The **Binomial Theorem** is a fundamental theorem that provides a formula for expanding the powers of a binomial expression.

Let **x **and **y **be two variables and **n **be non-negative interger then the **binomial theorem** for positive index **n **states that,

(a+x)C(n,0)a^{n} = ^{n }+ C(n,1)a^{n-1}x + C(n,2)a^{n-2}x^{2} +…..+ C(n,n)x^{n} |

**Note**: The quantities C(n,0), C(n,1), C(n,2), C(n,n) are called

**Binomial coefficient**

## ⁘ General Term of Binomial expansion of (a+x)^{n}

The** general term** of the binomial expansion **(a+x) ^{n}** is denoted by

**t**and given by:

_{r+1} tC(n,r) a_{r+1} = ^{n-r}x^{r} |

**Note**: if the expansion is **(a-x) ^{n}** , then its general term is given by:

tC(n,r) a_{r+1} = ^{n-r}x^{r} .(-1)^{r}= (-1) ^{r} C(n,r) a^{n-r}x^{r} |

## ⁘ Middle Term

• When we expand** (a+x) ^{n}** , then we always get

**n+1**term in the expansion of (a+x)

^{n}.

• Therefore, if

**n**is

**even**then

**(n+1)**will be odd and the expansion contains

**single middle term.**

• Similarly, if

**n**is

**odd**then

**n+1**will be even and the expansion contains

**two middle terms.**