Question Video: Expanding a Binomial of 𝑛th Degree

Expand (7 + 2𝑥)³.

04:57

Video Transcript

Expand seven plus two 𝑥 all cubed.

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 now.

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 factorial.

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 expression.

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 number.

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.

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