Binomial Expression: An expression consisting of two terms, connected by + or - sign is called binomial expression.

Binomial Theorem: If x and a are real numbers then for all n \in N , we have

(x+a)^n = ^{n}  C_{0} \ x^n \ a^0 + ^{n} C_{1} \ x^{n-1} \ a^1 + ^{n} C_{2} \ x^{n-2} \ a^2 + \ldots \\  { \hspace{5.0cm} + ^{n} C_{r} \ x^{n-r} \ a^r + \ldots + ^{n} C_{n-1} \ x^1 \ a^{n-1} + ^{n} C_{n} \ x^0 \ a^n}

\Rightarrow  (x+a)^n = \sum \limits_{r=0}^{n}  ^{n} C_{r} \ x^{n-r} \ a^r

This expression has the following properties:

i) It has (n+1) terms

ii) The sum of the indices of x and a in each terms is n

iii) The coefficients of terms equidistant from the beginning and the end are equal.

iv) The general term T_{r+1} =  ^{n} C_{r} \ x^{n-r} \ a^r

v) (x+a)^n = \sum \limits_{r=0}^{n}  ^{n} C_{r} \ x^{n-r} \ a^r can also be expressed as

(x+a)^n = \sum \limits_{r+s=n}^{} \frac{n!}{r! s!} x^r \ a^s

vi) Replacing a by -a in the expression (x+a)^n , we get

(x-a)^n = ^{n} C_{0} \ x^n \ a^0 - ^{n} C_{1} \ x^{n-1} \ a^1 + ^{n} C_{2} \ x^{n-2} \ a^2 - ^{n} C_{3} \ x^{n-3} \ a^3+ \ldots \\  { \hspace{5.0cm} +(-1)^r {\ ^{n} C_{r} \ x^{n-r} \ a^r} + \ldots + (-1)^n {\ ^{n} C_{n} \ x^0 \ a^{n}} }

The general term in the expansion of (x-a)^n is T_{r+1} =  (-1)^n {\ ^{n} C_{r} \ x^{n-r} \ a^r }

vii) Putting x = 1 and replacing a by x ,  in the expression ( x + a)^n we get 

(1+x)^n = ^{n} C_{0} + ^{n} C_{1} x + ^{n} C_{2} x^2+ \ldots + ^{n} C_{n} x^n = \sum \limits_{r=0}^{n} {^{n} C_{r} \ x^r}

viii) Putting a = 1 in the expression ( x+a)^n , we get 

(1+x)^n = ^{n} C_{0} x^n + ^{n} C_{1} x^{n-1} + ^{n} C_{2} x^{n-2}+ \ldots + ^{n} C_{n} x^0 = \sum \limits_{r=0}^{n} {^{n} C_{r} \ x^{n-r} }

This is the expansion of (1+x)^n in descending powers of x . In this case, T_{r+1} =  {\ ^{n} C_{r} \ x^{n-r} }

ix) Addition and Subtraction

( x+a)^n + ( x-a)^n = 2 \Big\{ ^{n} C_{0} \ x^n \ a^0 + ^{n} C_{2} \ x^{n-2} \ a^2 + \ldots   \Big\}

= 2 \{ \text{sum of the odd terms in the expression} (x+a)^n \}

Similarly,

( x+a)^n - ( x-a)^n = 2 \Big\{ ^{n} C_{1} \ x^{n-1} \ a^1 + ^{n} C_{3} \ x^{n-3} \ a^3 + \ldots   \Big\}

= 2 \{ \text{sum of the even terms in the expression} (x+a)^n \}

If n is odd, then \{ ( x+a)^n + ( x-a)^n \} and \{ ( x+a)^n - ( x-a)^n \}                      both have \Big( \frac{n+1}{2} \Big) terms.

If n is even, then \{ ( x+a)^n + ( x-a)^n \} has \Big( \frac{n}{2} +1 \Big) terms whereas \{ ( x+a)^n - ( x-a)^n \} both have \Big( \frac{n}{2} \Big) terms.

x) If O and E denote respectively the sum of odd terms and even terms in the expansion of (x+a)^n , then

(a) (x+a)^n = O+E and (x-a)^n = O-E

(b) (x^2 - a^2)^n = O^2 - E^2

(c) 4 OE = ( x - a)^{2n} - (x-a)^{2n}

(d) (x+a)^{2n} + ( x - a)^{2n} = 2 ( O^2 + E^2)

xi)     If n is even, then \Big( \frac{n}{2} + 1 \Big)^{th} term is the middle term.

If n is off, then \Big( \frac{n+1}{2} \Big)^{th} and \Big( \frac{n+3}{2} \Big)^{th} are middle terms