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

Video Transcript

Find the middle term in the expansion of 15𝑎 plus three 𝑏 squared.

Now, first thing we’re gonna do to actually find the middle term of this expansion is to actually rewrite our expression. And I’ve rewritten it as two pairs of parentheses because what we’ve got is 15𝑎 plus three 𝑏 squared. So it means that we’ve got 15𝑎 plus three 𝑏 multiplied by 15𝑎 plus three 𝑏. So now, the next stage is to actually expand our parentheses. And in order to do that, what I need to do is actually multiply every term in our first pair of parentheses by every term in second pair of parentheses.

So I’m gonna start by actually multiplying the first term in each parentheses. So we’ve got 15𝑎 multiplied by 15𝑎 which gives us 225𝑎 squared because 15 multiplied by 15 is 225. And 𝑎 multiplied by 𝑎 is 𝑎 squared. Then next, I’m multiplying 15𝑎 by positive three 𝑏 which will give us plus 45𝑎𝑏 because 15 multiplied by three is 45. And then 𝑎 multiplied by 𝑏 is 𝑎𝑏. Okay, great. So we’re finished with 15𝑎 in the first parentheses. Now, let’s move on to the second term in the first parentheses.

So now, we’ve got positive three 𝑏 multiplied by 15𝑎. And this, again, gives us 45𝑎𝑏. Okay, and now we move on to the last term in each of our parentheses. So we’ve got positive three 𝑏 multiplied by positive three 𝑏 which gives us our last term which is plus nine 𝑏 squared.

Okay, great. So now, we’ve expanded our parentheses. What we need to do is actually simplify. So therefore, this is equal to 225𝑎 squared plus 90𝑎𝑏 — because we had 45𝑎𝑏 plus 45𝑎𝑏, so that’s 90𝑎𝑏 — plus nine 𝑏 squared. So now we look back at the question. And the question asked us to find the middle term which we can see is positive 90𝑎𝑏 from our expansion.

So therefore, we can say that the middle term in the expansion of 15𝑎 plus three 𝑏 all squared is equal to 90𝑎𝑏.