In formulae, letters can be used to stand for unknown values or interchangeable values. These formulas can be written and equations solved for a range of problems in industries such as science and engineering. In algebra, letters are used to stand for values that can change (variables) or for values that are not known (unknowns). Collecting like terms means simplifying terms in expressions in which the variables are the same.
Factorising is the reverse process of expanding brackets. A factorised answer will always contain a set of brackets.
Algebraic expressions are the smallest building block that we really use in math. And it's a key way that we can represent real world information into something we can actually handle better in math. Generally, algebraic expressions are split into a few different pieces is of increasing difficulty. First is actually representing mathematical information, then ways that we can manipulate information, quadratics, which are a very particular kind of mathematical equation which we use a lot. And this thing called oddness and evenness over here, representing information, can generally be split down further into two different parts.
The first is that essentially as mathematicians, we're lazy. We might want to talk about how for many thousand pounds something costs, or we might want to talk about the fact that this requires 20 litres of fluid, but rather than every time writing £1000, which takes a while, instead we invent letters which mean this information instead. In the case of pounds, something like this, or in the case of a litre, it will be 30 L meaning litres. And for the most part, we represent these letters as things like x, which is a bit more common to be used in the classroom. But really this is just the way that we turn things into these letters and we can then manipulate that using operations.
We've spoken about different operations before and math, but these are things like addition, multiplication, subtraction and division. And in more complicated ways, brackets, indices and other ways that you can do that. We're going to come up with some more complicated examples this in a minute. But really, most of math comes down to taking real world problems and putting them in a way we can understand and then doing rigid operations to them. The next stage of actually solving these is factorization and multiplication, which are just the various ways that you might want to simplify mathematical problems for obvious reasons.
We want things to be simple. And finally quadratics, which we'll come on to lastly, which is a particular kind of very common mathematical problem. And this is one which very commonly reflects real life situations. One thing we're not going to have time to go into today is odds and evenness is oddness and evenness. Oddness and evenness just refers to whether a number is even ends in 02468, or odd, which ends in 13579.
This is an important property in math, but is one that could be quite easily studied by yourself. Rather than requiring much explanation, let's give some examples of representing information. For a start, we've got here this apparently enormous drink, requiring five apples, two bananas and four cranberries to be made, expresses information as an algebraic equation. Now, the thing is, you could write this down as a series of instructions and it would be absolutely fine. This is how someone might use it in a bar.
But when you're using it in a bar, you're only dealing with small quantities. Say for example, someone wanted to make 570 drinks because they worked in a warehouse. At that point, it's very difficult to be writing the word apples that many times. So instead we represent these. And normally we pick a letter.
So for apples, we'll pick, I don't know, a say, and then for bananas, B, and for cranberries, C. So at this point, each of these letters just means apples, bananas and cranberries. Now, we can make this a step further if we want to and call these XYZ, but for now, this is good enough. We can just call these whatever we want. And all this is basically saying is that if you add these together, because if we mix these together, that means to add them to introduce operator of addition, we're going to get one drink, but we need to make sure we've got the correct quantities of these things.
Now. We could write a plus. A plus? A plus. And that demonstrates right there why we bother to represent the information in shorthand way.
Because rather than writing A five times, instead we write five in front of A. And this is just a bit of standard notation. This is how we represent this information. Two for bananas, make sure the colour is consistent, and then after that, we end up having the cranberries with four. And this would be the way you do this calculation.
And the reason why we bother to do this is because if, as per previous example, we wanted to have 527 drinks, we know that we've to multiply everything here by five to seven. So this is why we bother to represent the information with letters in the first place. It just makes our lives a bit easier. Now, similarly, we might need to mix and match particular quantities and simplify them. So to dilute the string down for a party, each individual piece of fruit requires a litre of water.
How could this be simplified as an equation? So in this case, we've got the same one as before, where we've got this five A plus two B plus four C equals one drink. But if we want to dilute it, then we've got to add an additional litre of water for every bit of fruit. There's a few ways that we can actually write this. For example, if we say that a litre of water is represented by W, or one W, depending on preference, you can write one in front of the letter or not.
So one W, that means in front of all of these, we'd have to write a bracket which contained A plus W, and then B plus W and C plus W. And the bracket introduces the idea that anything in a bracket is grouped together. So we'd have five A plus five W, two B plus two W and four C plus four W. Or alternatively, we know for a fact that there's going to be five W. From here, we know it's going to be two w from here and four W from here.
So we could just add those together and instead we could have five A plus two, B plus four C plus. And then we count the number of WS. Four WS plus two WS plus five WS is going to make eleven W's. And that'll make one diluted drink. Okay.
And that gives you an idea of how you can represent this information. Already we're introducing so many different mathematical operators, so many of them, this next one uses indices so much like how previously we said that if you want to add something five times, like do five apples, we can represent that by saying, okay, that's the equilibrium is doing five times A. That's where multiplication comes in. What if instead we wanted to have five times five and then times that by five? And the way we represent that is using these things called indices.
And indices are where we end up getting, for example, in that five with a two up top means do five times five. And if you turn that into a three, it would then be five times five times five, four and so on. So this one here says it takes 30 minutes of a cultural bacteria to double in size. Given the culture begins with a thousand bacteria, write an equation showing how many bacteria there will be after a number of hours h. So we know that we want a number of bacteria, we usually worsen the capital N, and that's going to be equal to our starting number, 1000.
And then that's going to multiply by two for every time 30 minutes travels by, which is the equivalent of an hour divided by two. And that would be how you'd show this. The mathematical equation looks complicated, but this is the way you show it. Things like doubling is something that we call exponential growth. And it's actually quite common in things like nature, where you see that the more you have of something, the more it increases by.
So that would be things like populations of bacteria or number of fruit and things like that. And if we wanted to make this into an equation to calculate bacteria after 5 hours, I'm actually going to leave that for you guys to calculate a home. And you can see there's a very big number. Moving on to our next bit, this is now how we multiply things that are actually in brackets already. Now, these are just a few of the techniques that you'd use.
This is not based on anything more fundamental, we've already been through that. But in this example, this would be a case of saying if somebody has got this many pounds of something and this person is going to be given the same number of pounds minus two, what's the total number? To do that, you might want to multiply these out. In general, when we have brackets, every term in a bracket has to be multiplied by every other term in a bracket. So I tend to start with the XS and I write them as we go.
So we have X times X, which gives us X squared. It's usually good Practise when you're multiplying out brackets to have the biggest number of X at the front. By that I mean if we had X cubed, that would go first. In this case it's X squared. So next we then have three times X.
So plus three X. Then we have minus two times X minus two X, and then three times minus two. So that's minus six. Now you may notice the fact that we've then got X squared plus three X minus two X minus six. So we want to simplify these X's.
That then turns into X squared plus X minus six. Now, to show how we simplify this, you may note the fact that we've got a negative number at the end. That's fine. It's very possible to have a positive number here, but a negative number here, just depending on the relative values of the brackets here. Now, when you have got a bigger bracket like this, sometimes there really is no easier way other than just multiplying it out.
So we can simplify this one here because it's going to give us a cubic by doing this part first. In this case, it's going to give us X squared plus three, x plus four, x plus twelve, and then we can multiply that by two X plus five. So I'm going to simplify in this step here, x squared plus seven, x plus twelve, multiplied by two X plus five. And every term needs to be multiplied, so this will tend to create quite a long value. Now, in this case, we're going to end up with X cubed.
I always like to start with that. So we have two X times X squared, which is going to give us two X cubed. Then we're going to have seven X multiplied by two X. So that's going to give 14 X squared. Then after that we're going to have twelve times two X.
Now there's no particular order you need to pick these in. When you've got these, you'll notice I'm just going through each of the first term in this particular equation. That's fine to do. So that's then plus 24 X. Okay, so that's done everything for this two X component.
Now we're going to have plus five X squared.
After that we end up getting plus 35 X and there's only one term without X in it, which is then going to be plus twelve times five, which is 60. Simplifying it down the centre into two X cubed plus 19 X squared plus 59 X plus 60. So that's quite a mouthful. And you can tell why we want to simplify these values down as much as we possibly can. It's arguable as to whether this is a neater way of writing things in these brackets or this multiplied out being more useful.
I can see it both ways, but it's important to know to be able to go from one to the other. Going from one way to the other involves a process of actually solving these equations, which we call quadratics. At GCSE level, you would not be expected to solve anything higher than a cube. Not only do x cubes actually appear that much in nature, but they're also quite difficult to solve compared to everything else. So the way that we solve quadratics is by looking at what we just did before and noticing if there are any patterns in the numbers that appeared.
So when we did x plus three times x minus two, we may have noted that we ended up with a three here and a minus two here, exactly the same as these two numbers here, x plus three, x minus two. And then the last number was the two of them multiplied together. So what that means is if we look at say, x square plus seven, x plus twelve, what we're looking for is turning this into two brackets. They both have pluses. Now I know these both have pluses because this is a positive number here, and because there's no obvious negative number, they're not going to be both negatives either.
And we're looking for numbers in here which add up to equal seven, but that multiply to make twelve. Now we can go through the multiple common factors of twelve. So MCF of twelve is equal to one and twelve, two and six, three and four. So it's fairly evident that the ones that add up to seven are three and four. And that's how you factorise this.
Just listing that multiple common factor method does really seem to be the easiest way. If we go down to this one here, we've changed the letter to Y here, but it doesn't mean that we need to change anything else. We're going to turn this into two brackets. Now we notice that this is a positive number, but this is a negative. So what that means is we're going to have two negatives because those are both going to multiply out to make positive eleven.
But at the same time it also means that we're going to be having a negative total number of YS. So we need to think about the common factors of eleven. Fortunately, it's a prime number, so it's only eleven and 111 one. And that's where we get Y minus twelve. So that's where we get minus twelve Y, and we're going to have the Y squared and the plus eleven.
Finally, we have this somewhat interesting result, which is known as the difference of two squares, very common in maths, which is where we're missing the X part. It might be a bit of a clue though, if I write for you guys. X squared plus zero, x minus four. So we need two numbers and we know that one is positive and one's negative. Because this is a negative number, they're going to add up to zero.
I'll give you a second to try and think of it.
And some of you may already know this is x plus two and x minus two, which is where we have the same number. One is a positive, one is a negative. And those are going to cancel out these x's here and it's also going to make it sees multiply to their square number. This is called the difference of two squares and it's a very important quantity in mass and it appears quite a lot in Simplifying equations. Hopefully you guys enjoyed that video and it was good to you, show you through some of the more simplified ways of going through algebraic expressions.
These are skills you'll be building on constantly as you go throughout your maths career. So don't worry if you feel like you don't get it all now. Just try and do some practise for yourselves and I will speak to you guys next time. So take care. Bye.
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