1. ## Partial fraction decomposition

$\frac{1}{x^3(x^2+1)}=\frac{ax^2+bx+c}{x^3}+\frac{d x+e}{x^2}\\$
$ax^4+bx^3+cx^2+dx^4+ex^3=x^4(a+d)+x^3(b+e)+cx^2=1\ \$
$c=0\\$
$b+e=0\\$
$a+d=0\\$
i got 4 variables with 2 equations
what to do??

2. Hi transgalactic

$
\frac{1}{x^3(x^2+1)}\neq\frac{ax^2+bx+c}{x^3}+\fra c{dx+e}{x^2}\\
$
because if we put the denominator of RHS together, it won't become $x^3(x^2+1)$ but $x^3$ (different from LHS)

It should be :
$
\frac{1}{x^3(x^2+1)}=\frac{A}{x}+\frac{B}{x^2}+\fr ac{C}{x^3}+\frac{Dx+E}{x^2+1}$

3. Originally Posted by transgalactic
$\frac{1}{x^3(x^2+1)}=\frac{ax^2+bx+c}{x^3}+\frac{d x+e}{x^2}\\$
$ax^4+bx^3+cx^2+dx^4+ex^3=x^4(a+d)+x^3(b+e)+cx^2=1\ \$
$c=0\\$
$b+e=0\\$
$a+d=0\\$
i got 4 variables with 2 equations
what to do??
$\frac{1}{x^3(x^2+1)} = \frac{a}{x} + \frac{b}{x^2} + \frac{c}{x^3} + \frac{dx+e}{x^2+1}$

$1 = ax^2(x^2+1) + bx(x^2+1) + c(x^2+1) + (dx+e)x^3
$

$1 = (a+d)x^4 + (b+e)x^3 +(a+c)x^2 + bx + c$

equating coefficients ...

$a+d = 0$

$b+e = 0$

$a+c = 0$

$b = 0$

$c = 1$

from the above equations ...

$a = -1$ , $b = 0$ , $c = 1$ , $d = 1$ , $e = 0$

$\frac{1}{x^3(x^2+1)} = -\frac{1}{x} + \frac{1}{x^3} + \frac{x}{x^2+1}$

4. can you give me a link to the manual of this stuff
in order to know thurely all the laws

5. Originally Posted by transgalactic
can you gove me a link to this stuff
in order to know thurely all the laws
partial fraction decomposition - Google Search

6. thanks