Hello everyone, in this lesson we're just going to practice the difference of squares that we learned in the last lesson. So here's the first one. So x4 is a square number because x squared times x squared gives you x4.
9 is a square number because 3 times 3 is 9. So what we do is we just open up our brackets and we put x square and x square and we put 3 and 3 and then 1 gets a plus, 1 gets a minus, but it doesn't matter which. Which way around you do that? Here's an interesting one which a lot of people struggle with. It's because they think, what times what gives you 8?
But remember guys, when you are multiplying with exponents, what do you do with these numbers? For example, if I said a3 times by a2. Well remember, you don't multiply, you add those. So would you agree with me that x4 times x4 does give you x8?
Because we are adding. these numbers. We're not multiplying. I mean, we're not multiplying those numbers. So x4 can go here and x4 can go here.
Then what times what gives you 9? Well, that's easy. That's just 3 times 3. Then one of them gets a plus and one of them gets a minus. So please remember that, guys.
If I gave you x10, that's the same as x5 times x5. Why? Because we add these numbers. We don't multiply them. This one's quite good.
It's going to go on for quite a long time. We know that x4 times x4 gives you that, and 1 times 1 gives you that one. So we can do x4 minus 1, x4 plus 1. Now, if I look at this bracket, this one has two terms, there and there. There's a minus in between them, and each number is a perfect square, or each number is a square, so I can go even further. Okay, Kevin, but then what do we do with this one?
Why can't we do anything with that? Because it's got a plus, you can't go further. Remember, it has to be a minus. Okay, then what do we do with this? Do we just ignore it?
No, what you do is you just write it in the front, and then you carry on with this one. So we know that that will turn into two brackets, and it will be x2 and x2 and 1 and 1, and then one of them gets a minus, one of them gets a plus. So can you see what I've just done? This...
turned into this part. Because if I gave you this over here, you would say x2, x2, 1, 1, one of them will get a plus and one of them will get a minus. But now we're still not done, because this one has two terms, there's a minus, and each number is a perfect square. x times x gives you x squared, and 1 times 1 gives you that.
So what do we do with these two? Well, we add those into the front. We can't ignore them.
We just can't do anything with them. So we add them there. There we go. Now we do this part. So that means we're going to have two brackets.
It'll be x and x, 1 and 1. One of them gets a minus, one of them gets a plus. And now we are finished because these can't go any further. So that x8 minus 1 turned into that.
So here's another long one. Maybe pause the video and see how far you can get with this. So we know that this is the same as x8 and x8, okay?
But Kevin, 8 times 8 is not 16, it's 64. Yes, I know, but when you're multiplying these, then you're adding the exponents, and so that will give you 16. Aha! And then 1 times 1 gives you 1, so that'll just be a plus, that'll be a minus. Now, this one here with the minus can go further.
So what do we do with this part? You just leave it as it is. So that's just going to be x. I'm actually just going to move this up.
This one's going to get quite long. So we can say equals to x 8 plus 1. Then we open up our two brackets and that becomes x 4, x 4 and 1. One of them gets a plus, one of them gets a minus. This one over here can still go further because x 2 times x 2 gives you x 4 and 1 times 1 gives you 1. What do we do with the other stuff? You just leave it in the front.
You mustn't ignore it. You just keep rewriting it. Okay, now this is going to turn into two brackets, where it'll be x2 plus 1, x2 minus 1. Now, this one can go even further, because it's got a minus, and x times x gives you x squared, and 1 times 1 gives you 1. So that'll give you everything else I'm just going to write in the front. It's quite a long question, this one. Okay, and now we open up two brackets.
And so that becomes x and x, 1 and 1, plus 1, minus 1. At this point, we are finished because this part here, but the minus can't go any further. Now, when you look at this one, technically you can't do any difference of squares because 2 is not a square number. You can't say something times something gives you 2, and you can't say something times something gives you 16, I mean 32. But we said that whenever you get stuck like this, your first step must always be see what you can take out. That's what you should always try to do first.
So we know that the number 2 can fit into both of these, so we take it out as a common factor, and you'd be left with x squared minus 16. Now this, in the bracket, can become a perfect square. Why? Because x times x gives you x squared, and 4 times 4 gives you 16. And so we're going to end up with two brackets, x times x, 4 times 4. One of them gets a plus, one of them gets a minus. Always remember, guys, try take out a common factor first. For every single factorizing question that you do this year, try take something out first.
But Kevin, why didn't we do that in the last question? Yes, you guys are right. I should have tried, but I just knew that there was no number that could be taken out.
But you should always look for it first. Okay, so here there's no difference of square, but we should always try take something out first. The number 3 can go into both of those, so that just becomes x squared minus 9. Now x squared is the same as x times x, 9 is the same as 3 times 3, and there's a minus between them.
So we can open up our two brackets. The 3 that you take out must stay there. So it's going to be x and x, 3 and 3, one of them gets a plus, one of them gets a minus.
Okay, so here you can take out a 7, then you're left with x squared minus 4, and then x times x gives you x squared, and 2 times 2 gives you 4, so we open up two brackets. We say x and x, 2 and 2. One of them gets a plus, one of them gets a minus. Here's the same question with a plus in between.
So you can take out a 7, that's always allowed, x squared plus 4. But then you have to stop, because you can't go further, because this is a plus. Remember that that difference of square method only works if it's a minus. So you can take the 7 out, but then you stop over there.
Step 1, see what you can take out? Here you can't take anything out. But what's nice is that... 2 times 2 gives you 4, and x times x gives you x squared. So we can say 2x, and then this will be 3y.
So we can have two brackets, 3y. Why? Because 3y times 3y gives you 9y squared.
There we go, and then one of them gets a plus, one of them gets a minus. Okay, here's another one. So 10 times 10 gives you 100, and x times x gives you x squared.
3 times 3 gives you 9, and y times y gives you y squared. So we can open up two brackets. where we get 10x and 10x and then 3y and 3y.
One of them gets a plus, one of them gets a minus. Okay, so here there is no square number because you can't multiply any numbers together to give you a 3. But remember step 1, always see what you can take out. So the number 3 can fit into 3 and 12. And this one has 3x's and this one has 1x, so we can take an x out. Then look what we have left.
We have x squared left over for this one, minus. 4 now x squared minus 4 x times x gives you x squared 2 times 2 gives you 4 and there's a minus So we leave the 3x in the front open up 2 brackets x times x 2 times 2 one of them gets a plus and one of them gets a minus there we go guys. Thank you for watching