Ideas
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.
We can work the area out as follows:
| \displaystyle \left(5+8\right)\times 9 | \displaystyle = | \displaystyle 13\times 9 | Evaluate the addition in the brackets first |
| \displaystyle = | \displaystyle 117 \text{ cm}^2 | Evaluate 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+72 | Evaluate the multiplications |
| \displaystyle = | \displaystyle 117 \text{ cm}^2 | Evaluate 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.
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).
Example 2
Expand the expression -9\left(n-2\right).
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.