The Binomial Theorem

A binomial is a polynomial that has two terms. The Binomial Theorem explains how to raise a binomial to certain non-negative power.

The theorem states that in the expansion of ${\left(x+y\right)}^{n}$ ,

${\left(x+y\right)}^{n}={x}^{n}+n{x}^{n-1}y+...{+}_{n}{C}_{r}{x}^{n-r}{y}^{r}+...+nx{y}^{n-1}+{y}^{n}$ , the coefficient of ${x}^{n-r}{y}^{r}$ is

${}_{n}{C}_{r}=\frac{n!}{\left(n-r\right)!r!}$

The symbol $\left(\begin{array}{c}n\\ r\end{array}\right)$ is often used in place of ${}_{n}{C}_{r}$ to denote binomial coefficient.

The expansion is expressed in the sigma notation as ${\left(x+y\right)}^{n}={\sum }_{r=0}^{n}{}_{n}{C}_{r}{x}^{n-r}{y}^{r}$ .

Note that, the sum of the degrees of the variables in each term is $n$ .

Example:

What is the coefficient of ${a}^{4}$ in the expansion of ${\left(1+a\right)}^{8}$ .

Use the binomial theorem to determine the general term of the expansion.

The general term in the expansion of ${\left(x+y\right)}^{n}$ is $\frac{n!}{\left(n-r\right)!r!}{x}^{n-r}{y}^{r}$ .

Here, $x=1$ , $y=a$ and $n=8$ . The term that has the fourth power of the variable $a$ will be the fourth term in the expansion. Therefore, substitute $r=4$ in the binomial coefficient of the general term and evaluate.

$\begin{array}{l}\frac{8!}{\left(8-4\right)!4!}=\frac{8\text{\hspace{0.17em}}×\text{\hspace{0.17em}}7\text{\hspace{0.17em}}×\text{\hspace{0.17em}}6\text{\hspace{0.17em}}×\text{\hspace{0.17em}}5}{4\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=70\end{array}$

Therefore, the coefficient of ${a}^{4}$ in the expansion is $70$ .