Binomial Theorem and its Simple Applications
Binomial Theorem and its Simple Applications PDF Notes, Important Questions And Synopsis
Bionomial Theorem and its Simple Applications PDF Notes, Important Questions and Synopsis
- A binomial is a polynomial having only two terms. For e.g 2y2- 1
- (x + y)n can be expanded using the Binomial theorem without actually multiplying it n times.
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
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.
Binomial coefficients can be found using Pascal’s triangle given below.
We can also say that
Also, the binomial coefficient is given by
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
1. When n is even,
2. When n is odd,
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
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.
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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