# 4.04 The distributive law

Lesson

## The distributive law

Normally, when an expression has a multiplication and an addition or subtraction, for example 5+8\times 9, we evaluate the multiplication first. The exception is when the addition or subtraction is in brackets, for example, \left(5+8\right)\times 9.

It will help to visualise a rectangle with a height of 9 cm and a width of 5+8 cm.

The rectangle has an area of \left(5+8\right)\times 9 \text{ cm}^2.

We can work the area out as follows:

However, we can see that the rectangle is made up of two smaller rectangles, one with area 5\times 9 \text{ cm}^2 and the other with area 8\times 9 \text{ cm}^2. So we can also work out the total area like this.

So \left(5+8\right)\times 9=5\times 9+8\times 9. This can be extended to any other numbers.

If A, B, and C are any numbers then A\left(B+C\right)=AB+AC. This is known as the distributive law.

The distributive law is particularly useful for algebraic expressions where we can't evaluate the expression in the brackets.

### Exploration

The following applet explores the distributive law using algebra tiles.

The total number of +x tiles corresponds to the coefficient of the variable and the number of +1 tiles corresponds to the constant after multiplication.

### Examples

#### Example 1

Expand the expression 9\left(5+w\right).

Worked Solution
Create a strategy

Use the distributive law A\left(B+C\right)=AB+AC.

Apply the idea

#### Example 2

Expand the expression -9\left(n-2\right).

Worked Solution
Create a strategy

Use the distributive law A\left(B-C\right)=AB-AC.

Apply the idea
Idea summary

Expand means to write an algebraic expression without brackets.

We can use the distributive law to expand an algebraic expression brackets like so:A\left(B+C\right)=AB+AC

and if the second term in the brackets is negative:A\left(B-C\right)=AB-AC

where A,B and C are any numbers.

### Outcomes

#### MA4-9NA

operates with positive-integer and zero indices of numerical bases