Decomposition

Definition

Decomposition is the process of breaking down a complex rational function into simpler fractions, often called partial fractions. This technique is used to simplify the integration or solve equations involving rational expressions.

5 Must Know Facts For Your Next Test

1. Decomposition requires that the degree of the numerator is less than the degree of the denominator.
2. For distinct linear factors in the denominator, use separate constants for each factor in partial fractions.
3. For repeated linear factors, include terms for each power of the factor up to its multiplicity.
4. For irreducible quadratic factors, use a linear term in the numerator of each partial fraction.
5. The coefficients in partial fractions are typically determined by solving a system of equations obtained by equating coefficients or using substitution.

Review Questions

• What condition must be met regarding the degrees of the numerator and denominator before applying decomposition?
• How do you handle repeated linear factors when decomposing into partial fractions?
• What type of term should be used in the numerator for an irreducible quadratic factor?

Related terms

Partial Fraction: A fraction that represents part of a decomposed rational expression.

Linear Factor: A polynomial factor of degree one, typically in the form $(x - c)$ where $c$ is a constant.

Irreducible Quadratic Factor: A quadratic polynomial that cannot be factored into real linear factors, typically in the form $ax^2 + bx + c$ where $b^2 - 4ac < 0$.