Lesson video

Hello, Mr. Robson here.

Great choice to join me for more maths today.

We're gonna multiply algebraic expressions by algebraic terms today and it is beautiful.

So let's get started.

So our learning outcome for this lesson, is that we can use a distributed law to multiply multiple expressions by the respective terms and then simplify by collecting like terms. Quite a mouthful that, don't panic.

We'll break it down into small little chunks, keywords to simplify an expression, to write in a more efficient and compact form without affecting the value of the original expression.

Fully simplifying means expression cannot not be simplified.

Further, you'll hear me say "simplify" a lot during this lesson, like terms, are terms that have the same set of variables and corresponding exponents.

again, you'll hear me say "like terms" quite a lot throughout this lesson.

Two parts to our lesson today.

Firstly, we'll look at expanding more than one bracket and then we'll look at expanding and simplifying with negative terms because that can be a particularly difficult area.

Let's start by expanding more than one bracket.

Expanding brackets you've seen before, distributive law, you've seen before, distributive law in number, 3 lots of 105, well it's 3 lots of 100, 3 lots of 5.

you saw that same distributive law in expanding brackets.

2 lots of brackets, x plus 5, you need 2 lots of everything in that bracket.

2 lots of x, 2 lots of 5.

The expression expanded to 2 x plus 10 in the bottom example 3 lots of bracket x plus 6, 3 lots of x and 3 lots of 6 and we simplified that to 3 x and 18.

But what will happen when there are two or more expressions that can be expanded, like so, how do you think we deal with that? Sam said it becomes 2 x plus 10 plus 3 x plus 18, whereas alex said it becomes 5 x plus 28.

are they right? When we look at this expression, 2 lots of bracket x plus 5 closed bracket and 3 lots of bracket x plus 6, we know we can expand that first bracket.

2 lots of x, 2 lots of 5 to 2 x plus 10.

The second bracket we need 3 lots of x, 3 lots of 6, that's 3 x and 18.

Expand and simplify.

That's expanded.

Can we simplify? I think I've already said like terms. Do you see any like terms? I see x terms and constant terms. We can take the x terms, 2 x and 3 x, add those together, 5 x set the constant terms, 10 and 18 and they sum to 28.

In that first step we expanded and that second step we simplified.

They're the command words we frequently see and we seen multiple expressions.

you won't just be asked to expand, you'll be asked to expand and simplify.

So coming back to our Oak students work, both Sam and alex were correct.

Sam had expanded both expressions correctly.

However alex went on to simplify them.

I'd like to check, you've got that.

Could you fill in the gaps to expand and simplify these expressions, pause this video, copy this down and give that a go.

6 lots of that first bracket, 6 lots of x, 6 lots of 1, 7 lots of that.

Second bracket, 7 lots of x, 7 lots of 3.

They're the expansion.

To simplify that in like terms, I see x terms, 6 x and 7 x, that's 13 x, I see constant terms 6 and 21, that's 27.

So we've expanded and simplified, we end up with the expression 13 x plus 27.

Francis doesn't know Izzy expands and simplifies this expression.

Part a.

Izzy's made 2 errors.

How would you explain to Izzy the 2 errors she has made? Once you spotted those errors, explain those errors.

you should be able to correct Izzy's work.

Pause this video and give this a go.

you might have said, "you failed to multiply the 5 by the 3, incorrectly expanding the second term and then when simplifying you didn't collect all the like terms." So what would that have looked like if Izzy had done it correctly? Well, 2 x plus 8 is correct in that expression.

5 lots of x is right, 5 lots of 3, and that was an error.

We need to correct that to 15, and then when it came together in the like terms and simplifying, Izzy hadn't sum x in together.

When we do that we get 7 x.

Now that we've got the correct constant terms, 8, 15, make 23, you should have put those corrections on Izzy's work.

Moving on now, expanding and simplifying with negative terms. Expand and simplify.

2 lots of bracket, y plus 5 close bracket, 4 lots of bracket y plus 8.

Well the first bracket will expand to 2 y plus 10.

The second bracket will expand to 4 y plus 32.

That's an expansion.

Can I simplify? yes, there's some like terms, 2 y and 4 y, 6 y, 10 and 32 makes 42.

I did the top one for you.

The next three look quite similar.

That's really important differences.

Have a look at those next three expressions and spot what's different about them? The difference is we've got negative terms, between the first expression and the second expression.

That's y plus 8 has changed to y minus 8 and then in the third example, y plus 5 has become y minus 5.

In the final example, 4 lots of y minus 8.

We chose 4 lots of bracket, 8 minus y.

It's important to remember subtraction is not commutative for y minus 8, 8 minus y are very different things.

The second example, 2 lots of bracket y plus 5, we can expand to y plus 10.

4 lots of bracket y and negative 8, 4 lots of y, 4 lots of negative 8 and you see the difference between the top line of working and the second line of working.

Just that one negative third.

That's expansion with simplify that, 2 y and 4 y still positive 6 y and then positive 10, negative 32.

This is a sum but negative 22, 10 subtract 32, will take us down to negative 22.

We equal to 10 positives in 32 negatives we put together the zero pairs you end up at negative 22.

The next one, incredible similarities but crucial differences.

First bracket expands to 2 y and negative 10 and the next bracket expands to 4 y and negative 32.

The y terms are still positive in this case we sound like positive 6 y.

Both of the constants in this occasion are negative.

Don't 2 negatives make a positive.

'cause this could be positive 42.

No, not when we're adding negative 10, negative 32.

It's like having 10 pounds of debt in 32 pounds of debt.

From that debt together you've got 42 pounds of debt.

The last example, nice and positive in the first bracket.

2 y plus 10 and then we've got 4 lots of 8, 4 lots of negative y, 4 to 8 being 32, 4 to negative y, negative 4 y.

So in this example the constant terms are positive and we've got one negative y term, 2 y, negative 4 y, that's negative 2 y and 42.

Now that last example of the expression, negative 2 y plus 42, you might see it written as 42 minus 2 y.

you might see that written bothways.

Right, I'd like to check you've got that, here are some expressions to multiply to expand and simplify.

I've left out the operations in that top row between the expanded expression, the 2 x, the 6 the 4 x, the 20 I've missed out the operations.

all the operations that I've left out are either, plus or minus.

That's all I want you to look to insert into these gaps.

a plus or a minus, pause this video, see if you can insert positives and negatives into all of those spaces.

Okay, the top line, 2 lots of x, is 2 x, 2 lots of negative 3, negative 6 and the second bracket, 4 lots of x, positive 4 lots of positive x, that's positive.

Positive 4 lots of positive 5, positive 20.

and we come to some of those terms together.

2 x, 4 x, positive 6 x negative 6, positive 20, positive 14.

In the next line, that first bracket's nice and positive 2 multiplier, positive 10 in the bracket.

That'll give us 2 x plus 6.

Positive 4 lots of positive x that's positive.

4 x positive 4 lots negative 5.

Wasn't there something about when we multiply positive by a negative, didn't it give us a negative, negative 20 when we come to simplify that one positive 2 x, positive 4 x remains nice and positive, 6 x.

Positive 6 and negative 20 gives us, negative 14.

The third example, 2 lots of x is 2 x, 2 lots of negative 3, negative 6, 4 lots of x is positive 4 x, positive 4 lots of negative 5.

Oh it's the same as one above negative 20 and we going to sum those terms. Some of the like terms, simplified like terms, 2 x in 4 x means 6 x, negative 6 and negative 20.

That's like 6 pounds of added to 20 pound of debt, that's 26 pound of debt and the last example, 2 lots of 3, that's positive 6.

2 lots of negative x, that's negative 2 x.

4 lots of 5, that's positive 20.

4 lots of negative x, that's negative 4 x.

So, when we come to sum these terms together to simplify here, constant 6 and 20, that's 26.

The x terms negative 2 x negative 4 x is a total of negative 6 x.

Finally, to understand that expressions can be subtracted, as well as added.

In this case we're taking 6 lots of bracket x plus 5, we're subtracting 3 lots of bracket x plus 2.

The first bracket is nice and easy to deal with.

6 lots of x, 6 lots of 5, 6 x and positive 30.

The second bracket is a negative multiplier.

you just have to be careful with the positives and negatives again.

I need negative 3, lots of everything in that bracket.

Negative 3 lots of x, negative 3 lots of 2, that'll be negative 3 x and negative 6.

That's the expansion.

We just need to simplify, like terms the x terms, 6 x minus 3 x gives us 3 x, 30 minus 6 gives us 24 so it looks like a peculiar and difficult problem.

They've behaved the same way as the ones we saw earlier.

We're just being really careful with our multiplication of positives and negatives and our addition positives and negatives.

If I had to check you've got that, I'd like you to fill in the missing operations again.

We're now going to expand and simplify these sets of brackets.

I've omitted the positive signs and negative signs from my working.

Can you insert them? Pause this video, and fill in those gaps.

and the first line, 5 lots of positive y, 5 lots of positive 4, where positive, positive 20 and then we've got negative 2 lots of everything inside that bracket.

Negative 2 lots of y, negative 2 lots of 3 to add those like terms together.

5 y, subtract 2 y leaves us with 3 y and 20 minus 6, that's positive 20 minus 6 leaving us with 14, positive 14.

On the second line, 5 lots of y, 5 lots of positive 2, that's positive 10.

and then negative 3 lots of y, negative 3 y negative 3 lots of 4, negative 12.

Gathering the like terms. 5 y minus 3 y is 2 y, and 10 subtract 12 takes us down to negative 2.

On the last line.

again, positive expression, first bracket, positive multiplier, 2 lots of positive y, 2 lots of positive 3, positive 6.

Then we've got negative 4 lots of y negative 4 lots of 5.

and then we come to some like terms together.

2, subtract 4 y.

They give us negative 2 y, 6 subtract 20 will give us negative 14.

The distributed law applies to each expression regardless of what we're multiplying by.

This example looks particularly unusual, but nothing's changed.

Whether it's a decimal coefficient, we've got variables with exponents.

Distributed law is gonna apply to each expression.

Let's take that first expression.

a lots of bracket 3 x minus 2 a squared a lots of everything in the bracket.

a lots of 3 x, 3 a x, a lots of negative 2 a square, negative 2 a cubed, and put those terms there and then deal with that second expansion.

I need negative 7 x lots of everything in that bracket.

Negative 7 x lots of 5 a, that's negative 35 a x, negative 7 x.

Lots of negative 0.

1 a squared.

Well that's 2 negatives multiplied together.

So it's a positive.

7 lots of 0.

1 is 0.

7 and a squared multiplied by x is a squared x.

So we get negative 35 x plus 0.

78 a squared x.

Let's add that to our expression.

Now we're looking for like terms in cubed term, I can't see another a cubed term, a squared x term.

I can't see another a squared x term.

I think you are shouting at the screen now.

The a x terms, I can see two a x terms, 3 a x and negative 35 a x and they come together, they'll give us negative 32 a x.

So once we've simplified, we'd have 0.

7 a squared x subtract 32 a x minus 2 a cubed.

Let's check.

you've got that.

Let's check in a different way this time.

Instead of leaving out the operations, I've included them.

I've made some deliberate errors, can you spot them? Are any of these correct? How many errors have I made? Pause this video, take your time, see if you can find all my errors.

In the first example, and you're doing great.

Those brackets expanded to, w y, plus 2 w, minus 4 w, plus 20 w? No.

Negative 4 w lots of 5, or 5 lots of negative 4 w, that would've given me negative 20 w.

And the unfortunate when you make errors at this point is they compound beyond.

Because if that's my expansion when I go to simplify, well that can't be right anymore.

2 w plus 20 w is 22 w, but I haven't got positive 20 w I've got negative 20 w.

So that would've been 2 w.

Subtract 20 w, didn't mean negative 18 W.

In the second example, negative 9 lots of b, that's negative 9 b, negative 9 lots of negative 2 is negative 18? It's not.

It's positive 18.

2 negatives multiplied together, give us a positive negative 9 multiplied by negative 2.

They're giving us positive 18 and again, unfortunately we made that error in the first place.

It compounds into our final simplified expression.

That's also wrong.

Positive 14 b.

So we've made a mistake at the simplifying stage.

Wherever the b terms, negative 9 b added to negative 5 b would give us in total negative 14 B.

The last one, well we started off with a very common error, y lots of everything in that bracket, y lots of y, to begin, y add y is 2y.

But y multiply by y is y squared.

Unfortunately once we've made that error there, that compounds into a simplified expression because if that were a y squared, when we add it to negative 4 y squared, we'd get negative 3 y squared and it would leave us with a negative 2 y and a positive 20 y meaning the y term would be 80 y in the following expression.

This is why we have to take lots of time, care and attention and multiplying out multiple brackets, especially when dealing with positives and negatives.

Well.

Three brackets to expand and simplify.

The first bracket, I see I need 2 lots of a plus 3.

That's nice and easy.

That's 2 a plus 6.

The next bracket I see is a little more complex.

4 a, lots of bracket minus b, 4 a lots of 3, 12 a, 4 a, lots of negative B, that's negative 4 a B.

Final one's a bit trickier, negative b lots of bracket negative 2, negative 4.

5 b, positive 2 b and positive 4.

5 b squared.

So I need to take those terms and add them together.

2 a plus 6, 12 a, negative 4 a b plus 2 b plus 4.

5 b squared.

I'll add any a terms together.

2 a and 12 a making 14 a.

There's only one constant term, 6 There's only one a b term, negative 4 a b.

There's only one b term 2 b.

The b squared term has no other like terms, so it can't be simplified.

That's as far as we can simplify that expression.

your turn now, I'd like you to expand and simplify this again to each bracket.

1 and 1.

Your solution.

The first bracket should have expanded to 18 and 9.

6 b.

The second bracket multiplied by negative 4 b, should give us negative 4 b d plus 8 b.

And last bracket multiplying the contents of that bracket by b, would give us negative 2b negative 4.

5 b.

So let's write out all those terms and see if there's any simplification we can make, the constant terms 18 or b terms, add to 15.

6 b.

The b d terms add to negative 8.

5.

So you get the expression 18 plus 15.

6 b minus 8.

5 b d.

Practise time now.

Expand and simplify.

There's 5 questions for you to practise.

Pause this video and give them a go.

For question 2.

Lucas has expanded and simplified these expressions.

Mark his work for him, Has he made the errors? Can you correct them? Pause, and give it a go.

Question 3, Jacob isn't sure how to expand and simplify the following.

I can sympathise.

It's quite difficult.

How will you help him get started? How might you show him how to work that one out? How to expand and simplify? Pause this video and show Jacob, how it's done.

Some feedback now for a, after expanding and simplifying, we should have the expression 5 y plus 31.

And b, we should have the expression negative x minus 26 For c, we should end up with the simplified expression 3 a squared minus 2 a b plus 10.

a.

We should have 3 d cubed minus 28 d squared plus 28 d part e, a squared minus a d minus 3 a.

How did you do? Did you make a few errors? That's absolutely fine.

We're not gonna get everything right first time in mathematics.

If you've got some errors, it's a really good time to pause this video and look away your work if it's my work and see if you can identify where you went wrong.

Take some time to do that.

Now Question 2.

Marking in Lucas's work.

There's lots of good stuff here, Lucas.

Well done.

But a couple of common errors.

The first of which negative 2 e, it's only the 5 within that second bracket that's negative, not the e too.

So that would've been positive 2 e.

That then affects the simplification because instead of 3 e minus 2 e, we've now got positive 3 e, positive 2 e, which would've given us positive 5 e.

How about the second one? That's right.

He's expanded it correctly.

Has he simplified it correctly? Absolutely.

Well done Lucas.

The third one, the simplification, negative 21, negative 10.

Put those negatives together and we'd have negative 31.

Some wonderful work by Lucas there and some very common errors that we all need to watch out for.

On the third one, Jacob's unsure how to expand and simplify this expression, understandably so.

How might we help him? Well, we could say take each bracket turn by turn and expand it.

That would give us negative 7 c lots of 2, negative 14 c.

Negative 7 c lots of h, negative 7 c h.

Negative 2 c lots of 2, that's negative 4 C, et cetera, et cetera.

Which show Jacob that when we expand all the brackets, we get these terms. and then we taught Jacob about simplifying.

We've been looking like terms, I can see lots of c terms. I can see lots of c h terms. We'd be gathering those like terms and simplifying those together, giving us a simplified expression of negative 4 c plus 8 c h.

So here we are, at the end of the lesson.

To summarise, when we see multiple expressions, we still expand them.

Then we simplify by collecting new like terms. I hope you've enjoyed this lesson.

I have.

So I hope to see you again soon for more maths.

Bye for now.