Factor the denominator completely.
If the fraction is improper, perform polynomial long division first.
Write the expression as a sum of simpler fractions based on the factor types.
For distinct linear factors, use terms like (frac{A}{x-a}).
For repeated linear factors, use terms like (frac{A_1}{x-a}+frac{A_2}{(x-a)^2}+cdots).
For irreducible quadratic factors, use terms like (frac{Ax+B}{x^2+bx+c}).
For repeated irreducible quadratic factors, use terms like (frac{A_1x+B_1}{x^2+bx+c}+frac{A_2x+B_2}{(x^2+bx+c)^2}+cdots).
Multiply both sides by the common denominator.
Expand and combine like terms.
Set coefficients of matching powers equal.
Solve the resulting system for the unknown constants.
Substitute the constants back into the partial fractions.
Simplify the final expression.