A **Binomial** is an algebraic expression of two terms which are connected by the operation: + or −.

For example: x+2y, x−y, x^{3}+4y, a+b

Finding squares and cubes of a binomial by actual multiplication is not difficult. But the process of finding the expansion of binomials with higher powers becomes more difficult. There is a general formula which can help in finding the expansion of binomials with higher powers.

**Binomial Theorem for Positive Integral Index**

**General Term**

T_{r+1} = ^{n}C_{r} x^{n−r} a^{r}

### Properties

- There are (n+1) terms in the expansion of (x+a)
^{n}. - The degree of x in each term decreases while that of a increases such that the sum of the powers in each term is equal to n.
^{n}C_{0},^{n}C_{1},^{n}C_{2},^{n}C_{r},^{n}C_{n}are called**binomial coefficients**.- Since
^{n}C_{r}=^{n}C_{n−r}, the coefficients of terms equidistant from the beginning and the end are equal. - The binomial coefficients of the various terms of the expansion of (x+a)
^{n}for n = 1, 2, 3 form a pattern.

### Pascal's Triangle

The arrangement of the binomial coefficients is known as Pascal’s triangle. The first and last numbers are 1 each. The other numbers are obtained by adding the left and right numbers in the previous row.