Decompose Partial Fractions

Partial Fraction Decomposition is a method used to express a rational function as the sum of simpler fractions, making it easier to integrate or analyze.

The first step is to ensure that the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division.

For a linear factor like $$(x-a)$$ , the form is $$\frac{A}{(x-a)}$$ .

For a repeated linear factor like $$(x-a)^n$$ , the form is $$\frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n}$$ .

For a quadratic factor like $$(ax^2 + bx + c)$$ , the form is $$\frac{Bx + C}{(ax^2 + bx + c)}$$ .

The setup is $$\frac{A}{(x+1)} + \frac{Bx + C}{(3+x^2)}$$ .

The decomposition is $$\frac{3}{x+3} + \frac{3}{x-3}$$ .

It is a Repeated Linear Factor Method.

The setup is $$\frac{A}{x} + \frac{B}{(x+4)} + \frac{C}{(x+4)^2}$$ .

It is a Quadratic Factor Method.