Lesson video

Well done for taking that first step and choosing to load this video to learn today.

My name is Ms. Davis and I'm gonna be helping you as you work your way through this lesson.

Lots of skills that we're gonna be using, so feel free to pause the video and give things a go.

Give yourself time to think about what you are doing and then I'll help you out by adding any hints, any tips as we work our way through.

Let's get started then.

Welcome to this lesson on simplifying algebraic fractions.

By the end of the lesson you'll have explored all sorts of algebraic fractions and you'll be able to simplify them.

If you want a reminder of what we mean by simplifying and also what we mean by factorising pause the video and have a read through those now.

So we're gonna start by looking at simplifying using common factors.

So we can simplify fractions and I'm talking about any fraction here by identifying common factors in the numerator and denominator.

So let's say I had 30 over 45, Sam has spotted that the numerator and denominator both have a factor of five.

So using their observation, we could actually write 30 over 45 as five times six over five times nine.

Well that could also be written as five over five multiplied by six over nine.

Well that helps us, because five over five is the same as one.

So this is one times six over nine or just six nights.

Ah, Sam spotted something.

The numeration denominators still have a common factor of three.

Have they done something wrong? No, of course they haven't.

They've not fully simplified yet as they've not used the highest common factor, but we could take six over nine and we could simplify that further.

Okay, well we could write that as three times two over three times three and then of course that's the same as two thirds.

Now I have fully simplified as the only common factor in the numerator and denominator is one.

Okay, I should recognises the numerator and denominator have the highest common factor of 15.

Well that means we could write 30 over 45 as 15 times two over 15 times three.

That straight away simplifies our fraction to two thirds.

Yeah, that one was more efficient.

But Aisha's perfectly correct, they got the same answer.

So it's okay if you don't spot the highest common factor straight away.

It's reassuring to know that if you can't see the highest common factor, it's absolutely fine to do that in stages.

Right, we know that in algebra we use letters to represent numbers.

So the rules that apply with numerical fractions will be the same as the ones with algebraic fractions.

So let's simplify 9b over 12ab.

So what is the highest comp factor of the numerator and denominator? Can you spot it? Well done if you said 3b, if you just said three, you also need to look at your variable terms. So what that means is we can write this as three B times three over 3b times 4a 3b over 3b of course is equivalent to one.

So we can write that as three over four a.

Right how do we know that we have fully simplified? What do you reckon? Can you explain this in words? Right, the only common factor in the numerator and the denominator is one.

So we know we have fully simplified, that's gonna be important later on.

Just like sanded for that first question absolutely fine to simplify in stages and sometimes that stops us from getting in a muddle.

Let's look at this one.

What is the highest numerical factor of the numerator and the denominator? Now you possibly got this straight away.

I'm just gonna remind you of a method that can help you find the highest common factor.

So we can use prime factorization.

32 is two to the power of five.

40 is two to the power of three times five.

That means that the highest common factor is two to the power of three or eight.

We can write that then like this and that means that we can simplify this to 4x cubed Y to the five over 5xy cubed.

Right our numerator denominator still share common variable factors.

For example, both have a factor of X.

We can simplify this to 4x squared Y to the five over 5y cubed.

Can you see what other common factor the numerator and the denominator still have? Can you see the highest common factor remaining? If you want to challenge yourself, give it a go.

Right, if you said they both still have a common factor of Y you'd be correct.

Even better if you said that they have a common of Y cubed, we're gonna use our index laws to help us here.

So if we think about it, Y to the five divided by Y cubed is Y squared.

So we can write that numerator as y cubed times 4x squared y squared.

That simplifies then to four x squared y squared over five.

Now we did that quite slowly, we thought about each step.

We could have just used the highest common factor.

The highest common factor is 8xy cubed and done that in one step.

As always, as long as you show you are working out clearly it's okay to put in those extra steps.

Your turn then which of these is equivalent to 12ab squared over 20a squared B, give it a go.

You could have had 3B over 12a or 6B squared over 10ab.

The top one we've used the highest common factor, which is 4ab.

So that's fully simplified, but the bottom one is still equivalent because we've divided the numerator and denominator by 2a.

So we're gonna have a look then at algebraic fractions where the enumerating denominator have more than one term.

So we've got 6x plus three divided by 3x minus nine.

You might already have an idea of what a common factor might be.

All these terms have a common factor of three to help us see what we're doing, I'm gonna factorised the numerator and the denominator.

So the top one is three lots of 2x plus one and the bottom one is three lots of X minus three.

That means we can simplify this to 2x plus one over X minus three.

So Aisha has an answer of 2x plus two over 3x plus three has been asked to write her answer in simplest form.

I think this is already in simplest form.

Why do you think she has said this? Do you agree? So I think she said this, because there's no single term, which is a factor of all the terms in the numerator and denominator, you are looking for a factor of 2x two, 3x and three and there isn't a single term.

Just looking at that algebraic fraction, we cannot see a common factor.

You might have spotted that there is one.

If not, don't worry, I'm gonna show you how we can see it.

What we can do is we can factorised.

So factorising is a way of writing expression as a product of two or more expressions.

So by factorising we identify the factors of an expression, so let's have a look.

If I factorised the numerator and denominator, I get two lots of X plus one over three, lots of x plus one.

Now this fraction is not fully simplified because there's a common factor of X plus one.

Our fraction actually simplifies to two thirds.

So that more complex looking algebraic fraction that we started with is actually just the same as the fraction two thirds.

I'm gonna show you another example, then you're gonna give it a go yourself.

So looking at this example, I cannot identify a common factor easily, so I'm gonna factorised to help me.

So I can write that as Y, lots of 3y minus two and the denominator five, lots of 3y minus two.

And that's helpful for us, 'cause we can now see a common factor.

So this simplifies to Y over five.

Use my example to have a go at simplifying this fraction.

Give it a go.

It's not easy to see straight away a common factor.

So let's factorised the numerator becomes a multiply by A plus three and the denominator three B multiplied by A plus three.

Well that's helpful, we've got a common factor and we can simplify to A over 3b.

Well done if you did that one by yourself.

Gonna have a look at another one together.

Now this time you might be able to see some common factors already.

I'm gonna factorised just to make sure I don't miss any.

So this factor rises as 4x, lots of 2x plus one and 2x lots of x minus three.

Well the bracket terms aren't the same, but I have got a common factor.

I've got a common factor of 2x.

The numerator can be written as 2x times two, lots of 2x plus one.

And that means we can simplify this to two lots of 2x plus one over X minus three.

Of course we can expand the brackets as well if we prefer it in expanded form.

I'd like you time go at this one, off you go.

So factorising will help us see any common factors.

The only common factor this time is five.

So we can write that as 3x minus four over two lots of x plus two or of course we can expand the brackets and write that as 3x minus four over 2x plus four.

Right, Lucas and Jacob are discussing this algebraic fraction.

There is no single common factor for all the terms. The numerator and denominator cannot be factorised.

Therefore this is in simplest form.

Jacob thinks there's a common factor of negative one.

Have a think about what Lucas has said and what Jacob has said and then how could Jacob use this idea to write the fraction in another way? And this is a little bit trickier to see, so Jacob says there's a common factor of negative one.

We can see that clearly in the numerator we've got that as negative one, lots of 2x plus five and the denominator if we factorised out negative one, that would then be 3x plus four.

Now we can divide them both by negative one to get 2x plus five over 3x plus four.

Lucas is correct.

Now there are no common factors other than one, that's what makes this fully simplified.

There's a little bit of a strange case, but do look out for there being a factor of negative one, which could help you write expressions in a simpler way.

Time for a practise, I'd like you to write each fraction in its simplest form.

Give it a go.

Well done, you've got a second set to look at.

Some have been factorised for you where they haven't been factorised for you, you may find factorised helps you spot some common factors.

Give it a go.

Okay, Jacob has rushed the last question on his homework.

He is written this, he has made a little bit of a silly mistake.

Can you spot it? For question four, Lucas wants to write this algebraic fraction in a simpler form, he's thought about what the denominator would become.

Can you fill in the missing numerator? Once you're happy with your answers, come back and we'll look at them together.

Well done, I'd like you to pause video and check your answers.

Pay particular attention to those exponents in E and F.

Did you remember your laws of indices? For F that very complicated looking fraction, it's the same as a half.

Rewriting the variables so that they're in alphabetical order would help you see that you've got the same variables in the numerator and the denominator.

Pause then and check if you've got the rest of those right? Perfect, because I factorised A for you, we can see that that simplifies to two over five.

For B that's two six or a third in simplest form.

For C, you need to factorised.

You'll see that there's a common factor of three for D factor rising shows us there's a common factor of 3c plus one.

For E, we've got three terms on the numerator and denominator, but that's absolutely fine, we can factorised that as normal and we get an answer as two over 3c.

And finally F was written in a rather strange way which didn't really help us, well done if you spotted that 10x minus eight is two lots of 5x minus four.

So if you factorised out another two from the denominator, you get two times two lots of 5x minus four.

Of course two times two is four and the common factors cancel.

And Jacob, we know we shouldn't leave our homework to the last minute, 'cause he's made a silly mistake.

He has not found a common factor of the entire numerator and the entire denominator.

He's noticed that 8x and 4x share a common factor, but that's no use because that's not also a factor of negative three and five.

In order to spot a common factor he would need to factorised.

We can see that we can't factorised either of those expressions, so it's already in its simplest form, just take your time when you get questions like that to check, you might already have the simplest form of your fraction.

And then Lucas, Lucas has divided the denominator by negative two, so he needs to do the same to the numerator.

So that numerator could be written as negative two lots of negative three plus 2x divided the numerator and the denominator by negative two gives you negative three plus 2x over x plus five.

We often write our negative terms second.

So you might have written that as 2x minus three over x plus five.

Well done, we're now confident with how to simplify algebraic fractions.

We're gonna look at some now with quadratic expressions.

So Sam's looking at this one, I really cannot see what a common factor would be in this example.

And you know what just looking at it, I can't see it either.

What could Sam do to help spot any common factors? Of course they could factorise, the numerator is a linear expression, so we can write that as three lots of x plus two.

The denominator is a quadratic, So it may be possible to write it as the product of two binomials.

Remember not all quadratics factorised nicely.

Let's try it.

We need to have an X squared term and a constant term of 10.

So that means the constants in the binomials must have a product of 10.

So that could be one and 10, two and five negative one and negative 10 or negative two and negative five.

But remember the X terms have to sum to seven x.

So two and five is gonna be the only choice that might work.

Let's fill in our area model and check.

So if we have x plus five and x plus two, that'll give us x squared 5x, 2x and 10, which does give us x squared plus 7x plus 10.

Now that is helpful.

There is a common factor in the numerator and the denominator.

So we can now write this as 3 over X plus five.

Now that's a lot simpler than what we started with and if we were gonna go on to use this fraction in some way, it'd be definitely helpful to have it in that second form rather than that first form.

Let's have a look at another one.

The numerator is the easier one to factorised, so let's start there.

We can fraternise the numerator as five lots of 2x minus three.

Ooh.

Now we have a clue about what the denominator might be in factorised form.

What do you think Sam means by that? Well if it's possible to simplify then the denominator might have a factor of 2x minus three.

We've got a bit of a clue that that might be one of our factors.

So let's think about our area model.

Let's assume one of the is 2x minus three, so that it will cancel with the numerator.

That would mean that we'd have to have negative three times one to get our constant of negative three.

So that would have to be 2x one multiplied by 2x and that does work.

We can factor that denominator as X plus one, two X minus three.

Our simplified form then is five over X plus one.

Okay, true or false, the fraction X squared plus eight x plus 16 over X squared plus six X plus eight can be fully simplified to that form.

Have a read, is that true or false? And what's your justification? Well, I dunno if you said that's false, It's correctly factorised but it's not fully simplified.

There is a common factor in the numerator and denominator, X plus four all squared is the same as X plus four multiplied by X plus four.

We write it out in full, we can see there's a common factor.

We have X plus four ever X plus two.

Just keep your eye out for questions like that in the future.

Right, I'm gonna show you one on the left hand side and then you are gonna give it a go.

So looking at this one, the denominator's actually gonna be easier to factorised, 'cause it only has a coefficient of X squared of what? So that's gonna factorised to X plus six, X minus five.

Then that's gonna gimme a little bit of a clue as to what the numerator might factorised to, it's not a guarantee.

Let's assume that one of them is X minus five, that would make the other constant negative two.

Then if I was to expand that, I'd get negative 2x and negative 15x.

I'm just gonna check, does that give me the numerator? Yes it does.

So I can write that as 3x minus 2x minus five over X plus six, X minus five, simplify and then gives us 3x minus two over X plus six.

When you're happy with what I've done, try this one on the right.

Okay, we are relying on our factor rising skills here.

So the denominator is definitely easier again and that's gonna be X plus 3x minus six.

If this is going to simplify, then one of the factors will be X plus three, because obviously three isn't divisible by negative six.

We want to work with integers when we're factorising.

So if one of the bins was X plus three, the other one would have to be 4x plus one.

You can use an area model to help you or there's other methods to factorised quadratics that you could look into.

That simplifies then to 4x plus one over X minus six.

Something else to keep your eye out for is removing a common factor before factorised fully just help sometimes.

I'd like you to have a look at this one.

It's gonna be quite tricky to factorised in its current form.

What could we do first to help us? Can you spot anything? Right, we can remove a factor of two from the numerator and a factor of four from the denominator.

We can't just get rid of it, we can factorise them out.

So we've got two lots of X squared minus 5x plus six over four, lots of X squared plus six X plus five.

Now if you wanted to at this stage you could divide the numerator and the denominator by two, 'cause they both have a factor of two.

I'm gonna focus on factor rising and do that later.

So numerator can now be written as two lots of X minus 2x minus three and the denominator can be written as four lots of X plus 5x plus one.

Lucas says we must have done something wrong as none of the brackets are the same, do you agree? No, we have factorised correctly.

There are not any binomial factors in this case.

In some of our previous examples, we've assumed that there might be a common binomial factor to help us with our factorising, but it is not a guarantee.

So just because there's not a common binomial factor does not mean you've made a mistake.

The highest common factor, like we said earlier, is actually two.

So that is our simplified answer.

Of course, we could expand those brackets again if we wished.

That might have seemed like we wasted a little bit of our time, but we have shown, we've actually proven that there's no more common factors, so we could be confident that that does not simplify further now.

Alright, a quick check.

Which of these are equivalent to 2x squared plus 14x plus 20? What do you think? Right, there's two correct answers this time.

you could have 2x plus 4x plus five or 2x plus 10x plus two.

They are actually equivalent.

Jacob says this expression has a factor of two.

Can you factor this further to show that Jacob is correct and then have a go at simplifying that algebra fraction.

Take your time over this one, come back when you're ready to check.

If we remove a factor of two, we can write this as two lots of X plus 2x plus five.

The denominator can be written as six lots of X plus five.

So that can be written as two lots of X plus two over six or X plus two over three in simplest form.

Well done time for a practise on your own.

Give these ones a go.

When you think you've got the simplest form, come back for the next bit.

Right, there's a specific form I'd like this to be written in.

So can you show the algebraic fraction could be written in that form, so show that question.

So you need to make sure that you show every step of working to convince somebody that you are definitely correct.

Off you go.

And finally, we've got some slightly more challenging questions for you to try.

Think carefully about your factorised skills when you are happy that you fully simplified, come back for the answers.

Let's have a look.

I'd like you to pause the video and check that you've got the same final answers as me.

Where I've left something in factorised form is absolutely fine to expand that.

There's the second set.

Notice that last one does not simplify further, it's absolutely fine to leave it either as the original equation or as the factorised equation.

Then we've got, I'll show that question, so let's look at what I've put in each step.

So first step of factorised, the numerator and the denominator.

It's all right to do that in one step because I'm showing that this can be written like that.

It's important I can explain why this simplifies.

So I've written that as 2x plus one over X plus two times 5x plus three over 5x plus three.

And I've shown clearly now that because that's the same as one, this can simplify down to 2x plus one over X plus two.

And finally, I'd like you to check your factorising.

And then your final answers to these three, off you go.

And the second set for E, Notice that once I factorised out a six Y squared minus one is the difference of two squares, good spot if you got that one.

So it's six lots of Y plus one Y minus one.

And to help me with the denominator, I factored out a three first.

And then of course I know that the only things that multiply to get me negative one must be one and negative one.

So that helps me a little bit with factorising that denominator.

Check then you've got the correct final answer and again, if you've expanded your numerator to get 2y minus two, that's absolutely fine.

And again, just check through the last one.

I've started by factor rising out of four from the numerator and a 10 from the denominator just to make my life a little bit easier.

Check if you've got the right final answer and then we'll summarise what we've looked at today.

Well done those final questions, we're getting a little bit trickier.

We need really think carefully about our factorising skills.

So today we've seen that algebraic fractions can be simplified in the same way as numerical fractions.

They follow the same rules.

What we found with algebraic fractions though, is it's sometimes harder to spot a common factor.

If the common factor is actually a binomial or a expression with more than two terms, for example, we know then that our factorising skills is what's gonna help us spot any common factors.

Thank you for joining us today.

I'm hoping that you can now use those simplifying algebraic fraction skills to apply to other questions as you move forward with this topic, I look forward to seeing you again.