Why is the binomial theorem taught like this?

Question

If I look at any high school textbook or video for a binomial expansion like this

$(3+4x)^{-2}$

The first step will always be to write it like this

$(3+4x)^{-2} = 3^{-2}\left(1+\frac{4}{3}x\right)^{-2}$

and then use the $(1+X)^n$ formula.

Is there any reason why it is not more commonly taught to expand it directly using the $(X+Y)^n$ formula seen here:

https://en.wikipedia.org/wiki/Binomial_theorem#Newton.27s_generalized_binomial_theorem

Why a topic is taught in a certain way may not be the best question to ask in this forum but some people may have ideas. If you're familiar with the British A Level curriculum then you'll know that only the $(1+X)^n$ formula is taught and most students aren't aware of the other method.

Thanks.

Answer 1

When you express $(x+y)^n$ like $x^n\left(1+\dfrac yx\right)^n$, you only need to concentrate on the final term. This makes the expression easier to evaluate.

Answer 2

When $n\in\mathbb R\setminus\mathbb N$ the binomial theorem results in an infinite series. The series for $(1+x)^n$ is just the Maclaurin series for the function $f(x)=(1+x)^n$, which has radius of convergence $1$. These are familiar concepts.

Otoh the series for $(x+y)^n$ involves two variables; it's not "just a power series", and "radius of convergence equals $1$" becomes $|y|<|x|$. Not such familiar concepts.

(And why is it $|y|<|x|$ instead of $|x|<|y|$, since after all $(x+y)^n=(y+x)^n$? Of course there is one series that works if $|y|<|x|$, and if $|x|<|y|$ the same series, with $x$ and $y$ swapped, works. Oh, I'm so confused - simply saying the series for $(1+x)^n$ converges for $|x|<1$ seems simpler.)