Partial fraction decomposition.

Prove It

MHF Helper
Aug 2008
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Decompose into partial fractions: \(\displaystyle \frac{x}{x^4-a^4}\)
\(\displaystyle \frac{x}{x^4 - a^4} = \frac{x}{(x^2)^2 - (a^2)^2}\)

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

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


Now try using partial fractions with

\(\displaystyle \frac{A}{x - a} + \frac{B}{x + a} + \frac{Cx + D}{x^2 + a^2} = \frac{x}{(x - a)(x + a)(x^2 + a^2)}\).
 
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May 2010
14
2
1. \(\displaystyle \frac{A}{x - a} + \frac{B}{x + a} + \frac{Cx + D}{x^2 + a^2} = \frac{x}{(x - a)(x + a)(x^2 + a^2)}\)
2. \(\displaystyle x=A(x+a)(x^2+a^2)+B(x-a)(x^2+a^2)+(Cx+D)(x-a)(x+a)\)

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

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

How do I find C and D?
 
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Prove It

MHF Helper
Aug 2008
12,883
4,999
1. \(\displaystyle \frac{A}{x - a} + \frac{B}{x + a} + \frac{Cx + D}{x^2 + a^2} = \frac{x}{(x - a)(x + a)(x^2 + a^2)}\)
2. \(\displaystyle x=A(x+a)(x^2+a^2)+B(x-a)(x^2+a^2)+(Cx+D)(x-a)(x+a)\)

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

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

How do I find C and D?
Let \(\displaystyle x = 0\) to find \(\displaystyle D\).

Then you can use your information about \(\displaystyle A, B, D\) to find \(\displaystyle C\).
 
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