16a Binomial Expansion

Añadido: 2026-05-16

[00:00:00]Question 16. The binomial expansion of 2 - x^ 2 ^ 7 is given by that. Okay, that looks a bit scary but it is the binomial pattern and we just have to work it out. So almost like don't look at all that.

[00:00:18]Okay, let's just think about the pattern. So it's row seven of Pascal's triangle.

[00:00:30]And we might not know that. We probably don't know that, but it's in somewhere in here. It's probability, isn't it? I always forget on this calculator. If you're using the black and whites, it's just the shift and then it's either the times or the dividers got the NCR over it. Okay, so I want seven C. Now, I know row seven is going to start off with a one and then a seven. And that's 7 C 0 and that's 7 C1. So, I want 7 C2 and 7 C2 is 21.

[00:01:07]7 C3.

[00:01:11]No, 7 C3 is 35. So, if you don't want to write your triangle out all the way to the end, all the way to row seven, you can do this instead.

[00:01:24]Okay, it's going to carry on. I'm not actually going to need loads of terms, okay? All the way to the end. It's going to just start decreasing again. Okay, if I do work out another one. 7 C4 is 35 again, then 21, then seven, then one.

[00:01:39]Okay. So remember the pattern with binomial then okay as you multiply it out you get the first term to what I call full power.

[00:01:52]Okay. And your Pascal's triangle value is a one. So you have a 1 * a 2 to the^ 7. Your next term is going to be a seven from Pascal's triangle.

[00:02:05]The first term drops a power. So that's going to be power six. And then the next term, the second part including the minus comes in power one. So it's an x squared going in as power one like that. The next one will be a 21 from Pascal's triangle. The two will drop to power five and the minus x^ 2 in the bracket will now be up to power two.

[00:02:33]We can keep going. Okay? There's nothing wrong with writing the whole thing out if you especially if you've got time.

[00:02:41]Okay? Sometimes you do need to pick out a certain term. So you could just go all the way along until you get to the term you need. Sometimes it does ask you to do the whole expansion. Sometimes it asks you to do the end of the expansion.

[00:02:54]Okay? So you just have to adapt.

[00:02:57]Okay? I'm not going to write them all out. I'm just going to jump to the last term. Now in the last term, I'm going to be back to a one again.

[00:03:05]The two isn't there because it's run out. Okay, it's been dropping down in powers. Power three, power two, power one, and now it's not there. It's power zero. Okay, this one, the second part is going to be there to full power. So, that's now built up to power seven.

[00:03:24]Okay, as a little check, the sum of the two powers should always add up to whatever row you're on. So, six and one is seven. 5 and 2 is 7. 4 and 3 is 7.

[00:03:36]Obviously, there's an x squared in there, so when we tidy it up, it's not going to be so obvious, but let's have a look.

[00:03:46]We don't actually need to work out all of the terms. What we actually want is this term, this term, and this term. So, I'm not actually having to figure out every single term.

[00:03:58]So this one is just 2 to the^ 7 which is 128.

[00:04:06]I don't need this term.

[00:04:08]This one I need cuz that's going to be the power four. Can you see that's going to be a squared term? That will be a power four term because we're tsing these powers together. So we're going to have 21 * 2^ 5.

[00:04:24]That minus sign is going to count twice.

[00:04:26]So it is definitely going to be a plus whereas actually this term would have been a minus.

[00:04:31]Okay, that would have been a negative.

[00:04:38]This one is back to positive again because we've got two minus signs there.

[00:04:42]Okay, but we don't have a number to include in our calculation.

[00:04:47]So that's 672 x to the 4.

[00:05:05]Okay.

[00:05:07]And then we don't need to work out that one. That one again would be negative because there would be three minus signs there. We've just got to go all the way along till we get to here.

[00:05:18]We've got a one. No more numbers. So the coefficient is a one. But look, we've got a minus seven times. So actually that is not a plus. That is a minus. And x^2 to the^ 7 times those powers together.

[00:05:33]That would be - x 14.

[00:05:38]So we can say that a 0 is 128.

[00:05:43]A2 is 672 that's positive and A7 is minus one. Okay, the numerical values of the coefficients of the terms.

[00:05:58]Okay, bit strange but there we are.