GCSE Maths Simplifying Algebraic Fractions Explained

When it comes to simplifying algebraic fractions, which is really important before we get on to solving algebraic equations, we just need to think of these as normal fractions, right? So when we simplify a fraction, what do we normally do? Well, we look for something that we can divide the top and bottom both by, we look for the highest common factor. So for example, I can divide the top and bottom of this fraction by 2. And there you go, I have simplified the fraction because I can't, there is no other number I can divide 1 and 2 by, except by 1, which would be completely useless. Now the problem is, when we have algebra like this, it's a bit harder to see what we can divide them both by. So what we do first is we actually factorise them, to break them up into its factors. So let me give you another example of 6 over 9 here. If I break into its factors, 6 could be 3 times 2, and 9 could be 3 times 3. And then it's really clear that I can divide the top and bottom both by 3, right? In other words, I mean, you may have seen your teachers do this before, where you just cross out a 3, and you're left with just 2 over 3. So again, that's the overall concept, right? We're going to factorise this and then see if there's something we can divide the top and bottom both by, something that we can cancel out. So factorising the top, I can pull out an x to get x, open bracket x plus 5. The bottom is another quadratic, we're going to look for two numbers that times to 10, but add to give 7, which would be 5 and 2. So then we get x plus 5 and x plus 2. Now this means that I can divide the top and bottom by x plus 5. Or, again, the shortcut that your teachers may have taught you is just cancelling out both of those x plus 5, which leaves you with x over x plus 2, like so. Let's go on to a few other examples. So with this one, do, you know, try it yourself for a second, pause the video, have a go. But again, we're going to factorise the top and bottom. Now from the top, what I can do is I can take out a 3x, because both of these things can be divided by 3x. Then 3x times what gives me 3x squared? Well, I need another x. 3x times what gives me 9x? That'd be plus 3. And then the denominator, that is just the difference of two squares, right? x squared is a square number, 9 is a squared number, and there's a subtract. That means it must be a difference. So that means that would be x plus 3, x minus 3. And if you do difference of two squares, you literally just square root both things. So I square root x squared to get x, both brackets. Square root of 9 gives me 3, and we make one positive, one negative. And once again, I can see that I can get rid of the x plus 3 in the top and bottom, which leaves me with 3x over x minus 3, like so. We'll go through two more examples of questions that are slightly more tricky. For example, they might have a plus sign in between, or a subtract, or a times, or a divide. And all you do is you treat these like normal fractions again. What you're going to do is you're going to do the operation. So if it's times, you would times the fractions, divide, you divide the fractions, and so on. In this case, what do we do when we add two fractions? Well, the first thing we need to check is if the denominators are the same. And if they're not, what do you do? Well, again, if you go back to our example with numbers, by the way, this is exactly what I used to do in my exams. If I forgot how to add and break fractions, I would just try with normal ones. Well, all we need to do is times them by the other's denominator, right? If I times this denominator by 3, and this by 2, they'll both be 6. So here I can do the same thing. I'm going to times this fraction's denominator by 3, and this one's by 2. But remember, when you times the denominator, you have to times the numerator as well by the exact same thing to make sure that it's the same value. So here we have x plus 1 times 3, which would be 3x plus 3 over 6, plus 2x plus 6 over 6. And now we can actually add the numerators together. So we get 3x plus 2x, which is 5x. 3 plus 6 is 9 over 6. And then you need to ask yourself, can I factorise it to simplify it? Well, there's no number that divides by 5 and 9, so we can't do it. And here we go with one that's slightly more challenging with a few extra aspects to it. This time they give us that it can be written in the form 1 over x, x plus r, where r is an integer. So when we divide two fractions, what do we do? Keep, flip, change. We keep the first fraction the same. We change the divide to a times sign, and then we flip the fraction. So 3x minus 4 over 2x to the 4. Now here we could times the two fractions together, but in my opinion, it's easier to factorise anything we can. And if I look at this, I can factorise it, because this once again is the difference of two squares. 9x squared is a square number, because I can square root 9 and I can square root x squared. And 144 is also a square number. So this gives me 6x cubed over 3x minus 12, and then 3x plus 12. And that would be times 3x minus 4 over 2x to the 4. Now I can actually factorise a 3 out of both of these. So what I'm going to do there is I'm going to do 6x cubed over 3x minus 4 times 3x plus 4. And don't forget these are both 3s, so we can make that out of 9. So we've got 6x cubed over 9x minus 4x plus 4. Now by the way, you can cross-cancel some of this stuff, but I'm not going to use that in this particular example, just to show you the really full rigorous way to do it. But you could cancel the x minus 4 and the x minus 4 here. So now if we times it, we get 6x cubed times 3, which is 18x cubed. And that would be x minus 4 on the top. And on the bottom we get 9 times 2x to the power of 4, which is 18x to the 4. And we get x minus 4x plus 4. Now again, what's in the top and the bottom? Well, we have an x minus 4, so we can get rid of that. We also have an 18, so we can get rid of that. And if we look at the answer, there should just be a 1 on top. So we actually are going to cancel out x cubed here and change that to a 1. That leaves me with just 1 on the top, because everything else is cancelled, over x, x plus 4. Which if you look at the answer, 1 over x, x plus some number. So in this case, r is equal to 4. So hopefully you found that useful. If you did, please do consider subscribing. And again, if you want to join the group tuition sessions, then there is a link in the pinned comment and in the video description.