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)1 = x + a = 1C0 x1a0+ 1C1 x0a1
(x + a)2 = x2+ 2ax + a2
= 2C0x2a0 + 2C1x1a1 + 2C2x0a2
(x+a)3 = x3+3x2a+3xa2+a3
For n = 1, 2, 3, 4 the expansion of (x + a)n has been expressed in a very systematic manner in terms of combinatorial coefficients. The above expressions suggest the conjecture that (x + a)n should be expressible in the form,
(x + a)n
= nC0 xna0 + nC1xn − 1a1 + nC2xn − 2a2 + …..................
+ nCn− 1 x1an− 1 + nCn x0an
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)n 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
The number of terms in the expansion of (x + a)n 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
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