## Decending powers of x

I was in one of my colleagues lessons this week.and he was teaching the class to expand quadratic brackets. As the lesson went on he noticed that a number of pupils had been writing the X squared term, then the constant term then the X term so he pulled the class together to tell them that conventionally we write quadratic equations in decending powers of x. This is excellent practice and something we all should be encouraging, but it made me think “Why decending powers of x?”

When dealing with quadratic, cubics and quartics up to GCSE and A-Level level we use the convention of decending powers. This is common in all sorts from expanding through long division, even in linear function we use descending powers. However, when we start with series expansions, such as binomial or Taylor’s, we switch to ascending powers. I’m also fairly certain that everything in my second and third year university modules on polynomials was in ascending powers too.

Now don’t get me wrong, I’m happy with the fact we have different conventions here, and I’m not against using them. I’m just inquisitive, and would love to know if there is a reason, and if there is what that reason is. If you do know, I’d love to hear it!

It seems logical to have ascending powers when dealing with more advanced series, since often the series doesn’t even have an end. As far as normal quadratics go, maybe it’s just a nice way to emphasize that the expression is quadratic rather than linear. You don’t want the dominant term any less obvious, so if you have a convention (which is useful for me – if I can’t factorise a quadratic straight away, writing it in descending powers usually helps!) it might as well be this.

I compare it to our decimal number system where numbers are written in descending powers of 10 and that maybe, because it’s on the GCSE syllabus, we follow that convention for some form of familiarity. It also helps for a discussion (and hopefully deeper understanding) when you compare grid multiplication of two brackets and of two 2 digit numbers – how the diagonal cells are of the same power. Once you’re out of GCSE, there are no constraints.

Good question!

For Taylor’s Series, the series is usually infinite, so we would have no choice but to start with the constant.

As for why the convention is usually descending, I am not too sure too. Perhaps, it is ‘easier’ to see the degree of the polynomial since its the first term.