`text(Binomial Expression :)`
An algebric expression consisting of two different terms is called a binomial expression.
e.g. `(1) \ \ \ \ \ x+a`
`\ \ \ \ \(2)\ \ \ \ \ x^3+a^3`

But `(x + nx)` is not a binomial, it is called a monomial.

`text(Binomial Theorem :)`

This theorem was given by Newton.
Binomial Theorem `=> (x + a)^n` If `n in N` (From a finite series)
or Any index `n notin N` (From an infinite series)

`text(Statement of the Theorem :)`

`(x+a)^n=text()^nC_0x^n+text()^nC_1x^(n-1)a+text()^nC_2x^(n-2)a^2+...................+text()^nC_rx^(n-r) . a^r+...................+text()^nC_nx^(n-n)a^n........(i)`

We observe `T_1=text()^nC_0x^n`

`T_2=text()^nC_1x^(n-1) . a^1`

`=>` General term in the expansion of `(x + a)^n` is

`T_(r+1)=text()^nC_rx^(n-r) . a^r`

Where `text()^nC_r` is called as combinatonal or binomial coefficient also denoted by `(n/r)`

Also, `(x+a)^n=sum_(r=0)^n text()^nC_r x^(n-r) . a^r`



`text(Properties of Binomial Expansion)` `(x + a )^n : `

(i) This expansion has (n + 1) terms.
(ii) Since, we have `text()^nC_r = text()^nC_(n - r)`

`text()^nC_0 = text()^nC_n = 1`
`text()^nC_1 = text()^nC_(n-1) = n`
`text()^nC_2 = text()^nC_(n-2) = (n(n-1))/(2!)` and so on.

(iii) In any term, the suffix of C is equal to the index of a and the index of x = n- (suffix of C).
(iv) In each term, sum of the indices of x and a is equal to n.


`text(Important Result:)`

1. Replacing a by(- a) in Eq. (i), we get

`(x-a)^n=text()^nC_0x^(n-0)a^0-text()^nC_1x^(n-1)a^1+text()^nC_2x^(n-2)a^2-.............+...-...+(-1)text()^nC_rx^(n-r)a^r +...................+(-1)text()^nC_nx^(0)a^n....................(ii)`

or `(x-a)^n=sum_(r=0)^n(-1) text()^nC_r x^(n-r) . a^r`

2. On adding Eqs. (i) a.ncl (ii). we get

`(x+a)^n+(x-a)^n = 2{text()^nC_0x^(n-0)a^0+text()^nC_2x^(n-2)a^2+text()^nC_4x^(n-4)a^4 + ............}`

`= 2{ text(sum of terms at odd places)}`

The Last term is `text()^nC_n a^n` or `text()^(n-1)C_n a^(n-1)` according as n is even or
odd , respectively .

3. On subtracting Eq. (ii) from E:q. (i). we get

`(x+a)^n-(x-a)^n = 2{text()^nC_1x^(n-1)a^1+text()^nC_3x^(n-3)a^3+text()^nC_5x^(n-5)a^5 + ............}`

`= 2{ text(sum of terms at even places)}`

The Last term is `text()^nC_(n-1) a^(n-1)` or `text()^(n)C_n a^(n)` according as n is even or
odd , respectively .

4. Replacing x by 1 and a by x in Eq. (i) we get

`(1+x)^n=text()^nC_0x^0+text()^nC_1x^1+text()^nC_2x^2+...................+text()^nC_rx^r+...................+text()^nC_(n-1)x^(n-1) +text()^nC_nx^n `

or `(1+x)^n = sum_(r=0)^n text( )^nC_rx^r`

5. Replacing x by(- x) in Eq. (iii), we get

`(1-x)^n=text()^nC_0x^0-text()^nC_1x^1+text()^nC_2x^2-...................+(-1)text()^nC_rx^r+...................+text()^nC_(n-1)x^(n-1) +text()^nC_n(-1)^nx^n `

or `(1-x)^n = sum_(r=0)^n (-1)text( )^nC_rx^r`