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

Find the middle term in the expansion of (15π‘Ž + 3𝑏)Β².

02:19

Video Transcript

Find the middle term in the expansion of 15π‘Ž 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 15π‘Ž plus three 𝑏 squared. So it means that we’ve got 15π‘Ž plus three 𝑏 multiplied by 15π‘Ž 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 15π‘Ž multiplied by 15π‘Ž which gives us 225π‘Ž squared because 15 multiplied by 15 is 225. And π‘Ž multiplied by π‘Ž is π‘Ž squared. Then next, I’m multiplying 15π‘Ž by positive three 𝑏 which will give us plus 45π‘Žπ‘ because 15 multiplied by three is 45. And then π‘Ž multiplied by 𝑏 is π‘Žπ‘. Okay, great. So we’re finished with 15π‘Ž 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 15π‘Ž. And this, again, gives us 45π‘Žπ‘. 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 225π‘Ž squared plus 90π‘Žπ‘ β€” because we had 45π‘Žπ‘ plus 45π‘Žπ‘, so that’s 90π‘Žπ‘ β€” 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 90π‘Žπ‘ from our expansion.

So therefore, we can say that the middle term in the expansion of 15π‘Ž plus three 𝑏 all squared is equal to 90π‘Žπ‘.

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