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Good.

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Brackets.

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So we already mentioned brackets. If I want to emphasize that I'm taking this mathematical object, all of it,

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so it's not necessarily one number and one letter. I may have two letters, three letters, or I have an

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addition of two different terms, but I still want to take that as my mathematical object to consider.

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Then I put these brackets around it, okay?

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And then that becomes my new mathematical object.

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So there are rules. So this is a property. This is something. Yes, you want to remember it if you don't know it.

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If you know it, cool. But if you don't, then get, again, practice. And there is plenty of practice on doing those things.

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You have to be fluent in those little algebraic steps in our calculations. We really don't want to waste too much time on those.

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But it's important because when it comes to a complicated calculation, what is often happening is that we make, I make mistakes

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in those little algebraic steps like these.

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It's not that at the end of the second year, I will be explaining you how you can do image processing, you know, using a Fourier

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transforms, which is kind of cool. And how you take this image and you process it with the Fourier transforms, you stick it into the computers,

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and that will pixelize it and it will do calculations and you can do stuff with it because it's already in the form of a maths.

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But it's not going to be that complicated sophisticated method you're struggling with, you know, it will be this one plus one where we will drop the mistakes.

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And it's not because we don't know it, we just dropped the mistakes. It's not true. But more we practice it, better we will get.

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So that's the idea. You want to practice even those simple things to get better and make less mistakes.

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So the rules are listed here.

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And as before, note that there is an equal sign in between them.

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It works the other way. And it is often the case that we will be looking at the expression in the other way as well simplify the things, you know, you can see that on the left hand side here, for example, I have this simple thing,

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relatively simple, I have two brackets multiplying each other out.

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And on the right hand side of it, I will have this complicated expression.

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So it's often the case, I want to take this complicated expression and try to simplify it because then when I will give these answers to somebody, I want them to be as simple as possible so they can read them easily.

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So a couple of examples for multiplying three minus why that's for this. It's a four multiplying three minus why I don't really say the brackets when I am saying that out.

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But it is important to take a note of those things.

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So this is equals to four times three.

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Careful, there is a minus in between them.

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So it stays there minus four times why. Okay, so that is equals to 12 minus four why everybody follow the properties for the brackets.

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If you haven't seen them for a while, recap practice, I will send you an announcement later about what's what to do eight times to why that's a simple that is just a 16 why that is similar similar.

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What about this one.

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Five x minus two x plus one.

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What is happening there is something similar what was happening with the square root, you know, we know it's the second root of that object inside.

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We don't write it there.

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So in here, we know that the bracket is multiplied by minus one but we not really bothered to write the minus one there.

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Okay, so this is in fact five x minus one times two x plus one.

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Okay, so I can write it as a five x and now with the knowledge what we did before minus two x.

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That's a minus two x and minus one times plus one that is a minus one.

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Okay, so this is three x minus one.

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So many mistakes I have seen in the years I am teaching where students just forgot that there is a minus front of the bracket.

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And not, not like forgot completely, they will apply the minus to the first time the two x, but they will forgot that they have to do it to minus to the plus one.

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If you will practice it couple of times, you will be fine. Okay, any questions to this applying the law of brackets or properties of brackets.

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This one is important to keep in mind.

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So this is what we get.

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If we expand, this will be referred to it if you take two brackets and multiply them out, we expand those those brackets.

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If we have something complicated, we will try to put it back into the multiplication of some terms or, or some brackets, we will refer to it as a factorization.

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Okay, so multiplying brackets out is expanding the terms and trying to get these complicated terms as a product of less terms or even terms with the brackets.

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That's the factorization.

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There is an equal sign in here, both ways. It's true both ways. So the expansion is relatively easy.

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And we can just do one by one. So following the rule, following the layout, we can take the number of number three and multiply everything by the number three.

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We'll have a three times two plus four y. Careful, there is a minus.

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And it has to stay there. Okay, so that is my minus. And that's the x multiplying two plus four y again.

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So now I expand each of those separately. If I need to, so this is a six plus 12 y minus two x minus four x y.

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If you can do this in one go, then you're welcome to do it in one go. If you can do it without the mistakes.

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But if you do drop the mistakes, take the time, take the extra line and make the calculation when it comes to the assessment.

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It is designed in the way that you have all the time you need. So I'm accounting for everybody taking extra line in the calculation.

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Okay, there is a some time limit, but it's not the problem. The time limit assessment is not going to be the problem.

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We have some restrictions, you know, you can't just think, oh, I will look at it around Christmas. No, it's not that much of a time.