All Questions
Tagged with partial-fractions algebra-precalculus
157
questions
39
votes
4
answers
14k
views
Integration by partial fractions; how and why does it work?
Could someone take me through the steps of decomposing
$$\frac{2x^2+11x}{x^2+11x+30}$$
into partial fractions?
More generally, how does one use partial fractions to compute integrals
$$\int\frac{P(...
- 1,089
25
votes
5
answers
5k
views
How does partial fraction decomposition avoid division by zero?
This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example:
$$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$
Multiplying ...
- 25.4k
18
votes
9
answers
8k
views
Derivation of the general forms of partial fractions
I'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take
(1) If the denominator of the fraction has linear factors:
$${5 \over {(x - 2)(x + 3)}} \equiv {A \...
- 7,177
13
votes
3
answers
4k
views
Why does partial fraction decomposition always work?
Say you have a function $p(x)/q(x)$ for some polynomials $p(x)$ and $q(x)$ and $p$ has a lower degree than $q$.
Say $q$ has degree three and $p$ has degree two. If you partially decompose it, you'll ...
- 3,951
11
votes
6
answers
4k
views
The existence of partial fraction decompositions
I'm sure you are all familiar with partial fraction decomposition, but I seem to be having trouble understanding the way it works. If we have a fraction f(x)/[g(x)h(x)], it seems only logical that it ...
- 1,606
9
votes
4
answers
2k
views
Easy ways to find partial fraction representation? (via a concrete example)
In a homework assignment (about generating functions) the students find themselves having to expand $\frac{3-7x+9x^{2}-3x^{3}}{\left(1-x\right)^{4}}$ intro partial fractions. Using some automated tool ...
- 19.4k
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
7
votes
5
answers
378
views
Evaluate $\int{\frac{4x}{(x^2-1)(x-1)}dx}$
So I know I need to use the partial fractions method to solve this integral. However when I split it as:
$$\frac{4x}{(x^2-1)(x-1)} = \frac{Ax + B}{x^2-1} + \frac{C}{x-1}$$
I find that I can't solve ...
- 165
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
7
votes
4
answers
502
views
Why can't partial fractions expansion be "normally" done is this case?
I've learned partial fractions but I couldn't really understand one thing. When we have a case when one of the factors has multiplicity $> 1$, we got to make a kind of "stairs". e.g.
$$\frac{1}{(x-...
- 7,461
7
votes
3
answers
1k
views
Partial fractions and using values not in domain
I'm studying partial fraction decomposition of rational expression. In this video the guy decompose this rational expression:
$$ \frac{3x-8}{x^2-4x-5}$$
this becomes:
$$\frac{3x-8}{(x-5)(x+1)} = \...
- 813
7
votes
2
answers
816
views
The partial fraction decomposition of $\dfrac{x}{x^3-1}$
I was trying to decompose $\dfrac{x}{x^3-1}$ into Partial Fractions. I tried the following:
$$\dfrac{x}{(x-1)(x^2+x+1)}=\dfrac{A}{(x-1)}+\dfrac{B}{(x^2+x+1)}$$
$$\Longrightarrow A(x^2+x+1)+B(x-1)=x$$
...
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
7
votes
2
answers
2k
views
Integration of high order fraction; explanation of method.
I understand the cases where the order is less or equal to 3 (example where it is three we split numerator with A,B,C), but in this case(Example 8) I do not see why we split the numerator with A, Bx+C,...
- 2,130
6
votes
2
answers
531
views
Is this a valid partial fraction decomposition?
Write $\dfrac{4x+1}{x^2 - x - 2}$ using partial fractions.
$$ \frac{4x+1}{x^2 - x - 2} = \frac{4x+1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x-2} = \frac{A(x-2)+B(x+1)}{(x+1)(x-2)}$$
$$4x+1 = A(x-2)+B(...
- 13.6k