Binomial Theorem for any Index

Binomial Theorem for any Index

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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.