Partial Fractions Department of Mathematics University of Leicester

Content Quadratic factorsLinear factorsRepeated factorsIntroduction

Next Repeated factors Quadratic factors Linear factors Introduction

Linear factors The decomposition of a given fraction into partial fractions is achieved by first factorising the denominator Next Repeated factors Quadratic factors Linear factors Introduction

If the factors are linear then we will have partial fractions of this form Linear factors Next Repeated factors Quadratic factors Linear factors Introduction

Linear factors Next Repeated factors Quadratic factors Linear factors Introduction

Linear factors … and solving this equivalence Next Repeated factors Quadratic factors Linear factors Introduction

Cover-up method If we “cover-up” the factor associated with the value we want to fin and then evaluate at it’s zero we will achieve the value we were looking for. You can check this is correct with the previous method. This only works with linear factors! Next Repeated factors Quadratic factors Linear factors Introduction

Higher order factors Next Repeated factors Quadratic factors Linear factors Introduction If there is a quadratic factor we have partial fractions of this form.

Higher order factors Next Repeated factors Quadratic factors Linear factors Introduction Rearrange to gain this equivalence.

Higher order factors Next Repeated factors Quadratic factors Linear factors Introduction

Repeated factors Next Repeated factors Quadratic factors Linear factors Introduction If there is a repeated factor we have partial fractions of this form. The numerator of the linear factor was found using the cover-up method.

Repeated factors Next Repeated factors Quadratic factors Linear factors Introduction

Repeated factors Next Repeated factors Quadratic factors Linear factors Introduction

Summary A linear factor gives a partial fraction of the form: A quadratic factor gives a partial fraction of the form: A repeated factor gives a partial fraction of the form: Next Repeated factors Quadratic factors Linear factors Introduction