Binomial Expansion - Interpolation and Extrapolation, Business Mathematics and Statistics Video Lecture | Business Mathematics and Statistics - B Com

now it's quite common to find yourself
having to expand expressions say
something like this 3 plus 2x all to the
power 4 and to do that you might write
down all the 4 brackets you might then
go on to expand the last two brackets
get something like this a quadratic
expression you'd go on to simplify that
say giving you this and then multiply
the 3 plus 2x by everything in this
bracket starting to get quite long now
tidy up this expression here in the next
line it is tidied up and then finally
expand the 2 brackets again giving you
this horrendously long expression and
then just simplify that okay and finally
you get the right answer
imagine what this is going to be like
though if it was say to the power 5 5 of
these brackets few more lines to go what
happens here what's the power 10 say be
way down here and it'd be horrendous off
I would hate to do it this has got to be
an easier way
when deed there is it's called the
binomial expansion I'll take you through
it
if you've got a bracket that's got two
terms in it of the form let's say a plus
B raised to a positive integer I do
stress it must be to a positive integer
like this example up here this was a
positive integer for positive integers
remember our numbers like 1 2 3 4 5 and
so on then this is identical it can be
shown this is identical to something
called n C 0 times a to the power N and
then B to the power 0 plus another term
and I'll go through these terms in a
minute but just want to return back to
this function
nc0 often called NCR in general now what
I'm doing is I'm going to carry on with
this formula but I'm going to assume
that you know what this function is now
if you don't I've written the tutorial
on this showing you how to work this out
manually or how to do it on a calculator
and you'll find a link to this below
this tutorial okay so I'm going to
assume then that you know what NCR is so
our first term then is NC 0 you take a
raise it to the power n and you take
your second term and you raise it to the
power 0 we continue a pattern similar to
this ok for the next term we write NC
and then we increase the 0 by 1 and we
now reduce the power of a by 1 so it
becomes a to the N minus 1 and we
increase the power of B by 1 so this
becomes B to the power 1 we move on to
the next term which is going to be n C
to reduce the power of a by 1 it was n
minus 1 so reduce it by another 1 so it
becomes a to the N minus 2 and increase
the B power by 1 so it was B to power 1
it's now B to the power 2 B squared in
other words moving on to the next term
hopefully you can guess what it's going
to be just give you a moment just to
think about it okay
hopefully you thought that it would be
nc3 then it'd be a to the n minus 3 and
then increase the B power by 1 B cubed
it's going to go on like this all the
way down to the last term and the last
term would be NC n and the a would
eventually be a to the power n minus n
which will be really a 2 power 0 and B
would have gradually built up to B to
the power
so I want you to remember this formula
okay because it is incredibly efficient
at working out expansions like the one
that we have above okay and I'm going to
demonstrate that to you now so we'll
take the 3 plus 2 X to the power 4 3
plus 2 X then to the power 4 you can see
that the a is the 3 and the B value is
the 2 X and n is 4 so according to the
formula it's going to be NC 0 so in our
case it's going to be 4 C 0 and then
it's a to the power n so that'd be 3 to
the power n n being 4 3 to the power 4
and then it's the B term which is 2x
I'll put that in brackets because it's
made up of two bits they're all to the
power 0 moving on to the next term
becomes 4 see one reduce the power on
this 3 so we reduce it by 1 so it's 3 to
the power 3 then at the same time
increase the power over the 2 X that's
2x to the power 1 next term is 4 see to
reduce the power over the 3 so it
becomes 3 2 power 2 then increase the
other terms power so that's 2x now to
the power 2 next term 4 C 3 reduce the
power over the 3 by 1 and that's 3 to
power 1 increase the power on the 2x by
1 so that's now up to 2 X to the power 3
and then we come to the last term for C
4 reduce the power over 3 so that
becomes 3 to power 0 and increase the
power on the 2x that becomes 2x to the
power 4
okay now all we need to do is
get on the calculator and just work
these terms out now for c0 you may know
or if not just type it into your
calculator but it turns out to be one
and if you do 1 times 3 to the 4 and by
the way 2 X to the power 0 anything to
power 0 is 1 so we've just got 3 to the
power 4 here then 3 to the power 4 turns
out to be 81 the next term is going to
be a plus term and again use your
calculator to work out for c1 or you
might know this standard result it's 4
so you've got 4 times 3 cubed times 2 X
if you work that out again on your
calculator you should find you get 216
and then you got the X on the end the
next term is an x squared term typing
for c2 on your calculator or work it out
manually multiply it by 3 squared and
don't forget here it is really 2 squared
in here okay work that out you should
find you get another 216 but it would be
216 x squared this term for c3 for c3 do
it on the calculator or you could this
is a well known result it's for 4 times
3 times the 2 cubed 2 cubed comes out to
be 8 8 times 3 times this value turns
out to be 96 that's going to be 96 X
cubed and then you've got this last term
which is then going to come to 16 if you
do it on your calculator so that's going
to be 16 X to the power 4 now clearly
you can see how quick this is compared
to writing out all these lines here and
so whatever integer power this was if it
was 5 or 6 or whatever we could write
out something like this and then very
quickly work
count the terms and generate the
expansion now I've given you this
example just so you can compare the
methods but it is quite important that
you look at some of my other examples
where I've handled expansions where
we've got a minus here for instance
you're just got to be very careful that
you don't get caught out over how you
lay it out so do try and have a look at
that tutorial anyway I hope that gives
you some idea of how to use this
particular method