1. ## partial fraction

Express each of the following in partial fraction .

(a) $\displaystyle \frac{x^2-1}{x^2(2x+1)}$

Is $\displaystyle x^2$ (denominator) , considered as a repeated linear factor ?

(b)$\displaystyle \frac{x^3+3}{(x+1)(x-1)}$

i have tried long division for this question but it doesn't work .

Express each of the following in partial fraction .

(a) $\displaystyle \frac{x^2-1}{x^2(2x+1)}$

Is $\displaystyle x^2$ (denominator) , considered as a repeated linear factor ?
Yes.

So $\displaystyle \frac{x^2-1}{x^2(2x+1)} = \frac{Ax+B}{x^2} + \frac{C}{2x+1}$

Get A,B and C by cross multiplying and comparing coefficients.

(b)$\displaystyle \frac{x^3+3}{(x+1)(x-1)}$

i have tried long division for this question but it doesn't work .
It should. Divide $\displaystyle x^3 + 3$ by $\displaystyle x^2 - 1$. You will see that:
$\displaystyle \frac{x^3+3}{x^2 - 1} = x + \frac{ x+3 }{x^2 - 1}$

Now do a partial fraction decomposition for $\displaystyle \frac{x+3}{x^2 - 1}$

3. Originally Posted by Isomorphism
Yes.

So $\displaystyle \frac{x^2-1}{x^2(2x+1)} = \frac{Ax+B}{x^2} + \frac{C}{2x+1}$

Get A,B and C by cross multiplying and comparing coefficients.
Can i do it this way ..
$\displaystyle \frac{x^2-1}{x^2(2x+1)}$
=$\displaystyle \frac{A}{x}+\frac{B}{x^2}+\frac{C}{2x+1}$

$\displaystyle \frac{x^2-1}{x^2(2x+1)}$
=$\displaystyle \frac{A}{x}+\frac{B}{x^2}+\frac{C}{2x+1}$