The Binomial Expansion – AS Maths Revision – Core Mathematics (C2)

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

Pascal's triangle to 5 rows, the first row is ...

Pascal’s triangle to 5 rows, the first row is the zero row. (Photo credit: Wikipedia)

  • 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

About Matt

I like writing, filmmaking, programming and gaming, and prefer creating media to consuming it. On the topic of consumption, I'm also a big fan of eating.
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