C4 Chapter 1: Partial Fractions Dr J Frost Last modified: 30 th August 2015

Overview At GCSE you learnt how to combine a sum/difference of fractions into one. We now want to learn how to do the opposite process: split a fraction into a sum of simpler ones, known as partial fractions. ?

Method Q METHOD 1: Substitution METHOD 2: Equating coefficients ? ? ?

Test Your Understanding C4 Jan 2011 Q3 Notice we can move the “–” to the front of the fraction. Note that we don’t technically need this last line from the perspective of the mark scheme, but it’s good to just to be on the safe side ?

More than two fractions The principle is exactly the same if we have more than two linear factors in the denominator. Q ?

Test Your Understanding C4 June 2009 Q3 ?

Exercise 1B/1C 1 a c e g Exercise 1BExercise 1C 1 2 a b c a c ? ? ? ? ? ? ? ? ?

Repeated linear factors Q The problem is resolved by having the factor both squared and non-squared. ? ?

Test Your Understanding C4 June 2011 Q1 ?

Dealing with Improper Fractions The ‘degree’ of a polynomial is the highest power, e.g. a quadratic has degree 2. An algebraic fraction is improper if the degree of the numerator is at least the degree of the denominator.

Dealing with Improper Fractions Q Method 1: Algebraic DivisionMethod 2: Using One Identity (method not in your textbooks but in mark schemes) Bropinion: I personally think the second method is easier. And mark schemes present it as “Method 1” – implying more standard! ? ?

Test Your Understanding C4 Jan 2013 Q3 ?

Exercise 1E Express as partial fractions. 1a b c d 2a b ? ? ? ? ? ?

Summary Identify what identity you’d use in each case (no need to identify constants). ? ? ? ?