Binomial Theorem for any Index

Binomial Theorem for any Index

Binomial theorem for positive integral index

The rule by which any power of binomial can be expanded is called the binomial theorem.
If n is a positive integer and x, y ∈ C then
Binomial Theorem for any Index 1

Binomial theorem for any Index

Statement :
Binomial Theorem for any Index 2
when n is a negative integer or a fraction, where , otherwise expansion will not be possible.
If first term is not 1, then make first term unity in the following way,
Binomial Theorem for any Index 3

General term :

Binomial Theorem for any Index 4

Some important expansions

Binomial Theorem for any Index 5

Problems on approximation by the binomial theorem :

We have,
Binomial Theorem for any Index 6
If x is small compared with 1, we find that the values of x2, x3, x4, ….. become smaller and smaller.
∴ The terms in the above expansion become smaller and smaller. If x is very small compared with 1, we might take 1 as a first approximation to the value of (1 + x)n or (1 + nx) as a second approximation.

Three / Four consecutive terms or Coefficients

(1) If consecutive coefficients are given: In this case divide consecutive coefficients pair wise. We get equations and then solve them.
Binomial Theorem for any Index 7

Some important points

(1) Pascal’s Triangle
Binomial Theorem for any Index 8
Pascal’s triangle gives the direct binomial coefficients.
Example : (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4.

(2) Method for finding terms free from radicals or rational terms in the expansion of (a1/p + b1/q)N ∀ a, b ∈ prime numbers:
Find the general term
Binomial Theorem for any Index 9
Putting the values of 0 ≤ r ≤ N, when indices of a and b are integers.
Number of irrational terms = Total terms – Number of rational terms.

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