Lesson video

Well done for making the great decision to learn using this video today.

My name is Miss Davies and I'm going to be helping you as we work our way through all this exciting algebra.

There's lots to get done.

There's also lots of opportunities to learn new things, and to check that we're happy with things that we may have seen before.

Take your time.

It's really, really important with all of our algebra skills that we're writing our working out, so you'll want to make sure that you have something to write on, and you want to make sure you're taking the time to pause the video and check things as you go through to help you spot any mistakes that you might have made.

Thanks again for joining us.

Let's get started.

Welcome to today's lesson on the product of two binomials.

I'm really looking forward to this lesson 'cause it's one of those ones where it sounds like it's going to be quite complicated 'cause we've got some fancy words we might not have come across before, but actually we're just going to use skills that we're already fantastic at to make these complicated looking problems actually quite straightforward.

I'm sure you are then going to be able to impress people around you that you can use the distributive law to find the product of two binomials.

Let's get started.

If you're not happy with the distributive law, I suggest you pause the video and make sure that you know what we're talking about before we carry on.

If you haven't looked at multiplying a term by an expression, you might want to do that first as we're going to use the same skill.

The other word I want to bring your attention to is a partial product.

So a partial product refers to any of the multiplication results that lead to an overall multiplication.

So if we do two multiplications and then add the results, each one of those would be a partial product.

So our new word then is a binomial, and we are going to explore this in the lesson.

So that's where we're going to start.

We're going to explore the product of two binomials.

So a binomial is an algebraic expression representing the sum of exactly two unlike terms. We can say the sum or difference.

Remember, subtracting a value is the same as adding the negative value.

So here are some examples of binomials.

Notice we've got some with subtraction.

So five minus 6X is a binomial, because it has two unlike terms, X squared minus X is a binomial.

It has two unlike terms. Let's have a look at some non examples.

X squared plus 3X plus two has three terms. 5XY is a single term.

One over X plus two is not a binomial.

It cannot be written as the sum of two unlike terms. Also notice that 7X minus 5X, they are like terms. So really that's the same as writing 2X, so not a binomial.

We're going to use area models today to explore what happens when we multiply two binomials together.

Let's start by reminding ourselves how we can multiply two two digit numbers.

It might seem straightforward, but it's that concept that we're going to apply to our binomials.

So let's write 12 as 10 plus two.

So the calculation 12 times 13 could be written as 10 plus two times 10 plus three.

Jacob says, "That's easy then.

The answer is a 100 plus six, so 106." Is Jacob correct.

Now of course he isn't.

You can't just multiply the tens, multiply the ones and add them together.

He hasn't found the product of the two numbers.

In fact, he's worked out two partial products but not the complete product.

Let's use an area model to see where Jacob has gone wrong.

So 10 multiplied by 10 is 100.

He got that bit, but we also need to do 10 multiplied by three, two multiplied by 10, and two multiplied by three.

We should end up with four partial products.

Gives us a final answer of 156.

We need to multiply the whole of the 12 by the whole of the 13.

We can't just choose which elements to multiply.

We're going to do the same for the product of two binomials.

We're going to do X plus two times X plus three, so we've got X plus two as one of our dimensions, X plus three as our other.

We can use algebra tiles to help us, and we're going to split this into the four partial products.

So we need to do X multiplied by X.

That is X squared and we to do X multiplied by three.

That is 3X.

So so far we've shown that X lots of X plus three is X squared plus 3X.

We also need two lots of X plus three.

So two times X is 2X, two times three is six.

So pause video if you need to and check your happy with our area model and how we've used it.

Then we'll have a look at how we can write our final answer.

So what we can do is we can write our final answer by adding our terms together.

We've got an X squared, a 3X, a 2X, and a six.

So our final answer could be written as X squared plus 3X plus 2X plus six.

Have you spotted any like terms in our model? If you haven't, pause the video, see if you can find them.

There we go, the X's, we've got three and we've got two.

So we can simplify by adding those like terms. We can write our final product as X squared plus 5X plus six.

When we write the product of two binomials, we do not usually write the multiplication sign between brackets, so (X+3) times (X+1), we'd write as (X+3)(X+1).

We know what we're doing is we're multiplying those two brackets together.

What we're doing when we're multiplying the binomials is just writing the expression in a different format.

What the new format does is it shows the expression as a sum of terms rather than as a product, and that can sometimes be useful.

We can refer to this form as expanded form.

That means the process of multiplying the binomials is sometimes referred to as expanding the brackets.

I think multiplying the binomials sounds a lot fancier, right? Jacob's really getting into this.

He wants to try writing (X+3)(X+1) in expanded form.

Let's see if we can do it together.

So we started by drawing our area model.

We have X plus three on one dimension, X plus one on the other.

We can use our algebra tiles to help us if we want to.

X multiplied by X is X squared, X multiplied by one is 1X, three multiplied by X is 3X's.

Three multiplied by one is three.

We can write our final answer by collecting like terms. We've got an X squared 4X's, and three.

X squared plus 4X plus three.

Brilliant, what about if we write the product the other way around? Let's try it.

This time let's put X plus one on our height, and X plus three on our width.

X multiplied by X is X squared, X multiplied by three is 3X, one multiplied by X is 1X, one multiplied by three is three.

In total we have X squared plus 4X plus three again.

Of course, it didn't make any difference which way round we multiplied.

Multiplication is commutative.

We can write the binomials either way around.

Right, let's check we're happy with this.

I'm going to show you one on the left hand side.

I'd like you to do the same on the right.

In these early stages please take your time to draw your area models, use your algebra tiles if you wish, so you know that you are really confident with this before you move on Here we have our area model.

I'm going to use algebra tiles to help me.

They're my four partial products, and I can simplify.

I've got an X squared, 5X's, plus four.

I'd like you to try and do the same on the right hand side.

You can draw an area model to help us.

Both of our brackets were the same, so we can use a square.

So we have a final answer of X squared plus 4X plus four.

Well done if you've got that one.

If you didn't, it's really important that you're confident with that before moving on.

So just take some time to look back through what you've done and compare it to what's on the screen.

Let's try some binomials with negative terms. Remember X minus two is just the same as X add negative two.

So I can do X multiply by X, which is X squared, X multiplied by three, which is 3X, negative two multiplied by x, which is -2X, and negative two multiplied by three, which is negative six.

I can still collect like terms. I've got 3X and I've got -2X, that gives me a single X.

My final product can be written as X squared plus X minus six.

Right, Laura's doing some thinking.

"I think (X+2)(X-3) will be the same thing." Have a look at what we've done.

Do you think Laura is correct? Okay, so I'm going to put X plus two on my vertical height, instead of X minus two, and X minus three on my horizontal distance.

Instead of X plus three.

I have X squared minus 3X plus 2X minus six, right? There's definitely some similarities.

However, when I add like terms this time I've got -3X, positive 2X which leaves me with negative X, so I'll have X squared minus X minus six.

Laura has realised then that they are not the same thing.

Unlike before where we just wrote the same binomials but the other way around these binomials are different.

We've got (X-2)(X+3), and then (X+2)(X-3).

So here are the area models for the examples we've done so far.

There's (X+2)(X+3), there's (X-2)(X+3), there's (X+2)(X-3).

So I wonder what (X-2)(X-3) would look like.

Let's try it out.

So we've got X plus negative two and X plus negative three.

We have X squared, X times negative three is -3X, negative two times X is -2X, negative two times negative three is positive six.

Be careful with those negative multiplications.

We have then X squared minus 5X plus six.

Just take a moment to have a look at those four examples.

Spot the similarities, spot the differences.

You are going to have a go in a moment.

Well done, I would like you to match the expression to the correct area model.

Off you go.

Let's have a look.

You should have (X-1)(X-4) for A, B, (X+1)(X-4), C should be (X+1)(X+4), and (X-1)(X+4).

If you've got all of those correct, you've really understood how we're going to use these area models to find the product of two binomials.

If not, just go back through and check your happy before moving on.

Right, I'd like you to use these area models then to expand and simplify each expression.

What are we going to get? We should have X squared minus 5X plus four, working horizontally X squared minus 3X minus four, X squared plus 5X plus four, and X squared plus 3X minus four.

Check that you've got those with the correct model.

Fab, you've now learned that new skill of finding the product off two binomials.

What I'd like you to do now to show that you've got that is fill in these area models and all the missing terms. Come back when you're ready for the next bit.

Well done, this time I'd like you to use the area model to expand and simplify these products of two binomials.

So write your answers in their simplest form.

Give it a go.

Let's check then the first one.

You should have X squared plus 3X plus two.

Have a look at the area model to help you.

For B, you should have X squared plus 6X plus nine.

C, hopefully you wrote X plus four and X plus two on the correct dimensions of our rectangle.

And then for D, we were missing a one on both of our dimensions of the rectangle.

For question two we should have X squared plus 2X minus eight.

If you didn't draw the algebra tiles and you just wrote the terms in the right parts of the model, that's absolutely fine.

B, X squared minus 4X plus three, C, X squared minus X minus two D, X squared minus 4X plus four, fantastic.

You've mastered that new skill already.

I hope you're really proud of yourselves.

Now we're going to have a look at doing that efficiently.

If we don't want to be drawing our algebra towers out every time we're multiplying two binomials, it's not always possible to draw area models to scale.

We can still use the model as a representation.

Let's try doing this for (X+8)(X+12).

We can calculate our four partial products, and sum them to get the overall product.

So X multiplied by X is X squared, X multiplied by 12 is 12X, I don't want to draw 12 separate X's, but I do know that X times 12 is 12X, so that's going in that part of the box.

Eight multiplied by X is 8X, and eight multiplied by 12 is 96.

Where can I find my like terms? Okay, in this case they are the 8X and the 12X.

So I can simplify that by collecting like terms. I have an X squared plus 20X plus 96.

Right, Andeep's got a bit of confidence.

"I don't need an area model to expand (X+10)(X-15).

It's easy." Let's see what happens.

He's gone straight for it.

He reckons that it's going to be X squared minus 150.

Where has he gone wrong? I like Andeep's confidence and I hope you are feeling that actually this is quite straightforward.

However, he didn't use an area model, and he's made a big mistake.

He has only calculated two of the partial products.

He's missed out two of the other partial products that's going to make up his final answer.

So Jacob responds with, "The area model helps me remember to calculate all the partial products before collecting like terms." I agree with Jacob this time.

Drawing that area model is going to help us make sure we don't miss any partial products.

So let's try it.

We have X squared minus 15X plus 10X minus 150, but I sympathise with Andeep.

Maybe he doesn't always want to draw out an area model.

Let's see if we can think about another way that we're not going to miss any partial products, but what we can see is that we can split this actually into sort of two sections.

We've got X multiplied by X minus 15 and we've got 10 multiplied by X minus 15 and then we can add those all together.

So X multiplied by X minus 15 is X squared minus 15X, and we can add 10 multiplied by X minus 15.

So 10X minus 115.

Simplifying we've got X squared minus 5X minus 150.

So Andeep it says, "If I wanted to expand X minus 5X minus nine, I could do X lots of X minus nine, plus negative five lots of X minus nine." Let's show that this will work with an area model.

So there's our area models that we've been using and we want X lots of X minus nine, add negative five lots of X minus nine.

If we do all four of those we'll get our final product.

So we've got X squared minus 9X, plus negative 5X plus 45, or X squared minus 14X plus 45.

At this stage I still think it's great to be using that area model to really show that you understand the structure of multiplying two binomials.

If you don't want to use an area model, that is fine, but make sure you are laying out your working out clearly, and you calculate all four partial products before simplifying.

Let's have a look at how this would work then.

Using both methods side by side.

So we have X lots of X plus 20, plus negative six lots of X plus 20.

So we have X squared plus 20X, and then negative six lots of X plus 20 gives us -6X and negative 120.

In total then we can collect like terms. We have X squared, there's 14X minus 120.

I'd like you to do do the same with (X+7)(X-10).

You can either use an area model, or follow my working from the left hand side.

Off you go.

Let's check then.

We've got X squared negative 10X, 7X and negative 70.

Putting those together then we have X squared minus 3X minus 70.

Repeated multiplication of the same expression can be represented with an exponent.

So if you see X plus five all squared, that just means X plus five multiplied by X plus five.

When squaring a binomial it's helpful to write it as two brackets.

So let's do X plus seven all squared, that's X plus seven multiplied by X plus seven.

We can use our area model.

We've got our four partial products, so we have X squared plus 14X plus 49.

Time to check then, Lucas and Sophia are trying to write X minus five all squared in expanded form.

Can you explain why each of their answers must be incorrect? Then use an area model to work out the correct answer.

Give it a go.

So for Lucas's, our final partial product is going to be negative five times negative five.

That's going to give positive 25.

So he must have made a mistake 'cause he's got negative 25 as a term.

Sophia has only worked out two of the partial products.

There's another two partial products that she's missing.

Let's use an area model then to work out the correct answer.

So we need to write that as X minus five multiplied by X minus five, or X plus negative five multiplied by X plus negative five.

There are four partial products, so we should have X squared minus 10X plus 25.

Fantastic, for each of these then I'd like you to work out the product of two binomials in expanded form.

You might want to fill in the gaps in the area models to help you.

Off you go.

Well done, you have six more to do here.

For I and J, I've given you two of the partial products.

You are going to have to use those to work out what the original binomials were as well.

Off you go.

Let's check our answers.

So pause the video and make sure you are happy with the area models as well as the final answers.

Few things to draw your attention to.

In B, you've got seven X and you are adding an X.

That should be 8X.

For D, we've got 4X and we're adding -15X, so we should have -11.

Have a look at these answers as well.

I'm going to draw your attention to certain parts.

So for F, negative five multiplied by negative six should be positive 30.

For G and H you need to write them out as two binomials.

So we've got X plus six multiplied by X plus six for G, and X minus six multiplied by X minus six for H.

Notice that both end in a plus 36, 'cause negative six squared is positive 36.

And six squared is positive 36.

For I, you needed to work backwards, and hopefully you could see that we needed a plus three and a plus nine as part of our two binomials.

And for J we needed a negative three, and then we needed to get 24 as a product, and it's negative three times negative eight that gets positive 24 as a product.

Fantastic work with all of those, and I hope you enjoyed learning that new skill.

So we've done loads of area models today, and we can see how useful they are to find the product of two binomials.

It's important that we use them correctly, so we do not miss any partial products.

Both terms in one bracket must be multiplied by both terms in the other bracket.

I hope after all that you agree with me that this is a topic which sounds like it's going to be really complicated 'cause of all the fancy words, but actually we're just using our algebra multiplication skills.

I hope you enjoyed that today, and I look forward to learning with you again.