All Questions
12
questions
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
1
answer
64
views
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
0
votes
2
answers
61
views
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
$$
\int x+3\,dx + \int\frac{10x-8}{(...
- 115
0
votes
1
answer
82
views
About a statement of partial fraction in an answer
I'm reading this answer of The logic behind partial fraction decomposition, I think my question is too basic and not directly related to the answer so I don't comment there. I don't understand why:
...
- 229
0
votes
1
answer
64
views
Polynom and decomposition
I need to know I can decompose into simple elements
$$\frac X{(X+1)^4 (X^2 +1)}$$
What is the easiest way?
2
votes
2
answers
501
views
What is this change of variable in a polynomial?
The autocovariance generating function, $\gamma(z)$, is defined as the $z$-transform of the autocovariance function, $\gamma_\tau$:
$$
\gamma(z) = \gamma_0 + \gamma_1(z+z^{-1}) + \gamma_2(z^2+z^{-2}) +...
- 167
2
votes
1
answer
180
views
1967 HSC 4 unit Mathematics Question 2
Screenshot from the examination paper
[...asking about partial fraction decomposition of $$\frac{1 - abx^2}{(1-ax)(1-bx)} $$ and related formulas...]
This question is taken from the New South Wales ...
- 41
0
votes
1
answer
64
views
Algebraically solve for reciprocal of result of polynomial long division
How can I show the following relationship algebraically?
$$\frac 1 {4-\frac2 x}=\frac1 4+\frac1 {8x-4}\ \ \ \ \forall x\ne\frac1 2$$
I tried to multiply by the conjuage
$$\left(\frac 1 {4-\frac2 x}\...
- 1,381
1
vote
1
answer
249
views
When are we allowed to match coefficients?
Related to this answer: Find k in $(1−2k)x^2−(3k+4)x+2=0$ given facts about the roots.
In the partial fraction decomposition of $\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$, we have:
$0x + ...
- 13.7k
25
votes
5
answers
5k
views
How does partial fraction decomposition avoid division by zero?
This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example:
$$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$
Multiplying ...
- 25.4k
7
votes
2
answers
565
views
Partial fraction of $\prod_{j=1}^{N}\frac{1}{(x-a_{j})^{n_{j}}}$
Does anybody know the partial fraction decomposition of
$$
\prod_{j=1}^{N}\ \frac{1}{(x-a_{j})^{n_{j}}}
$$
with all $a_{j}$ different and $n_j$ positive integers? I know you can get it with the ...
- 2,624
9
votes
3
answers
2k
views
Partial fraction of $\frac{x^n}{(1-x)(1-2x)(1-3x)\ldots(1-nx)}$
How should you break $\displaystyle \frac{x^n}{(1-x)(1-2x)(1-3x)\ldots(1-nx)}$ into partial fractions?
- 2,039