In this video, we will be looking at algebraic fractions. First of all, let's remind ourselves what an algebraic fraction is. An algebraic fraction is a fraction where the numerator and denominator are both polynomials. In order to simplify an algebraic fraction, we need to do two things: first of all, cancel any common factors in the numerator and denominator; secondly, reduce the fractions if possible.
This is a great tutorial for AQA GCSE maths students, but it can be useful for anyone who wants to learn more about algebraic fractions.
We are looking at algebraic fractions and there are three main subtopics to us. There are simplifying fractions, timing and dividing fractions and adding and minus fractions. So let's first look at simplifying fractions. A very simple version of this could be something like 25 over five x. What we can see here is we've got a common factor of five in both cases.
And so I can divide the top and the bottom by five. And what I end up with is going to be 25 of five, gives me 5512 five. And so that's a very simple example of simplifying. In general, it's a bit more complex, right? We often end up having to deal with quadratics, I guess, but it's the same principle.
We just want to find a common factor in the top and the bottom and then divide by that factor. And so in this case, if I want to find a common factor, well, the key is factor, right? What do I want to do when I want to find a factor is I have to factorise. And so I'm going to factorise this top, get this numerator.
What I need to do is I'm going to have my standard factorised format. I want to find two numbers that can times together to get six and adds together to get five. Well, that's just going to be plus three and plus two and my denominator stays the same. What we see is that we can therefore cancel that common factor, just like we did when we divided the top and the bottom by five and we cancelled five from the top and the bottom. And our final answer is going to be just x plus.
Tittle trick that I often teach is that when we're given something like this in an exam, we know we're going to probably be cancelling out some facts from the top and the bottom. And so I know that I'm most likely I'm going to have an x plus three bracket on top because I know that I'm eventually going to probably cancel that x plus three with the bottom x plus three. So I could just start off by filling in x plus three in that first bracket to save myself some time. Well, let's go into timing and dividing. So timing is very nice.
We'll cover dividing in a second. If I'm timing fractions together, it's just top times top, bottom times bottom, and so I end up with x squared times seven, seven x squared and five times x. I get five x. And I could simplify this further, I've got a common factor in the top and the bottom, just like we had. Again with our simple example.
In this case, I've got a common factor of x and so I can cancel out x from the top and the bottom. So I'm just left with x on the top rather than x squared and only for five on the bottom because again, I cancelled out that X and That's Our Answer. That's Our Final Answer. Right, moving on to the final sub topic, adding and minusing. So this is slightly trickier counterintuitively.
It takes a bit longer to do this. What we need when we're minus fractions. Just like with regular fractions, we need a common denominator. And there's a trick that we can use. We can basically use these two existing generators to turn this problem into a nicer problem.
So I'm going to take this whole fraction and I'm going to times it by that denominator. I'm going to take this whole fraction and times it by that denominator. And you'll see why in a second. When I do the first fraction times x, I end up with x cubed over five x and I take the second whole fraction. I times it by this denominator by five and I end up with five times seven to get 35 over five times x to get five x.
Then you can see right now that the tricks worked, right? We've ended up with the same denominator in both fractions. And so this can then be combined. We can just combine the numerators together now and end up with x squared sorry. X cubed -35 over five x and that is our final answer.
Let's go on to some practise exam style questions and put into practise what I just learned. So, yeah, we're going to first tackle a simplified question. So in this case, again, I can see I'm probably going to have to find a common factor in the top and the bottom. So I'm probably going to have to factorise the bottom. And I've therefore got to think of when I have my standard format for factorising, it's going to be the standard format there.
And I've got to think of two numbers that can times together to get -16 And I'm actually going to rewrite this as x squared plus zero x -16 because we can add in a zero x if you want to I know my two numbers here and here have to add together to get zero. Well, these two numbers are going to have to be four and minus four, right? Minus four times four gives me -16 four minus four gives me zero. So that's going to be my factor as former and again, we could have cheated and used the fact that I knew I probably needed an express four term in here. Right.
Because I know I'm probably going to be cancelling out a common factor. What we end up with is X minus four in our denominator. Right, moving on to dividing. Next. So just before we do any dividing, I'm actually going to want to simplify this slightly and turn it into a time zone question.
And just to look at a much nicer example, I can either do kind of a quarter of 20 and I can get five, or I can do 20 divided by four and get five. And I can actually take this four, and I can divide it by one. I'm not changing anything, but I'm just writing it in a different way. And what we see here is I can either divide by four over one or I can times by the flip or the reciprocal of that fraction. I can flip the top and the bottom and turn it into a times question instead.
So I'm going to do that here. I'm going to turn this question into a times question.
So it's going to be times X squared minus four, x over two, x squared plus nine, x plus flip the top and bottom and put a times in instead of a divide. And we're now going to factorise everything as we always do in this case. Well, I'm going to look at the first fraction. Can I factorise this? I've got a common factor of three in these two terms, so that numerator can probably be factorised.
I can pull out a three and write this as X plus two in our denominator. I can't do anything with that. That's just going to be my standard term, x minus four. And with the second fraction, again, I've got a common factor in these two terms, so I can pull out X. I'm going to be left with X minus four in the brackets, and I've got a standard quadratic on the bottom.
I'll hopefully be able to factorise that as well. So again, I'm going to be dealing with two X plus and minus something and X plus or minus something, because I know I need to have that two X squared term. So I've got to have a two X here and an X here. And we can use our cheat here, right? I know that I've got to find two numbers at times together to get ten.
So it's probably going to be one and ten or two and five. Well, I know I've got an X plus two here, right? So my first guess is going to be that I'm probably going to have an X plus T bracket here. I was going to have the T numbers. I'm going to times this T by T and then add it on to this number here to get the coefficient in front of X, right?
So I'm going to end up with four X plus something. X should give me nine X. Well, I know that something has to be ten. Sorry. And it all works out because I can double cheque.
It does. Two times five, give me ten. Yes, it does. So I know that's going to be my factorised format and I can now start cancelling some stuff. Right.
I've got X minus four, X minus four. I've got X plus T, x plus T. And so I end up with just top times, top, bottom times, bottom. I end up with three X on the top and end up with just one times TX plus five on the bottom. And that's my final answer.
Right, one more to finish off with an adding question. So I've got to solve for X, but I first of all got to take care of the left hand side. I want to add these two fractions together. So do our standard trick. I'm going to take the whole first fraction and times it by the second denominator.
So I'm going to times it by X plus seven on the top and the bottom.
And I'm going to take the second fraction and times it by this first denominator. So that's going to give me three brackets, five minus X, fall over again, five minus X times the original denominator, x plus seven. And what we can see here is I've now turned this into a much nicer question. I've got the same denominator in both cases, times that want to put plus sign there. And now what I need to do is I just add these two fractions together.
I just add the two numerators together and so I get two X plus seven plus three, five minus X all over five minus X, brackets X plus seven, when I'm solving for X, right? So I want to simplify this down as much as I can, and I don't want to have any X's in the denominator. So I'm going to times both sides by the denominator to get this whole thing out of the denominator, first of all. So I get just the numerator left where it is, and obviously one times this denominator is just going to be that same denominator.
What's next? Well, we probably want to expand out everything now because I want to eventually get all of my X's onto one side. And so if I expand this out, I get two X, and then I get minus three X. So I get minus X, and I've got two times seven is 14 plus three times five plus 1514 plus 15 is going to give me positive 29. So that's my left hand side.
My right hand side I'm going to end up with minus X times X. So minus X squared, I'm going to get X times five to get plus five X. I've got minus X times seven to get minus seven X, and I've got five times seven to get plus 30, 35 plus 35. Well, I know that five X minus seven X is just minus two X, and I generally want to have a positive X squared. It's quite tricky to solve quadratic when I've got a negative X squared.
So I'm going to actually get everything on the right hand side and I'm going to bring it over onto the left hand side. So I'm going to add X squared to both sides. That's going to give me X squared there. I'm going to add two X to both sides to take care of these two terms. So I've got minus X plus two X to go, plus X and I'm going to -15 from both sides.
So 29 -15 is going to give me minus six and I'm left with nothing on the right hand side. I've cancelled out all those four terms now. Well, now we're just solving a basic quadratic, right? So hopefully we can factorise this. I'll put this into my standard factorised format, and if we look at how we can solve this, I need to find two numbers that times together to get six adds together to get one that's going to be plus three and minus two.
And so x therefore, to make this whole thing zero, or if this brackets going to be zero, x has to be minus three. If this bracket's going to be zero, x has to be two. And that's the end of algebraic fractions.
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