Partial fraction decomposition

Definition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This is particularly useful for integrating rational functions.

5 Must Know Facts For Your Next Test

1. Partial fraction decomposition applies to proper rational functions where the degree of the numerator is less than the degree of the denominator.
2. The first step in partial fraction decomposition is to factor the denominator completely into linear and/or irreducible quadratic factors.
3. For each distinct linear factor $(ax + b)$ in the denominator, include a term $\frac{A}{ax+b}$ in the partial fraction decomposition.
4. For each repeated linear factor $(ax + b)^n$, include terms $\frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + ... + \frac{A_n}{(ax+b)^n}$.
5. For each irreducible quadratic factor $(ax^2+bx+c)$, include a term $\frac{Ax+B}{ax^2+bx+c}$ in the partial fraction decomposition.

Review Questions

• What is the purpose of partial fraction decomposition?
• How would you decompose a rational function with a repeated linear factor like $(x-1)^3$?
• Explain how to handle an irreducible quadratic factor in partial fraction decomposition.

Related terms

Rational Function: A function that can be expressed as the ratio of two polynomials.

Integration by Partial Fractions: A method for finding integrals by expressing a rational function as a sum of simpler fractions and then integrating each term separately.

Irreducible Quadratic: A quadratic expression that cannot be factored into real linear factors.