Question Video: Finding the Coefficient of a Term in the Expansion of a Binomial Mathematics

Video Transcript

Find the coefficient of 𝑎 sub five in the expansion of nine 𝑥 plus two to the power of six.

So what 𝑎 sub five tells us is that we’re looking for the fifth term of our expansion. And what we’re gonna use is the binomial expansion to help us solve this problem. And with a binomial expansion, we have a general form. And that tells us that if we have an expansion where we’ve got 𝑎 plus 𝑏 all to the power of 𝑛 — so we’ve got it in this form. Then this is equal to 𝑎 to the power of 𝑛 plus 𝑛 choose one 𝑎 to the power of 𝑛 minus one 𝑏 plus 𝑛 choose two 𝑎 to the power of 𝑛 minus two 𝑏 squared. All the way up to 𝑛 choose 𝑛 minus one multiplied by 𝑎 multiplied by 𝑏 to the power of 𝑛 minus one plus 𝑏 to the power of 𝑛. So we can see that our powers or exponents of 𝑎 are decreasing each time. And our powers or exponents of 𝑏 are increasing.

So if we take a look at our example in our question, we’ve got nine 𝑥 plus two all to the power of six. Then our 𝑎 is nine 𝑥, our 𝑏 is positive two, and our 𝑛 is six. So we can use these and substitute them into the binomial expansion to help us find out what the coefficient of 𝑎 sub five or the fifth term is going to be. Now, we don’t have to necessarily work out every term because what we’d need to do is work out the fifth time. But I’m gonna work through it just so we can see how it would come together.

So first of all, we’ve got our 𝑎, so nine 𝑥. And then, this is to the power of six. And then, we’ve got six choose one multiplied by nine 𝑥 to the power of five. And that’s because we reduced the exponent or power by one, then multiplied by two because two was our 𝑏. Then, we’ve got plus six choose two multiplied by nine 𝑥 to the power of four multiplied by two squared. So once again, we reduced the power of our nine 𝑥 and increased the power of our 𝑏 which is our two. So that was our third term. So we’ve got two more terms until we get to the term that we want.

So then, the fourth term is gonna be six choose three nine 𝑥 cubed multiplied by two cubed. So great, now we’ve reached the fifth term. And this is the term that we’re looking for. So we can use this to work out the coefficient while quickly is just finish the expansion. So then, we’ll have plus six choose five multiplied by nine 𝑥 multiplied by two to the power of five. And then, finally plus two to the power of six or our 𝑏 to the power of our 𝑛, which was six.

Okay great, so we’ve done this. Now, let’s work out the value of our fifth term’s coefficient. Well, the first thing we can notice if we’re trying to work out the value is that we’ve got this notation, six 𝐶 four or six choose four. But how do we calculate what this is? Well, first of all, you could use your calculator. So there’s a little button that might say 𝑛 𝐶 𝑟. So what you’ll do is you press, say, for instance, six and then 𝑛 𝐶 𝑟 and then four. And this gives you six choose four. And that will give you the value. But what does that actually mean?

Well, if you’ve got 𝑛 choose 𝑟, in our case six choose four, what this is equal to is 𝑛 factorial over 𝑟 factorial multiplied by 𝑛 minus 𝑟 factorial, where the exclamation mark, which we call factorial, means that number multiplied by each positive integer down to one. So, for example, three factorial will be equal to three multiplied by two multiplied by one. So let’s use this to work out what six choose four would be equal to. So it’ll be equal to six factorial over four factorial multiplied by six minus four factorial, which would be equal to six factorial over four factorial multiplied by two factorial.

Well, as we’ve already established, six factorial is six multiplied by five multiplied by four multiplied by three multiplied by two multiplied by one. Or we could rewrite this as six multiplied by five multiplied by four factorial, which can be very useful when we’re trying to work out the value we’ve got here. And that’s because our four factorials would cancel. So what we’ll be left with is 30 over two. And that’s cause six multiplied by five is 30. And two factorial is just two multiplied by one, which would give us a value of six choose four of 15. Okay, great, just so that we could see where that’s come from, I’ve shown you how to do that.

Now, let’s get back on and find out what our coefficient of the 𝑎 sub five is going to be. Well, we’re gonna get 15, cause I’ve just shown that, multiplied by 81𝑥 squared. And that’s cause nine 𝑥 multiplied by nine 𝑥 is 81𝑥 squared multiplied by 16. So this is gonna give us the fifth term is equal to 19440𝑥 squared. Well, we’re just interested in the coefficient. So therefore, the answer to our problem is 19440.