Expand seven plus two 𝑥 all
In order to expand this expression,
we’re going to be using the binomial theorem. So here’s the binomial theorem and
we’re actually gonna have a look at how we use it. Actually, it looks quite
complicated, but what it actually means is that our 𝑎 term will be descending in
power and our 𝑏 term is going to be ascending in powers. I’m gonna look at how that works
But before I do, I wanna quickly
draw your attention to this part which is a binomial coefficient. And what this tells us is what each
term in our expansion is actually gonna be multiplied by, and it’s gonna be worked
out in this way. But this shows us that our binomial
coefficient is equal to 𝑛 factorial over 𝑛 minus 𝑘 factorial multiplied by 𝑘
And a factorial is the product of
all positive integers less than or equal to 𝑛. So I’m gonna show you a quick
example of how we actually find a binomial coefficient. Okay, so here’s an example just
using that formula above are for the binomial coefficient four, two. So it’s got four factorial, which
is four multiplied by three multiplied by two multiplied by one, over 𝑛 minus 𝑘
factorial. So in this case that’s going to be
four minus two factorial, so that’s two factorial, so two multiplied by one, and
then multiplied by 𝑘 factorial.
So in this case, it’s two factorial
so that’s two multiplied by one, which gives us a binomial coefficient of six. It’s also though worth mentioning
this other notation, 𝑛 𝐶 𝑘, or on some calculators, 𝑛 𝐶 𝑟. And this is just another way of
representing the binomial coefficient.
So if you’re actually using a
calculator, what you type in is four then your 𝑛 𝐶 𝑟 button and then two, and then
what that will give you is your binomial coefficient of six. Great! So now we actually know what the
binomial theorem is and the parts within it. We can now go on and expand our
So we’ve got seven plus two 𝑥 all
cubed is equal to, well, here’s our first term. Well we’ve got our binomial
coefficient — three, zero — where we’ve got three because that is the actual power
of our brackets, so that’s our 𝑛, and then we’ve got zero because that’s our term
What’s also worth bearing in mind
is that any binomial coefficient with a zero at the bottom is always equal to one,
and we have that multiplied by seven cubed. And then the next part — which we
actually probably wouldn’t usually write down, but I’ve just put it in to show you
what actually happens — is we’ve got two 𝑥, our two 𝑥 term, to the power of
zero. Well anything to the power of zero
is just one so it says that it’d just be like multiplying it by one.
And that’s because our powers begin
at zero and then they work their way up. And then we can see our next term
where, as you can see, the two 𝑥 has had its powers raised. It’s now just two 𝑥 or two the
power of one and our seven term has actually decreased in power since instead of
being seven cubed, it’s now seven squared.
And the same with our next term,
our power of seven has decreased yet again so it’s now just seven, seven to the
power of one, and our power of two 𝑥 has increased so it’s now two 𝑥 squared,
which then brings us to our final term — which, again, I wouldn’t usually write
this, but I’m just trying to show you what’s happening. We’d have seven to the power of
zero, which again will be one, and then we’d have two 𝑥 cubed.
And then by either using a
calculator or the formula we have here for binomial coefficient, we can actually
simplify and work out each term. And then we get our first two
terms, which is 343 because that’s seven cubed, plus 294𝑥. And then I draw your attention to
the third term, which is plus 84𝑥 squared. Well this is actually where the
most common mistakes are actually made.
Okay, so first of all, we get
three, because that’s what our binomial coefficient worked out to be, multiplied by
seven and then multiplied by four 𝑥 squared, not two 𝑥 squared; this is where
people make the most common mistake because they forget to square the coefficient of
𝑥 as well as the 𝑥 itself.
So it’s multiplied by four 𝑥
squared, which gives us 84𝑥 squared. That then leaves us with our final
term, which is eight 𝑥 cubed. So then we can say our final answer
is, when we expand seven plus two 𝑥 all cubed, it is equal to eight 𝑥 cubed plus
84𝑥 squared plus 294𝑥 plus 343. And I’ve actually written it that
way because convention says that we usually work in descending powers of 𝑥.