Sometimes a linear factor appears more than once. For example in
the factor occurs twice. We call it a repeated linear factor . The repeated linear factor produces two partial fractions of the form . In general, a repeated linear factor of the form generates two partial fractions of the form
This is reasonable since the sum of two such fractions always gives rise to a proper fraction:
A repeated linear factor in the denominator produces two partial fractions:
Once again the unknown constants are found by either equating coefficients and/or substituting specific values for .
Express in partial fractions.
First factorise the denominator:
There is a repeated linear factor which gives rise to two partial fractions of the form
Multiply both sides through by to obtain the equation to be solved to find and :
Now evaluate the constants and by equating coefficients:
Equating the coefficients gives so . Equating constant terms gives from which .
Finally express the answer in partial fractions:
Express the following in partial fractions.