Improper Rational Function

Definition

An improper rational function is a rational function where the degree of the numerator polynomial is greater than or equal to the degree of the denominator polynomial. This type of function can be further decomposed using the technique of partial fractions to simplify its expression and make it easier to evaluate.

5 Must Know Facts For Your Next Test

1. Improper rational functions can be classified into two types: those with distinct linear factors in the denominator and those with repeated linear factors in the denominator.
2. The process of partial fractions involves finding the constants or coefficients that allow the improper rational function to be expressed as a sum of simpler rational functions.
3. Partial fractions can be used to integrate improper rational functions, as the simpler rational functions are easier to integrate than the original improper rational function.
4. Improper rational functions can also arise in the context of solving differential equations, where the technique of partial fractions is often employed.
5. The decomposition of an improper rational function into partial fractions is a crucial step in many calculus applications, including integration, series expansions, and solving certain types of differential equations.

Review Questions

• Explain the relationship between the degree of the numerator and denominator polynomials in an improper rational function, and how this affects the decomposition using partial fractions.
  • The defining characteristic of an improper rational function is that the degree of the numerator polynomial is greater than or equal to the degree of the denominator polynomial. This means that the function cannot be expressed as a proper rational function (where the degree of the numerator is less than the degree of the denominator). The technique of partial fractions is used to decompose the improper rational function into a sum of simpler rational functions, each with a denominator of lower degree than the original. This decomposition allows for easier evaluation and integration of the function.
• Describe the two main types of improper rational functions and how the partial fractions decomposition differs between them.
  • Improper rational functions can be classified into two types: those with distinct linear factors in the denominator and those with repeated linear factors in the denominator. For improper rational functions with distinct linear factors in the denominator, the partial fractions decomposition involves expressing the function as a sum of simpler rational functions, each with a linear denominator. For improper rational functions with repeated linear factors in the denominator, the partial fractions decomposition is more complex, as it involves expressing the function as a sum of simpler rational functions, each with a denominator that may include linear factors raised to a power. The specific approach to the partial fractions decomposition depends on the structure of the denominator polynomial.
• Analyze the role of partial fractions in the integration and evaluation of improper rational functions, and explain why this technique is crucial in many calculus applications.
  • The decomposition of an improper rational function into partial fractions is a crucial step in many calculus applications, as it allows for the integration and evaluation of these functions. Improper rational functions are often difficult to integrate directly, but by expressing them as a sum of simpler rational functions through partial fractions, the integration process becomes much more straightforward. Additionally, partial fractions can be used to solve certain types of differential equations and to obtain series expansions of improper rational functions. The ability to decompose and simplify improper rational functions using partial fractions is a fundamental skill in calculus, as it enables the solution of a wide range of problems involving these types of functions.

Related terms

Rational Function: A rational function is a function that can be expressed as the ratio of two polynomial functions.

Partial Fractions: Partial fractions is a technique used to decompose an improper rational function into a sum of simpler rational functions, which can then be more easily integrated or evaluated.

Degree of a Polynomial: The degree of a polynomial is the highest exponent of the variable in the polynomial.