In this video we're going to look at how you can simplify algebraic fractions. The way we do this is by factorising. I'm going to first of all run through all of the factorising you'll need to be able to do to do this topic.
First of all we have some single bracket factorisations, like this one here. In both terms here I have a common factor of 3, so I can factorise that out. Then inside the brackets to get 3a I would need to multiply by a, and then to get the one I would need to multiply by 7. For this one here we can factorise out a b.
So it would be b, and then inside the brackets, to get from b to b squared I need to multiply by b, and to get the negative 5b I would need to multiply by negative 5. And then we've got this one here. For this one there's lots we can factorise out. First of all look at the numbers 8 and 4, and the highest common factor of these is 4. Then if we look at the a's we've got a and a cubed, so the most we can factorise out there is a, and then we've got b to the 4 and b squared. So the most we could factorise out is b squared.
Then we need to work out what goes inside the brackets. To get from 4ab squared to 8ab to the power 4, we need to multiply by 2b squared. To get from 4ab squared to 4a cubed b squared, we need to multiply by a squared.
Now there's also some double bracket factorisations. If we take this quadratic expression here, you should recognise that we can factorise this into two brackets. You need to look for a pair of numbers that multiply to make 12, but add to make negative 8. For this one it would be negative 6 and negative 2, so the brackets would be x subtract 6, x subtract 2. You sometimes have some harder quadratics to factorise when the coefficient of x squared is not 1, for example this one here.
Now I'm not going to run through how you would factorise this one in detail, that will be in a separate video, but to factorise this one it would still be two brackets, the first would be 3x plus 4, and the second would be x plus 5. Then we have expressions like this. You need to recognise that this one is the difference of two squares. The giveaways are the square number, 16, and also that there's no x term. This will still factorise into two brackets.
If we square root 16 we get 4, so this is x plus 4 and x minus 4. You can also get difference of two squares that look like this. Once again you can see the square number 25, but we also have a square number in front of the x squared, 4 is the square number 2. So this one will factorise into two brackets as well. If you square root the 4 you get 2, and if you square root the 25 you get 5. This one's 2x plus 5. and 2x minus 5. Before attempting this topic you want to make sure you're comfortable with all of the factorizations I've shown you.
If not I suggest you go and revise some factorizing first. Now let's get stuck into this topic then. If we were asked to simplify this fraction here we want to do this by factorizing and looking for common factors. If we factorize the top this will be a single bracket factorization there's a common factor of 6. Inside the bracket to get 12m we would need 2m and to get to 18 from 6 we need to multiply by 3. On the bottom it's also a single bracket factorization.
This time we can factorize out the m. To get 2m squared from m we need to multiply by 2m, and to get 3m from m we need to multiply by 3. You can now see that we have a 2m plus 3 on the top and on the bottom. These are common factors so we can cancel them. So we're just left with 6 on the top and m on the bottom, so this one would simplify to 6 over m. Now let's try another one.
So on the top here we've also got a single bracket factorization, we can factorize out c to the power 3. To get from c to the power 3 to c to the power 4 we need one more c, and then we just need c to the power 3 again so we can multiply this by 1. On the bottom though we have a quadratic expression, this can be factorized into two brackets. We look for the numbers that multiply to make 5 but add to make 6, and that's 5 and 1. So this is c plus 5 and c plus 1. You can now see we have common factors on the top and bottom of c plus 1, so we can cancel those and we're just left with c to the power 3 over c plus 5. And let's try another. So for this one on the top we can factorise out a 6, so it would be 6, then brackets, x squared, and to get to 54 from 6 we need 9, so subtract 9. On the bottom we can factorise out 2x, to get from 2x to 2x squared we need an x, and to get from 2x to 6x we need plus 3. Now at first glance it doesn't look like there's much simplifying we can do. We can certainly simplify the 6 over 2 but it doesn't look like anything else cancels. But that's because there's a difference of two squares here.
So this bracket itself can also be factorised. x squared minus 9 is the same as x plus 3 x minus 3. Now we can cancel out the x plus 3 from the top and bottom and we can also simplify the 6 over 2. If you simplify 6 over 2 you get 3 over 1. So what we're left with on the top is a 3 and then the bracket x-3 and on the bottom we're just left with x. If you wanted to at this point you could expand out that bracket but this would be considered fully simplified still.
And let's try one more. So for this one I recognise a difference of two squares on the top so we can factorise that into a double bracket. It would be 3y plus 1 and 3y minus 1. Now on the bottom we have one of those tricky factorisations to do.
because the coefficient of y squared is not 1 anymore, it's 3. Now these can be quite tricky to do, however there's a big clue in how we factorise the top of this fraction. We're expecting that one of these brackets on the top will match one of the ones on the bottom, so that they can cancel out. And since all of the terms in the quadratic on the bottom are positive, we can suspect that one of the brackets might well be 3y plus 1. Now if it were 3y plus 1, what would the other bracket need to be?
Well to get the 3y squared we would multiply 3y by y. and to get the final term, the 6 at the end, we need to multiply the 1 by 6. And you can check this one and it does indeed work, because when you multiply 3y by 6 you get 18y, and when you multiply 1 by y you get 1y, and 18y plus 1y gives you 19y. So even though this quadratic would normally be quite difficult to factorise, it was made a little bit easier by looking at the brackets we had when we factorised the numerator. So we can now go ahead and cancel.
These two brackets are the same so they'll cancel out. So we're left with on the top 3y-1 and on the bottom y plus 6. Sometimes we get questions where there's more than one fraction. You can see these two have been multiplied together.
To do a question like this we're going to start by factorising everything we can. So on the left fraction on the top we can factorise out 5a and to get 20a squared and plus 5a inside the bracket we'd need 4a plus 1. Then if we move to the bottom there's a common factor of 3 so we could factorise out a 3 and then inside the bracket we just need a plus 2. Then if we move to the right fraction we can factorize a 6 from the top and then inside the bracket we'd need a squared minus 4. Now if we look inside that bracket we've got a squared minus 4 which is the difference of two squares, so we can factorize that again to give a plus 2 a minus 2. Now we move on to the bottom of this fraction, this is one of those difficult factorizations again so it's going to be two brackets. If you have a look around everything we factorized so far you can see we have a 4a plus 1. So you could bet this one will probably include a 4a plus 1. And indeed it does, and the second bracket would be a minus 2. Now that we've factorised everything, we can multiply the fractions together. To multiply fractions, you multiply the numerators and then the denominators. So if we multiply the numerators first, so we've got 5a and 6, that will multiply to give 30a, and then we've got three more brackets.
We've got 4a plus 1, a plus 2, and a minus 2. On the bottom, we've just got 3a plus 2 on the left one. and then two more brackets, 4a plus 1 and a minus 2. And now we come to the point of simplifying. If we look at the numbers we've got 30 divided by 3, that will just give you 10 over 1, or 10, and then for the brackets there's plenty that match, in fact the 4a plus 1s match, the a plus 2s match, and so do the a minus 2s, so they all cancel out. So all we're left with on the top is 10 and a, and on the bottom it's just 1, and 10a over 1 is just 10a. And we can even get questions that look like this.
In this one we've been told the form we need to present the answer in. The way I'm going to approach this one is by factorising everything I can first. So I'm going to leave that 9 alone and then try and factorise all of these bits of the fractions. So let's start with the left fraction on the top, we can factorise out a 2 there, so that's 2x plus 5. On the bottom we can factorise x to the power 3, inside here we need 3x and then plus 1. and on the right fraction I can see a difference of two squares on the top so that's x plus 5 x minus 5 and on the bottom we can factorize out x to the power 4, inside we would need x, subtract 5. Now in this question there's an addition and also a division. The order of operations says we should divide before we add, so we're going to leave that plus alone and work on dividing these fractions.
To divide fractions we turn that divide into a multiply, but take the reciprocal of the second fraction, so we need to flip that fraction upside down. Now we can go ahead and multiply. So we're going to leave the 9 and the plus alone and then we'll multiply these two fractions together. So if we multiply the numerators we have a 2, then we have an x to the power 4, and then we have two brackets which are x plus 5 and x minus 5. On the bottom we've got x to the power 3 and then we've got three brackets 3x plus 1, x plus 5 and x minus 5. Here you can see we have some common factors so we can get rid of those x plus 5s and those x minus 5s. We can also cancel the x's, we've got x to the power 4 on the top and x to the power 3 on the bottom.
So if we cancel those out we'll just be left with one x on the top. So we can now rewrite this as 9 plus and then on the top we have 2x, and on the bottom we've just got that one bracket 3x plus 1. Now we're trying to write it in that form on the right hand side, so we need to get these into one fraction. So what we're going to do is rewrite them both over a common denominator. The denominator we're given is 3x plus 1, so we need to write the 9 over 3x plus 1 as well.
If we were to write the 9 as a fraction over 3x plus 1, we would multiply the top and bottom by 3x plus 1. So it's 9 lots of 3x plus 1 over 3x plus 1. that's the same as 9. And then the second fraction is already over 3x plus 1. Now we can combine these into one fraction since they have the same denominator. So on the top we have 9 lots of 3x plus 1 and then plus 2x. We can expand out this bracket. 9 lots of 3x plus 1 is the same as 27x plus 9. And then we can just simplify the top. We have like terms 27x and 2x.
If you add 27x and 2x you get 29x. And then there's plus 9. you can now see that this answer here matches the form given in the question. The value of a is 29, b is 9, c is 3 and d is 1. Thank you for watching this video, I hope you found it useful. Check out the one I think you should watch next, subscribe so you don't miss out on future videos, and also now go and try the exam questions that are linked in this video's description.