5.03 The distributive property with integers
Introduction
In the previous year, we have learned how to apply the distributive property to find equivalent expressions. This time, we will apply distributive property to add, subtract, factor and rewrite algebraic expressions.
The distributive property with algebraic terms
Remember that the order of operations calls for us to evaluate expressions inside grouping symbols (such as parentheses) first. However, sometimes the terms inside the grouping symbol cannot be combined. Is there a different way that we might rewrite the expression?
Exploration
Let's use the applet below to rewrite a few expressions with parenthesis. Set the sliders to get the expression in the first column of the table. Then, count the number of +x tiles and +1 tiles to complete the rest:
What patterns exist between the number of +xs, +1s and the expression in the last column?
If someone says that 2(3x+3) = 6x+3, should we agree or disagree? Why?
| \text{Expression} | \text{Number of} + x \text{ Tiles} | \text{Number of} + 1 \text{ Tiles} | = |
|---|---|---|---|
| 1(4x+3) | 4 | 3 | 4x+3 |
| 2(4x+3) | 8 | 6 | 8x+6 |
| 3(4x+3) | |||
| 4(4x+3) | |||
| 1(2x+1) | 2 | 1 | 2x+1 |
| 2(2x+1) | 4 | 2 | 4x+2 |
| 3(2x+1) | |||
| 4(2x+1) |
If we complete the table using the applet above, we can see that we multiply everything inside the parenthesis by the first number.
The expression 2(3x+3) should be equal to 2 groups of 3x+3, or 2 groups of 3x and 2 groups of 3. This means it is equal to 6x+6.
Here's another way we might look at rewriting the expression using the associative and commutative properties of addition.
| \displaystyle 2(3x+3) | \displaystyle = | \displaystyle (3x+3)+(3x+3) | Two groups of (3x+3) |
| \displaystyle = | \displaystyle 3x+3+3x+3 | Associative property (addition can be grouped differently) | |
| \displaystyle = | \displaystyle 3x+3x+3+3 | Commutative property (we can add in any order) | |
| \displaystyle = | \displaystyle 6x+6 | Simplify |
The property that we have just demonstrated is called the distributive property. We can summarize it using the following:
For all numbers a,b, and c, a(b+c)=ab+ac and a(b-c)=ab-ac.
For example,
| \displaystyle 5(x+6) | \displaystyle = | \displaystyle 5x+5 \times 6 |
| \displaystyle = | \displaystyle 5x+30 |
Since the distributive property is true for all numbers, we can also apply it to expressions with negative integers and rational numbers.
Examples
Example 1
Rewrite the expression -2(3x-1) using the distributive property.
Example 2
Rewrite the expression 4(t+6) using the distributive property.
Example 3
Rewrite the expression -(5-s) using the distributive property.
Example 4
A student incorrectly used the distributive property and wrote 7(4x+3)=28x+3.
Which one of the following is the best explanation to help the student correct their error?
For all numbers a,b, and c, \begin{aligned} a(b+c) &= ab + ac \\& \text{and} \\ a(b-c) &= ab - ac\end{aligned}
This is known as the distributive property.
For example,
| \displaystyle 5(x+6) | \displaystyle = | \displaystyle 5x+(5)(6) |
| \displaystyle = | \displaystyle 5x+30 |