💻 Bionomial

Bionomial.

This lesson covers core applied mathematics concepts and their agricultural applications for BSc Agriculture learners.

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