About "Binomial theorem"

Binomial theorem :

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

For example, x + 2y, x − y, x3+ 4y, a + b etc.. are binomials.

Expansion of Binomials with positive Integral Index:

We have already learnt how to multiply a binomial by itself. Finding

squares and cubes of a binomial by actual multiplication is not difficult.

But the process of finding the expansion of binomials with higher powers such as (x + a)10, (x + a)17, (x + a)25 etc becomes more difficult. Hence we look for a general formula which will help us in finding the expansion of binomials with higher powers.We know that

(x + a)= x + a = 1C0 x1a0+ 1C1 x0a1

(x + a)= x2+ 2ax + a2

= 2C0x2a+ 2C1x1a+ 2C2x0a2

(x+a)= x3+3x2a+3xa2+a3

  =   3C0x3a0+3C1x2a1+3C2x1a2+3C3x0a3

(x+a)4 =x4+4x3+6x2a2+4xa3+a4

  =   4C0x4a0+4C1x3a1+4C2x2a2+4C3x1a1+4C4x0a4

For n = 1, 2, 3, 4 the expansion of (x + a)has been expressed in a very systematic manner in terms of combinatorial coefficients. The above expressions suggest the conjecture that (x + a)should be expressible in the form,

(x + a)n

= nC0 xna+ nC1xn − 1a+ nC2xn − 2a+ …..................

+ nCn− 1 x1an− 1 + nCn x0an

Remarks in binomial theorem

In the above expansion the power of x will get reduced and power of a will get increased.

the general term is nCr xn − r ar

Since this is nothing but the (r + 1)th term, it is denoted by Tr + 1

Tr + 1 = nCr xn − r ar.

2. The (n + 1)th term is Tn + 1 = nCn xn − n an = nCn an, the last term.

Thus there are (n + 1) terms in the expansion of (x + a)n

3. The degree of x in each term decreases while that of “a” increases such that the sum of the powers in each term is equal to n.We can write 

4. nC0, nC1, nC2, …, nCr, … , nCn are called binomial coefficients. They are also written as C0, C1 , C2, … , Cn.

5. From the relation nCr = nCn − r , we see that the coefficients of terms equidistant from the beginning and the end are equal.

6. The binomial coefficients of the various terms of the expansion of (x + a)for n = 1, 2, 3, … form a pattern.

This arrangement of the binomial coefficients is known as Pascal’s triangle after the French mathematician Blaise Pascal (1623 – 1662). The numbers in any row can be obtained by the following rule. The first and last numbers are 1 each. The other numbers are obtained by adding the left and right numbers in the previous row.

1, 1 + 4 = 5, 4 + 6 = 10, 6 + 4 = 10, 4 + 1 = 5, 1

How to find the middle term in binomial expansion ?

The number of terms in the expansion of (x + a)depends upon the index n. The index is either even (or) odd.

Let us find the middle terms.

Case (i) : n is even 

The number of terms in the expansion is (n + 1), which is odd. Hence, there is only one middle term and it is given by T(n/2) + 1

Case (ii) : n is odd

The number of terms in the expansion is (n + 1), which is even. Hence, there are two middle terms and they are given by T(n + 1)/2 and T(n + 3)/2

After having gone through the stuff given above, we hope that the students would have understood "Binomial theorem". 

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