All Questions
Tagged with partial-fractions algebra-precalculus
157
questions
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If $S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}$, then calculate $14S$.
If $$S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}\,$$ find the value of $14S$.
The question can be simplified to:
Find $S=\sum\limits_{k=1}^n\,t_k$ if $t_n=\dfrac{n}{1+n^2+n^...
- 2,642
7
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3
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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
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1
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64
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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?
0
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0
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43
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substract functions
Im trying to substract a set of functions
$$f(x)-g(x)-h(x)-i(x)$$
where
$$ f(x)=\frac{1}{(x+3)(x+4)^2(x+5)^3} $$
$$ g(x)=\frac{-\frac{1}{2}}{(x+5)^3} $$
$$ h(x)=\frac{-1}{(x+4)^2} $$
$$ i(x)=\...
- 151
2
votes
2
answers
501
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What is this change of variable in a polynomial?
The autocovariance generating function, $\gamma(z)$, is defined as the $z$-transform of the autocovariance function, $\gamma_\tau$:
$$
\gamma(z) = \gamma_0 + \gamma_1(z+z^{-1}) + \gamma_2(z^2+z^{-2}) +...
- 167
2
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1
answer
180
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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
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2
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41
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Converting into partial fraction.
For the equation,
x-1/(x+1)(x-2)^2 = A/(x+1) + B/(x-2) + C/(x-2)^2
x-1 = A(x-2)^2 + B(x+1)(x-2) + C(x+1)
Using x = -1 gives A = -2/9
Using x = 2 ...
1
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1
answer
559
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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
$$12\frac{x^3+4}{(x^2-1)(x^2+3x+2)} = \frac{A}{x-1} + \frac{B}{x+2} + \frac{C}{x+1} + \...
- 1,283
2
votes
2
answers
50
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Calculus 2: Partial fractions problem. Finding the value of a constant
I encountered the following problem.
Let $f(x)$ be a quadratic function such that $f(0) = -6$ and
$$\int \frac{f(x)}{x^2(x-3)^8} dx $$
is a rational function.
Determine the value of $f'(0)$
Here'...
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1
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64
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Algebraically solve for reciprocal of result of polynomial long division
How can I show the following relationship algebraically?
$$\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
7
votes
3
answers
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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:
$$\frac{3x-8}{(x-5)(x+1)} = \...
- 813
1
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0
answers
248
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Why did this incorrect partial fraction decomposition produce the correct answer?
I was reviewing a classmate (call him Bob)'s work on an integration of a rational expression (although integration is involved, it's beyond the scope of this question).
The problem was:
$$\int\frac{...
- 113
2
votes
1
answer
654
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Understanding Why Partial Fractions Works [duplicate]
I was wondering why, not how, partial fractions work the way we are normally taught to do. To be specific:
We are told that, when we have a second degree expression on the bottom that can't be ...
user304152
3
votes
1
answer
115
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Partial fraction decompostion
Solve the partial fraction.
Starting out with...
$${x^2+1\over x^3-1}={x^2+1\over (x-1)(x^2+x+1)}$$
Then the partial fraction formula part of $\displaystyle {A\over x-1}+{Bx+C\over x^2+x+1}$.
...
- 458
1
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1
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65
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integral problem $\int \frac{2 \lambda a}{\mathbf{ (e^{at}-1)\lambda \sigma^2+2ae^{-at}}}dt $
Does anybody know how to tackle the below integral? I am analyzing a formula derivation where this appears as the final calculation, but I don't know how to get it solved
$$\int \frac{2 \lambda a}{\...
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