# Partial fraction decomposition.

• May 19th 2010, 12:00 AM
ChristopherDunn
Partial fraction decomposition.
Decompose into partial fractions: $\frac{x}{x^4-a^4}$
• May 19th 2010, 12:09 AM
Prove It
Quote:

Originally Posted by ChristopherDunn
Decompose into partial fractions: $\frac{x}{x^4-a^4}$

$\frac{x}{x^4 - a^4} = \frac{x}{(x^2)^2 - (a^2)^2}$

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

$= \frac{x}{(x - a)(x + a)(x^2 + a^2)}$.

Now try using partial fractions with

$\frac{A}{x - a} + \frac{B}{x + a} + \frac{Cx + D}{x^2 + a^2} = \frac{x}{(x - a)(x + a)(x^2 + a^2)}$.
• May 19th 2010, 07:45 PM
ChristopherDunn
1. $\frac{A}{x - a} + \frac{B}{x + a} + \frac{Cx + D}{x^2 + a^2} = \frac{x}{(x - a)(x + a)(x^2 + a^2)}$
2. $x=A(x+a)(x^2+a^2)+B(x-a)(x^2+a^2)+(Cx+D)(x-a)(x+a)$

let x=a;
1. $a=A(a+a)(a^2+a^2)$
2. $a=4a^3A$
3. $A=\frac{1}{4a^2}$

let x=-a;
1. $-a=B(-a-a)((-a)^2+a^2)$
2. $-a=-4a^3B$
3. $B=\frac{1}{4a^2}$

How do I find C and D?
• May 19th 2010, 09:02 PM
Prove It
Quote:

Originally Posted by ChristopherDunn
1. $\frac{A}{x - a} + \frac{B}{x + a} + \frac{Cx + D}{x^2 + a^2} = \frac{x}{(x - a)(x + a)(x^2 + a^2)}$
2. $x=A(x+a)(x^2+a^2)+B(x-a)(x^2+a^2)+(Cx+D)(x-a)(x+a)$

let x=a;
1. $a=A(a+a)(a^2+a^2)$
2. $a=4a^3A$
3. $A=\frac{1}{4a^2}$

let x=-a;
1. $-a=B(-a-a)((-a)^2+a^2)$
2. $-a=-4a^3B$
3. $B=\frac{1}{4a^2}$

How do I find C and D?

Let $x = 0$ to find $D$.

Then you can use your information about $A, B, D$ to find $C$.
• May 19th 2010, 09:50 PM
ChristopherDunn
That'll do it -- thanks!