Rational functions

Definition

Rational functions are ratios of two polynomials. They are often used in calculus to perform integration using partial fractions.

5 Must Know Facts For Your Next Test

1. A rational function is expressed as $\frac{P(x)}{Q(x)}$, where both $P(x)$ and $Q(x)$ are polynomials.
2. To decompose a rational function into partial fractions, the degree of the numerator must be less than the degree of the denominator.
3. If the denominator has distinct linear factors, each factor will have its own term in the partial fraction decomposition.
4. For repeated linear factors, each power of the factor will have a separate term in the decomposition.
5. Quadratic factors in the denominator require terms of the form $\frac{Ax + B}{(quadratic factor)}$ for decomposition.

Review Questions

• How do you determine if a rational function can be decomposed into partial fractions?
• What is the form of a partial fraction for a quadratic factor in the denominator?
• Why must the degree of the numerator be less than that of the denominator for partial fraction decomposition?

Related terms

Polynomial: An algebraic expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents.

Partial Fraction Decomposition: The process of breaking down a complex rational function into simpler fractions that can be more easily integrated.

Integration: \text{The process of finding an integral or antiderivative. It is fundamental to calculus and is used to calculate areas under curves.}