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 x5, x4, x3, 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.
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...