Binomial Expansion with Examples and Solution

Binomial expansion:

$\boxed{\left(1+x\right)^n=1+nx+\frac{n\left(n-1\right)}{2!}x^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^3+…….}$

 are called meaningless when  is negative or fraction

 and  are exponents and  is called index.

 index is always less then one.

 exponent is always less then one.

This series is called Binomial series.

Example:

using Binomial expansion solve this:

$\boxed{\left(1-x\right)^\frac12}$

HERE

$\boxed{1=1}$

$\boxed{x=-x}$

$\boxed{n=\frac12}$

using Binomial expansion formula

$\boxed{\left(1+x\right)^n=1+nx+\frac{n\left(n-1\right)}{2!}x^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^3+…….}$

$\boxed{\left(1-x\right)^\frac12=1+\left(\frac12\right)\left(-x\right)+\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle\frac12}-1\right)}{2!}\left(-x\right)^2+\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle\frac12-1}\right)\left({\displaystyle\frac12}-2\right)}{3!}\left(-x\right)^3+….}$

$\boxed{\left(1-x\right)^\frac12=1+\left(\frac12\right)\left(-x\right)+\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle\frac12}-1\right)}{2!}\left(x\right)^2-\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle\frac12-1}\right)\left({\displaystyle\frac12}-2\right)}{3!}\left(x\right)^3+….}$

$\boxed{\left(1-x\right)^\frac12=1+\left(\frac12\right)\left(-x\right)+\frac{\left({\displaystyle\frac12}\right)\left(-{\displaystyle\frac12}\right)}2x^2-\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle-\frac12}\right)\left({\displaystyle\frac{-3}2}\right)}6x^3+….}$