Binomial Theorem

Binomial Expression

Any algebraic expression consisting of only two terms is known as a binomial expression.

It’s expansion in power of x is known as the binomial expansion.

e.g. (i) a + x (ii) a2 + 1/x2 (iii) 4x − 6y

Binomial Theorem
Such formula by which any power of a binomial expression can be expanded in the form of a series is known as binomial theorem.

For a positive integer n , the expansion is given by :

(a + x)n = nC0an + nC1an−1 x + nC2 an−2 x2 + . . . + nCr an−r xr + . . . + nCnxn

where nC0, nC1 , nC2 , . . . , nCn are called Binomial co-efficients and there are (n + 1) terms in the expansion.

The value of nCr is defined as :

Similarly,

(a−x)n = nC0annC1an−1 x + nC2 an-2 x2 − . . . +( −1)r nCr an−r xr + . . . + ( −1)n nCnxn

Illustration : Expand

Solution:

7C0x7 + 7C1x6 + 7C2x5 + 7C3x4 + 7C4x3 + 7C5x2 + 7C6 x + 7C7

= x7 + 7x5 + 21x3 + 35 x + 35/x + 21/x3 + 7/x5 + 1/x7

General Term in the Expansion:

The general term in the expansion of (a + x)n is the (r + 1)th term given as :

tr+1 = nCr an-r xr .

Similarly the general term in the expansion of (x + a)n is given as :

tr+1 = nCr xn-r ar .

The terms are considered from the beginning.

The (r + 1)th term from the end = (n − r + 1)th term from the beginning.

Corollary: Coefficient of xr in expansion of (1 + x)n is nCr

Illustration : Find the co-efficient of x24 in

Solution:

General term ((r + 1) th term) in

15Cr(x2)15−r(3a/x)r

= 15Cr x30-2r(3rar/xr)

= 15Cr 3rar x30-3r

If this term contains x24 .

Then 30−3r = 24

=> 3r = 6 => r = 2

Therefore, the co-efficient of x24 = 15 C2 . 9a2

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