partial fractions

Topics

• partial fractions - linear factors
• partial fractions - repeated linear factors

Partial fractions are the reverse of adding/subtracting fractions. The goal is to decompose a rational function (a fraction where the numerator and denominator are polynomials) into a sum of simpler fractions.

This is useful in calculus, especially when finding anti-derivatives.

General Form:

A rational function P(x) / Q(x) where the degree of P(x) is less than the degree of Q(x) (i.e., a proper rational function) can be decomposed into partial fractions. If the degree of P(x) is greater than or equal to the degree of Q(x), perform polynomial long division first.

$$\frac{3x+5}{2x^2-5x-3}=\frac{2}{x-3}-\frac{1}{2x+1}$$

And for improper:

$$\frac{x^2+3x+2}{x+1}(\text{Improper}⟹\text{divide first})$$