Lesson video

Hello everyone.

This lesson is on factorising single brackets using more complicated expressions.

Hi everyone, I'm Mr Lund, I've got two questions at the start of this lesson and the skills that you need to feel confident with, if not revisit lessons on dividing or multiplying indices.

So, first question to you is, true or false? A to the power of 5 divided by A to the power of 2, does that equal A to the power of 3? I hear you cry true, it is true.

Yes, when dividing numbers of the same base, we subtract the powers.

Look at this question, true or false? It's true.

So, very similar to the last question, however, we have coefficients of 10 and 5, divided give you 2, and A to the power of 5 divided by A to the power of 2 gives you A to the power of 3.

That was true.

Let's use multiplication grids to help us factorise more complicated expressions.

What's the highest common factor of A to the power of 5 and A to the power of 2? What would I need to multiply A to the power of 2 to find me A to the power of 5? What would I multiply A to the power of 2 by to find me A to the power of 2? Have a look at this next expression.

What's different in this example? Do you see? In this example, we have coefficients of a 10 and a 5.

The algebraic terms have not changed.

We have A to the power of 5 and A to the power of 2.

But the highest common factor of both these expressions is 5A to the power of 2.

To complete our multiplication grid, there are our solutions.

So let's factorise 4A to the power of 5, plus 8A to the power of 3.

We can think of the coefficients, and the algebra almost as separate things to help us factorise.

So, the highest common factor of the coefficients is 4.

And the highest common factor of the algebra is A to the power of 3.

So it is the lowest power that is common to both A to the power of 5 and A to the power of 3.

That means our highest common factor is 4A to the power of 3.

We need two terms inside the brackets, which when expanded my the highest common factor gives our original expression.

And there we go, boom! Let us double check, because as these get more complicated, it is quite necessary to check.

So 4A to the power of 3 times by A to the power of 2, gives you 4A to the power of 5.

Plus 8A to the power of 3.

It's the same as our original, we factorised correctly.

Let's have a look, example two.

How is it different and how is it the same? Well, the coefficient numbers seem to have flipped, we have a negative sign, and also we have a different letter for the variable.

But also, there is something raised to the power of 4.

Let's have a look at the highest common factor of the coefficients.

That has not changed.

But the highest common factor of the algebra, this time around is T to the power of 4.

Ah, highest common factor of the entire expression is 4T to the power of 4.

And inside we need two terms that will multiply us to give us the original expression if we multiply it by the highest common factor.

Let's check by expanding.

So 4T to the power of 4, multiplied by 2T gives me 8T to the power of 5.

Remember, 2T is the same as saying 2T to the power of 1.

4T to the power of 4, multiplied by negative 1, gives you negative 4T to the power of 4.

Yeah, they're the same! Well done everyone.

Let's factorise.

Here, the highest common factor of the coefficientS is 4.

I could rewrite my expression to look like this.

Maybe that helps you to see that A is common, and also that B is common, which means in terms of the algebra, the highest common factor is AB, therefore the highest common factor of 8A to the power of 2 B minus 4AB to the power of 2 is 4AB.

Let's check that we factorised correctly by expanding.

As these expressions get more complicated, I would say that it is a very good practise to get into to expand.

4AB times by 2A, gives you 8A to the power of 2 B.

4AB times by negative B gives you negative 4AB to the power of 2.

And that is the same as our original expression, we've factorised correctly.

Has this expression been fully factorised? No, in this case it is no.

Maybe you saw that inside the brackets we still have a common factor of A.

The actual answer should have been that.

Check by expand.

Here are some examples for you to try.

Pause the video, and return when you want to check your answers.

Here are the solutions to questions one and two.

In question two B, you could have used a multiplication grid to show whether it was true or false.

In this example, I've used my multiplication grid to show that the answer for two B should have been 2B plus AB squared.

It was false.

Here's some examples for you to try.

Pause the video, and take your time with these examples.

Come back when you want to check your answers.

Here are the solutions to question number three.

Always check your answers by expanding, especially when it gets a little bit more complicated.

It's well worth it.

Here are some examples for you to try.

Well done for getting this far.

Pause the video and come back and check your answers.

Here are the solutions to questions five and six.

Did you see Amir hadn't spotted that he could have factored out another number 2 from the brackets? That would have given him a highest common factor of 4X squared.

Here are some very last questions for you to try.

Pause the video and come back to check your answers.

Well done everyone, doing fantastic to get this far.

Here are the solutions for question seven and eight.

Question seven there are some tricky expansions, to check your answers with negative terms, so take your time with those ones.

In question eight, why not use, instead of algebra, a number, say for example any square number, two lots of any square number, let's take the number 9.

So two lots of 9.

Check 18 has 6 factors.