6.11 Expansion
Normally, when an expression has a multiplication and an addition or subtraction, for example $$5+8×9, we evaluate the multiplication first. The exception is when the addition or subtraction is in brackets, for example, $$(5+8)×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 $$(5+8)×9 cm2. We can work this out as follows.
| $$(5+8)×9 | $$= | $$13×9 |
Evaluate the addition in the brackets first |
| $$= | $$117 cm2 |
Evaluate the multiplication |
However, we can see that the rectangle is made up of two smaller rectangles, one with area $$5×9 cm2 and the other with area $$8×9 cm2. So we can also work out the total area like this.
| $$5×9+8×9 | $$= | $$45+72 |
Evaluate the multiplications |
| $$= | $$117 cm2 |
Evaluate the addition |
So $$(5+8)×9=5×9+8×9. This can be extended to any other numbers.
If $$A,B, and $$C are any numbers then $$A(B+C)=AB+AC. This is known as the distributive law.
The distributive law is particularly useful for algebraic expressions where we can't evaluate the expression in the brackets.
Worked Example
Expand $$7(x−12).
Think: Expand means to write an algebraic expression without brackets. We can expand this expression using the distributive law.
Do:
| $$7(x−12) | $$= | $$7×x+7×(−12) |
Use the distributive law, $$A(B+C)=AB+AC. Here, $$A=7,B=x, and $$C=−12 |
| $$= | $$7x−84 |
Evaluate the multiplication |
Reflect: Because of the distributive law we know that both sides of the equation are equal. But now we have a way to write an equal expression without brackets.
We had to be careful of the negative sign here. Because $$A is positive and $$C is negative, $$AC is negative.
To solve the previous example we could also use a slightly different version of the rule that accounts for the negative sign: $$A(B−C)=AB−AC. Notice that in this case we are assuming $$C is positive, but we are taking away $$AC.
We can use the distributive law to expand an algebraic expression brackets like so:
$$A(B+C)=AB+AC,
and if the second term in the brackets is negative:
$$A(B−C)=AB−AC
where $$A,B, and $$C are any numbers.
Practice questions
Question 1
Expand the expression $$9(5+w).
Question 2
Expand the expression $$(y+8)×7.
Question 3
Expand the expression $$−8(c−5).