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

Statement :

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,

General term :

Some important expansions

Problems on approximation by the binomial theorem :

We have,

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.

Some important points

(1) Pascal’s Triangle

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 (a^1/p + b^1/q)^N ∀ a, b ∈ prime numbers:
Find the general term

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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