SL 1.9—Binomial theorem where n is an integer

The binomial theorem provides a systematic method for expanding expressions of the form $(a+b{)}^n$, where $n$ is a positive integer. Rather than multiplying out brackets repeatedly (imagine expanding $(a+b{)}^7$ by hand!), the theorem gives us a direct formula.

Binomial Theorem: For any positive integer $n$, the expansion of $(a+b{)}^n$ is given by:

$$(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k$$

This produces $n+1$ terms in total.

Written out fully, this looks like:
$(a+b{)}^n=(\genfrac{}{}{0ex}{}{n}{0})a^n+(\genfrac{}{}{0ex}{}{n}{1})a^{n-1}b+(\genfrac{}{}{0ex}{}{n}{2})a^{n-2}{b}^2+\cdots +(\genfrac{}{}{0ex}{}{n}{n-1})a{b}^{n-1}+(\genfrac{}{}{0ex}{}{n}{n})b^n$

Notice the pattern:

• The powers of $a$ decrease from $n$ down to $0$
• The powers of $b$ increase from $0$ up to $n$
• The powers of $a$ and $b$ in each term always sum to $n$
• The coefficients $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ are called binomial coefficients