1.Algebra Booster

CHAPTER
6 Binomial Theorem
CONCEPT BOOSTER
1. DEFINITION
An algebraic expression which contains two and only two
terms, is called a binomial expressions.
2
1 3 x
For examples, a + b, 2x + 3y, 4p + 5q, x + ,
x x
- 4
, etc.,
are called binomial expressions.
In general, expressions containing more than two terms
are known as multinomial expressions.
2. FACTORIAL OF A NATURAL NUMBER, n.
It is a continued product of first n natural numbers. It is generally
denoted as n! or (n) and is defined as
n! = 1.2.3 … (n – 1).n.
= n(n – 1)(n – 2) … 3.2.1
= n(n – 1)!
= n(n – 1)(n – 2)!
= n(n – 1)(n – 2)(n – 3)! and so on
(i) (–n)! is not defined
(ii) (0)! = 1
(iii) (1)! = 1
(iv) (2)! = 1
(v) (3)! = 6
(vi) (4)! = 24
(vii) (5)! = 120
(viii) (6)! = 720
(ix) (2n)! = 2 n ¥ (n!) ¥ {1.3.5 … (2n – 1)}
3. BINOMIAL CO-EFFICIENTS
For n Œ N, r Œ W and r £ n, the expression n C r
is called a
binomial co-efficient and it is defined as
n n!
Cr
=
.
r! ¥ ( n-r)!
(i) n C r
= n C n–r
(ii) n C 0
= 1 = n C n
(iii) n C 1
= n = n C n–1
(iv) n nn ( - 1) n
C2= = Cn
- 2
2
n nn ( -1)( n-2)
n
(v) C 3 = = Cn
-3
6
4. BINOMIAL THEOREM (FOR POSITIVE INTEGRAL INDEX)
For a, b ΠR, n ΠI + , then
(a + b) n n n-0 0 n n-1 1 n n-2 2
= Ca b + Ca b+
Ca b
0 1 2
n n-r r n n-n n
Ca r b Ca n b
+ + + + ,
where
n
C 0
, n C 1
, n C 2
, …, n C n
are called the binomial co-efficients.
This theorem can also be described in summarized form as
n
n n n-
r r
r
r = 0
( a + b)
=Â C a b
…(i)
Some special cases:
1. Replacing b by –b in Eq. (i), we get
n
n r n n-
r r
r
r = 0
( a - b) = Â (-1)
C a b
…(ii)
2. Replacing a = 1, b = x in Eq. (ii), we get
n
n n r
r
r = 0
(1 + x)
=Â C x
…(iii)
3. Replacing x by –x in Eq. (iii), we get
n
n r n r
 Cr
x
r = 0
(1 - x) = (-1)
4. Adding Eqs (i) and (ii), we get
(a + b) n + (a – b) n
n n n-2 2 n n-4 4
2 4
= 2( a + C a b + C a b + )