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# Binomial Theorem and its Simple Applications

## Bionomial Theorem and its Simple Applications PDF Notes, Important Questions and Synopsis

SYNOPSIS

1. A binomial is a polynomial having only two terms. For e.g 2y2- 1
2. (x + y)n can be expanded using the Binomial theorem without actually multiplying it n times.
3. Properties of Binomial Expansion (x + y)n

i.Total number of terms in this expansion is n + 1.

ii.The exponent of x decreases by 1, while the exponent of y increases by 1 in subsequent terms.

iii.The first term is   and the final/last term is

iv.The general term in this expansion is given by

4. Binomial Coefficients:
Binomial coefficients in the expansion of (x + y)n are simply the number of ways of choosing x from the brackets and y from the rest.

5. Pascal’s Triangle:
Binomial coefficients can be found using Pascal’s triangle given below.

We can also say that
.
Also, the binomial coefficient is given by

6. Terms in the binomial expansion of (x + y)n

i.General Term (rth term): General term in the expansion of (x + y)n is given by

ii.Middle Term(s):

1. When n is even,

2. When n is odd,
and

iii.Greatest Term:
In any binomial expansion, the values of the terms increase, reach a maximum and then decrease.

So, to find the greatest term, find the value of r till

So, the greatest term occurs when
r =
iv.Term independent of x:

The term independent of x is the term not containing x.
So, find the value of r such that the exponent of x is zero.

7. Applications of Binomial Expansion:
i. We have a very important result

, n ≥ 1, n ∊ N
ii. Finding the remainder using Binomial Theorem:
To find the remainder when pn is divided by q, adjust the power of p to pm which is very close to b, say with difference 1 and then divide by taking the remainder always positive.

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