📄 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 
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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 
Solution
In the expansion of
, the general term is
Let 
be the Constant term then,
= 0Þ r = 2
\ The constant term 
= 
= 180
---|---
Summary Cheat Sheet
- Focus on core formulas, definitions, and solved patterns from this lesson.
- Practice stepwise derivations and numerical substitutions carefully.
- Connect each concept to practical agricultural problem-solving contexts.
References
1 source
References
Primary classroom notes and standard BSc Agriculture applied mathematics references.
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