Yes, The Binomial Expansion

- The binomial expansion seems to be a shortcut for expanding (something + something)^something brackets without having to go through hundreds of steps
- Its general form is given in the formula booklet in the Core 2 exam
- So is another form, for when you have to expand (1 + something)^something [but the general form works just as well for those, too]
- Ignore the vertical brackets for now – those are just how you calculate the coefficients
- You’ll notice that it goes ?a^n + ?a^(n-1)b + ?a^(n-2)b^2… all the way to b^n
- The power of a decreases each time, and the power of b increases
- Usually, you’ll just be asked for the first three or four terms
- If it says “in ascending powers of x”, you know that b in the general form represents x, and a represents the other thing

The Coefficients

- The coefficient (number at the beginning of, basically) of each term in the expansion depends on the power the expansion is to
- You can use Pascal’s triangle to work out the coefficients, but that’s not fast enough
- There’s a quicker way in the form of the nCr button you hopefully have on your calculator
- The C can be pronounced as “choose”, because the result tells you how many ways you could choose r objects out of the total of n objects (where choosing the same objects in different orders doesn’t count)
- So, the vertical brackets in the binomial expansion’s general form want you to use the top one as n, and choose r
- nCr = n!/(r!(n-r)!) …but… do it on your calculator, if you can, to save time. (I’m pretty sure Core 2 is a calculator paper!)
- If you must use the above method, you’ll want to know that the exclamation mark, !, can be pronounced “factorial” (in other words “multiplied by all the positive integers below it”. I don’t know what would happen if you used ! on a number that wasn’t an integer…)
- 5! = 5*4*3*2*1

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