1. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions PROGRAMME F8 PARTIAL FRACTIONS

2. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions Denominators with repeated and quadratic factors

3. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions Denominators with repeated and quadratic factors

4. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions Consider the following combination of algebraic fractions: The fractions on the left are called the partial fractions of the fraction on the right.

5. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions To reverse the process, namely, to separate an algebraic fraction into its partial fractions we proceed as follows. Consider the fraction: Firstly, the denominator is factorized to give:

6. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions Next, it is assumed that a partial fraction break down is possible in the form: The assumption is validated by finding the values of A and B.

7. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions To find the values of A and B the two partial fractions are added to give:

8. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions Since: And since the denominators are identical the numerators must be identical as well. That is:

9. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions Consider the identity: Therefore:

10. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions For this procedure to be successful the numerator of the original fraction must be of at least one degree less than the degree of the denominator. If this is not the case the original fraction must be reduced by division. For example:

11. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions Denominators with repeated and quadratic factors Programme F8: Partial fractions

12. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Partial fractions Denominators with repeated and quadratic factors Programme F8: Partial fractions

13. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Denominators with quadratic factors A similar procedure is applied if one of the factors in the denominator is a quadratic. For example: This results in:

14. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Denominators with quadratic factors Equating coefficients of powers of x yields: Three equations in three unknowns with solution:

15. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Denominators with quadratic factors

16. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Denominators with repeated factors Repeated factors in the denominator of the original fraction of the form: give partial fractions of the form:

17. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Denominators with repeated factors Similarly, repeated factors in the denominator of the original fraction of the form: give partial fractions of the form:

18. STROUD Worked examples and exercises are in the text Programme F8: Partial fractions Learning outcomes Factorize the denominator of an algebraic fraction into its prime factors Separate an algebraic fraction into its partial fractions Recognise the rules of partial fractions