**MATHS :: Lecture 03 :: Bionomial**

**BINOMIAL THEOREM**

A Binomial is an algebraic expression of two terms which are connected by the operation ‘+’ (or) ‘-‘

**For example**, *x+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, 17-4*r* = 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

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