All Questions
18
questions
1
vote
0
answers
30
views
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 ...
0
votes
0
answers
180
views
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
1
vote
1
answer
393
views
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
vote
5
answers
75
views
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
0
votes
1
answer
53
views
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
0
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 ...
2
votes
4
answers
79
views
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
1
vote
0
answers
54
views
A quick way for decomposing fractions
The complete method for decomposing fractions is obvious . For example $y = \frac{2x+1}{(x-1)(x+3)} = \frac{a}{x-1} + \frac{b}{x+3} = \frac{a(x+3) + b(x-1)}{(x-1)(x+3)} \Rightarrow $$
\left\{
\begin{...
- 4,359
0
votes
3
answers
102
views
partial fraction decomposition of $\frac{k^4}{(a\, k^3-1)^2}$
I have to perform complex partial fraction decomposition of the following term:
$$\frac{k^4}{(a \, k^3-1)^2}$$
where $a$ is a real positive number.
and I would like to know if it is possible to ...
- 563
0
votes
3
answers
212
views
Partial Fractions help!?
$A + C = 0$
$-4A + B - 8C + D = 1$
$3A + 16C - 8D = -29$
$-12A + 3B + 16D = 5$
How do I equate the coefficients? Please provide steps an an explanation.
- 33
7
votes
3
answers
226
views
Is $\frac{a^4}{(b-a)(c-a)}+\frac{b^4}{(c-b)(a-b)}+\frac{c^4}{(a-c)(b-c)} $ always an integer?
In a textbook I found the rather strange identity:
$$ \frac{2^4}{(5-2)(3-2)}+\frac{3^4}{(5-3)(3-2)}+\frac{5^4}{(5-3)(5-2)}= \frac{414}{6}=69 $$
just kind if out of nowhere and I wonder if it ...
- 24.5k
0
votes
0
answers
43
views
substract functions
Im trying to substract a set of functions
$$f(x)-g(x)-h(x)-i(x)$$
where
$$ f(x)=\frac{1}{(x+3)(x+4)^2(x+5)^3} $$
$$ g(x)=\frac{-\frac{1}{2}}{(x+5)^3} $$
$$ h(x)=\frac{-1}{(x+4)^2} $$
$$ i(x)=\...
- 151
1
vote
1
answer
559
views
Partial Fraction Expansion of $12\frac{x^3+4}{(x^2-1)(x^2+3x+2)}$
Find the vector $(A,B,C,D)$ if $A$, $B$, $C$, and $D$ are the coefficients of the partial fractions expansion of
$$12\frac{x^3+4}{(x^2-1)(x^2+3x+2)} = \frac{A}{x-1} + \frac{B}{x+2} + \frac{C}{x+1} + \...
- 1,283
4
votes
4
answers
400
views
Write a formula as a sum of fractions with constant numerators
I'm supposed to write this formula:
$$\frac {9a + 43}{a^2 + 9a + 20}$$
As a sum of fractions with constant numerators as:
$$\frac {7}{a+5} + \frac {2}{a+4}$$
The first step is of course:
$$\frac {...
- 211