## 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 x^{2}, x^{3}, x^{4}, ….. 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} = x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4}.

**(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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