# Binomial expansion

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## Binomial expansion

To expand an expression like (2x - 3)^{5} takes a lot of time to actually multiply the 5 brackets together. Instead we use a fast way that is based on the number of ways we could get the terms x^{5}, x^{4}, x^{3}, etc. and is calculated as follows.

- Write down (2x) in descending powers - (from 5 to 0)
- Write down (-3) in ascending powers - (from 0 to 5)
- Add Binomial Coefficients.

This gives:

* Note:* You don't have to write down the powers of zero (and the multiples of 1) as they both equal 1.

When expanding (a+b)^{n}, the **binomial coefficients** are simply the number of ways of choosing 'a' from a number of brackets and 'b' from the rest, and are found using...

**1.** * Pascal's Triangle* which looks like this:

and so on. (Each value is found by adding the two above it.)

**2.** The formula for the * coefficients in Pascal's Triangle* is written as: 'n choose r' is

**This means we can write the sequences in Pascal's triangle as,**

**This formula allows us to now calculate the coefficients for any binomial expansion.**

Using the formula for binomial coefficients it is possible to now expand any bracket in the form, (1+x)^{n} where n can be any real number (i.e. n ∈ ℜ). This gives us the formula for the **general binomial expansion** as:

This series carries on forever (unless n is a positive integer. i.e. n ∈ Ζ, in which case we use the earlier rules).

As the series is infinite it can only converge if -1 < x < 1, (normally written as |x|<1).

* Note:* If the bracket does not start with 1 then rearrange to get (1 + ...) as follows:

And use the expansion to get,

and so on...