As I teach Further Maths I haven’t really considered how I would teach partial fractions, and normally just do them in my head without writing down any workings.

However, I have recently started providing some last minute tuition and one of the things they wanted explaining was partial fractions. To be honest I have forgotten how I was taught this, I have a feeling it was the “substitute different x values in to knock out terms” method. I went through two slightly different methods for the partial fraction shown below

Personally I prefer Method 1 but I think Method 2 would probably be better for the weaker students as it shows explicitly what is happening.

How do other people teach this?

*Related*

I was taught method 2. It does also have the logical advantage of not encouraging substitution of values of x which the question will potentially have banned! A rigorous version of method 1 might need to consider x ~= -3 etc… which is far too messy. So perhaps for self-validation reasons I prefer method 2. The big advantage of method 2, which you elude to is it also is genuinely proving that the original two statements are equivalent FOR ALL x. The idea that to make two equations in x the same for all x can be done by equating coefficients (assuming they’re both nice finite power series!) is an excellent side effect. Whereas Method 1 has what looks to me to be a very serious flaw… how do we know the formula works at other values of x?

Nice post Tom, I teach both these methods, I actually prefer method 2, but my students tend to prefer method 1.

Nice post Tom, I teach both these methods, I actually prefer method 2, but my students tend to prefer method 1. I teach method 2 first to build understanding and show them substitution later when they have grasped the concept.

Thanks. Nice to know your preference. I think I prefer method 2 for its validity and generality, but also show method 1 as it is “quick and dirty” when you want to put something on partial fractions and o find it easier to do in my head than method 2.