**Binomial theorem :**

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

For example, x + 2y, x − y, x^{3}+ 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 = 1C_{0} x^{1}a^{0}+ 1C_{1} x^{0}a^{1}

(x + a)^{2 }= x^{2}+ 2ax + a^{2}

= 2C_{0}x^{2}a^{0 }+ 2C_{1}x^{1}a^{1 }+ 2C_{2}x^{0}a^{2}

(x+a)^{3 }= x^{3}+3x^{2}a+3xa^{2}+a^{3}

= 3C_{0}x^{3}a^{0}+3C_{1}x^{2}a^{1}+3C_{2}x^{1}a^{2}+3C_{3}x^{0}a^{3}

(x+a)^{4} =x^{4}+4x^{3}+6x^{2}a^{2}+4xa^{3}+a^{4}

= 4C_{0}x^{4}a^{0}+4C_{1}x^{3}a^{1}+4C_{2}x^{2}a^{2}+4C_{3}x^{1}a^{1}+4C_{4}x^{0}a^{4}

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}

= nC_{0} x^{n}a^{0 }+ nC_{1}x^{n − 1}a^{1 }+ nC_{2}x^{n − 2}a^{2 }+ …..................

+ nC_{n− 1} x^{1}a^{n− 1 }+ nC_{n} x^{0}a^{n}

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

the general term is nC_{r} x^{n − r} a^{r}

Since this is nothing but the (r + 1)^{th} term, it is denoted by T_{r + 1}

T_{r + 1} = nC_{r} x^{n − r} a^{r}.

2. The (n + 1)^{th }term is T_{n + 1} = nC_{n} x^{n − n} a^{n} = nC_{n} a^{n}, 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. nC_{0}, nC_{1}, nC_{2}, …, nCr, … , nC_{n} are called binomial coefficients. They are also written as C_{0}, C_{1} , C_{2}, … , C_{n}.

5. From the relation nC_{r} = nC_{n − 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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