What are the rules for decomposition of partial fractions?

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What are the rules for decomposition of partial fractions?

There are some very useful rules when dealing with partial fractions

1) For every linear factor such as ax + b in the denominator, there will be a partial fraction of the form A / ax + b.
2) For every repeated factor such as (ax + b)² in the denominator, there will be two partial fractions: A/ ax + b and B/ (ax + b)² . For higher powers there will be correspondingly more terms.
3) For quadratic factors in the denominator e.g. ax² + bx + c there will be a partial fraction of the form: Ax + B / ax² + bx +c.

For example, let's decompose (7x + 2) / [(x + 2)^2 * (x - 2)]. Using the rules above, (x + 2)^2 would give us A / (x + 2) + B / (x+2)^2. (x - 2) in the denominator would give C / (x - 2). Therefore, the partial fraction (7x + 2) / [(x + 2)^2 * (x - 2)] can also be written in the form: A / (x + 2) + B / (x+2)^2 + C / (x - 2).

In fact, having equated appropriate coefficients we find out that:
(7x + 2) / [(x + 2)^2 * (x - 2)] = - 1/ (x + 2) + 3 / (x+2)^2 + 1 / (x - 2).

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