Question Video: Finding a Certain Term in a Binomial Expansion

Video Transcript

Find the third term in the expansion of four 𝑥 plus three all cubed.

Well, if we actually think about our expansion as 𝑎 plus 𝑏 to the power of 𝑛, then what we can actually think of is actually we’ve got a general term formula. And it’s gonna actually help us find any term of our expansion. And this general term formula tells us that if we have a term 𝑟 plus one, then this is equal to 𝑛 choose 𝑟 multiplied by 𝑎 to the power of 𝑛 minus 𝑟 multiplied by 𝑏 to the power of 𝑟.

Okay, so now we have our formula. Let’s try and apply it to our expansion to try find our third term. Well, we want to find the third term. So therefore, we’re saying 𝑇 three. But this means that our 𝑟 is gonna have to equal two.

So what about our 𝑛? What will our 𝑛 equal? Well, if we see that actually our parentheses are raised to the power of three, then we know that 𝑛 is gonna be equal to three. So we can now start writing out our formula. So we can say that the third term is equal to three choose two multiplied by our first term, which is four 𝑥, to the power of three minus two, because that’s 𝑛 minus 𝑟, then multiplied by three to the power of two again because three is our 𝑏 and two is our 𝑟.

Okay, great! So now let’s try and simplify. Well, first of all, we’re actually gonna look at three choose two because we can actually find this on our calculator. But what does this actually mean? Well, what this actually means is actually is the number of combinations of two items that can be selected of a set of three items.

But how do we calculate it? Well, we calculate it using a general formula, which is that 𝑛 factorial over 𝑟 factorial multiplied by 𝑛 minus 𝑟 factorial. So in this case, we have three factorial over two factorial multiplied by three minus two factorial, which is just gonna give us three multiplied by two multiplied by one over two multiplied by one multiplied by one. And that’s because we find the factorial by multiplying each of the integers greater than zero that are less than or equal to the value that we have. So in this case, on the top here, be three factorial would be three multiplied by two multiplied by one.

Okay, great! So this would just give us a result of three because it’ll be three over one, which is just three. Okay, and if you check that on a calculator, you’ll get the same value. Right now, we know what three choose two is and where it comes from. Let’s get back on and find our third term. So our third term is gonna be equal to three multiplied by four 𝑥 multiplied by nine. So therefore, we can say that the third term in the expansion of four 𝑥 plus three all cubed is gonna be equal to 108𝑥.