I'm wondering what (a+b)^x is called. I will need this information to study for a test, but I could not manage to Google this one out. Specifically, I need to know (a+b)^2 and (a+b)^3 are.
While we're on the topic, I'm wondering how you expand the latter. Is it something like:
(a+b)(a+b)(a+b)= a^3+a^2b+a^2b+ab^2+b^3+b^2*a+b^2+a*ba^2
Or am I wrong?
Thanks, H. Z.
P.S. I know that ^^ isn't completely simplified (if it's even correct)
This could be called the binomial expansion or the binomial theorem. When $x$ is a positive integer, we have the well-known relationship
$$(a+b)^x = \sum_{k=0}^x \begin{pmatrix} x \\ k \end{pmatrix} a^k b^{x-k}.$$
The quantity $\begin{pmatrix} x \\ k\end{pmatrix} = \frac{x!}{k!(x-k)!}$ is known as the binomial coefficient.
When $x$ is non-integral, it gets substantially more complex. We replace the binomial coefficient with a form using a Pochhammer symbol and turn the series into an infinite series, which generalizes the expansion to all reals but non-negative integers (and in fact complex numbers, too).