ...
- 1,796
1
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1
answer
393
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Skepticism concerning Heaviside's "Cover-up Method" for partial fraction decomposition
I was reading this paper from MIT and it introduces Heaviside’s Cover-up Method for partial fraction decomposition. In that paper in Example $1$ it solves a problem using that method and just when ...
user964242
1
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1
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95
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Find the partial fraction for: $\frac{1}{(x^2+1)^2(x-1)}$
I want to find the partial fraction of: $$\frac{1}{(x^2+1)^2(x-1)}$$
Although I want this expressed through the format of finding the complex number that goes into $x$ so that I can get this equation ...
- 361
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2
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87
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Find the partial fraction: $\frac{x^2}{(x-1)^2(x-2)^2(x-3)}$
I'm trying to find the partial fraction for the following $\frac{x^2}{(x-1)^2(x-2)^2(x-3)}$ by the process of long-division, here's what I have tried:
Leaving $(x-3)$ as it is in the denominator and ...
- 361
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2
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90
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Partial Fraction Decomposition of $\frac{z+4}{(z+1+2i)(z+1-2i)}$
I have the fraction $\frac{z+4}{(z+1+2i)(z+1-2i)}$. I want to partial decompose this fraction, but I am not seeing how to do it. I know that the answer is a=$\frac{1}{2}$+$\frac{3i}{4}$ and b=$\frac{1}...
- 49
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3
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52
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Clarification on partial fraction expansion
I would like to use the cover up method for the following equation.
$$\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)}$$
I realize that you ...
- 23
1
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3
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284
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Why does this partial fraction decomposition work, even with division by $0$? [duplicate]
Here is how partial fraction decomposition works. First, take a fraction like $\frac{1}{n(n+1)}$. You can express this fraction as the sum $\frac{A}n + \frac{B}{n+1}$ for some constants $A$ and $B$. ...
- 2,685
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3
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Compute $\sum_{n=1}^\infty (\frac34)^n \frac{7n+32}{n(n+2)}$
Question: Compute $$\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
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5
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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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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
$$\frac{1}{x^4+4}=\frac{1}{(x-1+...
- 1,239
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1
answer
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Evaluation of a telescoping sum
I have come to a problem in a book on elementary mathematics that I don't understand the solution to. The problem has two parts :
a.) Factorize the expression $x^{4} + x^{2} + 1$
b.) Compute the ...
- 595
1
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1
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Decomposing fractions
I am not sure how these two terms are equal from this wiki:
$$
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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652
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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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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:
$$\frac{24}{(x-4)(x+3)}$$
$$=\...
- 327
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1
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597
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Why do some partial fractions have x or a variable in the numerator and others don't?
Why do rational expressions like $\left(\frac{1}{(x-2)^3}\right)$ do not have x in the numerator of the partial fraction but a rational expression like $\left(\frac{1}{(x^2+2x+3)^2}\right)$ does have ...
0
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0
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52
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Partial Fractions $\frac{2}{s^2+4} = \frac{As+B}{s^2+4}$
I have this relatively simple partial fraction
$$\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 ...
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