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4.04 The distributive law


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.

A rectangle with a height of 9 centimetres and a length of 5 plus 8 centimetres. Ask your teacher for more information.

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:

\displaystyle \left(5+8\right)\times 9\displaystyle =\displaystyle 13\times 9Evaluate the addition in the brackets first
\displaystyle =\displaystyle 117 \text{ cm}^2Evaluate the multiplication

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.

\displaystyle 5\times 9+8\times 9\displaystyle =\displaystyle 45+72Evaluate the multiplications
\displaystyle =\displaystyle 117 \text{ cm}^2Evaluate the addition

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.


The following applet explores the distributive law using algebra tiles.

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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.


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
\displaystyle 9\left(5+w\right)\displaystyle =\displaystyle 9\times 5+9\times wExpand the brackets
\displaystyle =\displaystyle 45+9wEvaluate the multiplication

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
\displaystyle -9\left(n-2\right)\displaystyle =\displaystyle -9\times n-\left(-9\times 2\right)Expand the brackets
\displaystyle =\displaystyle -9n-\left(-18\right)Evaluate the multiplication
\displaystyle =\displaystyle -9n+18Combine the adjacent signs
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.



operates with positive-integer and zero indices of numerical bases

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