Decomposition

Definition

Decomposition involves breaking down a complex fraction into simpler partial fractions. This is often used to simplify integration or solve algebraic equations more easily.

5 Must Know Facts For Your Next Test

1. Partial fraction decomposition is particularly useful for integrating rational functions.
2. To decompose a fraction, the degree of the numerator must be less than the degree of the denominator.
3. If the denominator has repeated linear factors, each factor must have a separate term in the decomposition.
4. For irreducible quadratic factors in the denominator, use terms with linear numerators in the decomposition.
5. Decomposition requires solving a system of equations to determine the coefficients of the partial fractions.

Review Questions

• What conditions must be met for a rational function to be decomposed into partial fractions?
• How do you handle repeated linear factors when performing partial fraction decomposition?
• Describe how to decompose a rational function with an irreducible quadratic factor in its denominator.

Related terms

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

Linear Factor: A factor of a polynomial that is first degree (e.g., $(x - a)$).

Irreducible Quadratic Factor: A quadratic polynomial that cannot be factored over the real numbers (e.g., $x^2 + bx + c$ where $b^2 - 4ac < 0$).