What is (a+b)^x called?

Question

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)

See the Binomial Theorem: https://en.wikipedia.org/wiki/Binomial_theorem — mvfs314, Jul 12 '16 at 02:54
$(a+b)^2$ expands to $a^2+2ab+b^2$. Often needed. And $(a+b)^3$ expands to $a^3+3a^2b+3ab^2+b^3$. You had gotten most of the terms but not all. — André Nicolas, Jul 12 '16 at 03:05

score 4 · Accepted Answer · answered Jul 12 '16 at 02:55

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).

I love me some complex exponents without Euler's famed exponential formula. ;-) — Simply Beautiful Art, Jan 14 '17 at 02:18

What is (a+b)^x called?

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