Partial Fractions A rational function is one expressed in fractional form whose numerator and denominator are polynomials. A rational function is termed proper when the degree of the numerator is less than the degree of the denominator. It is termed improper otherwise. are all proper functions. are all improper functions.

Improper rational functions can be simplified by algebraic division. The fractional part is now a proper rational function.

1. Distinct Linear Factors in the Denominator When we encounter a rational function whose denominator is the product of two or more distinct linear functions, we can decompose the fraction into two or more distinct fractions called Partial Fractions. This is a pair of simultaneous equations which can easily be solved. OR

Starting from Let x = 2 Let x = -1 For larger numbers of unknowns this is an easier method. Page 18 Exercise 2 - But not Just YET TJ Exercise 1

A repeated linear factor in the denominator. The general solution for a repeated linear factor in the denominator is OR

Let x = 1 Let x = -2 Let x = 0

Page 19 Exercise 3 TJ Exercise 2

An irreducible quadratic factor in the denominator The general solution for an irreducible quadratic factor in the denominator is OR

Let x = -1 Let x = 0 Let x = 1

Page 20 Exercise 4 TJ Exercise 3 Page 21 Exercise 5

Improper Rational Functions As demonstrated at the start of this topic, an improper rational function can be reduced by algebraic division. This can then be further reduced into partial fractions.

Let x = -1 Let x = 0 Page 22 Exercise 6 Page 23 review