What is the convention for binomial expansion?

Question

When doing a Binomial expansion, in general for:

$\\(a+b)^n$

Is there a convention for which term you would raise to the power of n first, the a or the b?

And does this convention stay or change for the expansion of:

$\\(1+x)^n$

Which I guess is a second question; do you write powers of x in ascending or descending order? With polynomials it is in descending, but is it the same for binomial expansion?

One of the reasons I ask is I have some students who have come across exam questions asking for the '3rd' or '8th' term in an expansion.

Answer 1

There is no need for a convention. Addition and multiplication are commutative, so no one cares. Just a matter of convenience and context.

Also note that polynomials are not necessarily written in descending degree.

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Addendum:

Asking for the $n^{th}$ term makes sense provided the convention has been announced. A less error-prone formulation would be like to ask for "the term of degree $4$ in $b$".

Answer 2

There is no standard convention, because of commutativity of addition and multiplication. But in most of the standard texts, we usually write the binomial expansion as $$(a+b)^n=\sum_{r=0}^n\binom{n}{r}a^{n-r}b^r$$ $$(1+x)^n = \sum_{r=0}^n\binom{n}{r}x^r$$ And therefore, the first term is taken to be $a^n$, second term $na^{n-1}b$ and so forth.