## MATHS :: Lecture 03 :: Bionomial BINOMIAL THEOREM
A Binomial is an algebraic expression of two terms which are connected by the operation  ‘+’ (or) ‘-‘
For examplex+siny, 3x2+2x, cosx+sin x etc… are binomials.
Binomial Theorem for positive integer:
If n is a positive integer then ----(1)

Some Expansions
a) If we put a = -a in the place of   a in  b) Put  x =1 and a = x in (1)  ----------(2)
c) Put  x = 1 and  a = -x in (1)  -----------(3)
(d) Replacing n by – n in equation (2) ---------(4)
e) Replacing n by – n in equation (3) -----------(5)
Special Cases
1. 2. 3. 4. Note:
1. There are n+1 terms in the expansion of (x+a)n.
2. In the expansion the general term is . Since this is the (r+1)th term, it is denoted by Tr+1 i.e. .
3. are called binomial coefficients.
4. From the relation , we see that the coefficients of terms equidistant from the beginning and the end are equal.
Note:  The  number of terms in the expansion of (x+a)n depends upon the index n. the index is  either even (or) odd. Then the middle term is
Case(i): n is even
The number of terms in the expansion is (n+1), which is odd.
Therefore, there is only one middle term and is given by Case(ii) : n is odd
The number of terms in the expansion is (n+1), which is even.
Therefore, there are two middle terms and they are given by and Examples
1. Expand   (i) 2. Find 117.
Solution:
117=  (1+10)7  = 1+ 70 + 2100 +35000 + 350000 + 2100000 + 7000000 + 10000000
= 19487171

2. Find the coefficient of x5 in the expansion of Solution
In the expansion of , the general term is  Let be the term containing x5
then, 17-4r = 5 Þ r = 3 =  = 680 x5
\coefficient of x5 = 680.
3. Find the constant term in the expansion of Solution
In the expansion of , the general term is   Let be the Constant term then, = 0Þ r = 2
\ The constant term = = 180 Download this lecture as PDF here 