 ## What you need to know

Things to remember:

• To expand a single bracket, we multiply what is inside a bracket by what is outside the bracket.
• When multiplying by a negative number it changes the sign.
• A negative multiplied by a negative is positive.
• A positive multiplied by a negative is negative.

Expand and simplify $4(3x-4)$

Before we can do this, we need to answer 3 questions:

• What is multiplication?
• What does $4(3x-4)$ mean?
• What does “expand” mean?
• What does “simplify” mean?

So, what is multiplication? Multiplication is a quick way of writing lots of additions.

$$4\times3=3+3+3+3=12$$

$$4\times x=x+x+x+x=4x$$

What does $4(3x-4)$ mean? Well, something is missing. Let’s start by thinking about what $4x$ means. Well, if we look at the last equation we did, we can see that it means “4 multiplied by $x$”, there was an invisible $\times$ symbol. Well, there is an invisible $\times$ symbol between the 4 and $(3x-4)$ as well.

$$4(3x-4)=4\times(3x-4)$$

What does “expand” mean? Well, this just means to do the multiplication!

Hint: Remember, multiplication just means adding the same things multiple times.

$$4(3x-4)=4\times(3x-4)=(3x-4)+(3x-4)+(3x-4)+(3x-4)$$

Finally, what does “simplify” mean? This just means that we want it in its simplest form, which we do by collecting like terms!

$$(3x-4)+(3x-4)+(3x-4)+(3x-4)=3x+3x+3x+3x-4-4-4-4$$

$$3x+3x+3x+3x-4-4-4-4 =12x-16$$

$$4(3x-4)= 12x-16$$

Notice how the final answer is actually just what is inside bracket multiplied by what is outside!

$$4\times3x= 12x$$

$$4\times-4= -16$$

So, we can actually expand and simplify a bracket by just multiplying what is inside the bracket by what is outside it.

$$3(8x+2)=3\times8x+ 3\times2=24x+6$$

$$7(11x-5)=7\times11x-7\times5=77x-35$$

$$a(b+c)=a\times b+ a\times c=ab+ac$$

## Example Questions

$$12(2-3x)=12\times2-12\times3x=24-36x$$

$$a(3b-2a)=a\times3b-a\times2a=3ab-2a^2$$