Partial Fractions

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We may have to express expressions such as

in the form

Notice first that factorises into We can write,

Ignore the first term here and multiply throughout by We get

after cancellation.

The procedure now is to eliminate the A term by putting x=2

Eliminate the B term by putting x=-4

Then

If we have a denominator which includes an term which does not factorise, we have to include a term, term in the answer. In general we have one term for every term in the denominator which does not factorise, and for that term, the highest power of in the numerator is one less than the highest power of in the numerator.

Example:

Multiply out by and cancel to give

(1)

Put into (1) to make the first term on the right hand side zero. We get,

If we putwe eliminate

Now put

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