The distributive property says that the sum or difference of two numbers multiplied by a third number is equal to the third number multiplied by each of the two numbers followed by the addition or subtraction.
Example:
5 x (4 + 3) = 35
(5 x 4) + (5 x 3) = 35
2 x (5 - 2) = 6
(2 x 5) - (2 x 2) = 6
Essentially, what the distributive property allows us to do is take the term outside the parentheses and distribute it over each term separated by a plus (+) or minus (-) sign contained within the parentheses.
In the first example above we are taking the 5 and distributing it to the 4 and the 3:
Distributive Property and Variable Expressions
Much like with plain old numbers, the distributive property can also be used with variable expressions. Let's take the following expression:
3(x + 5)
We can use the distributive property here to distribute the 3 to both the x and the 5:
We can distribute a value over as many terms as need as long as those terms are all contained with in the parentheses being multiplied by the term we are going to distribute. Let's take a look at another example:
3a(2b + 4c - 8)
In this problem we can distribute the 3a over each of the three terms:
3a(2b + 4c - 8) = 3a(2b) + 3a(4c) - 3a(8).
Now we can simplify:
3a(2b + 4c - 8) = 3a(2b) + 3a(4c) - 3a(8) = 6ab + 12ac -24a
Be careful about what you distribute!
Let's look at the following problem:
7 + y(x + 2)
In this problem it may look tempting to distribute the 7 + y to both terms, however that would be incorrect. We can only distribute terms that the expression in the parentheses are being multiplied (or divided) by. In the case of this problem we can only distribute the y:
7 + y(x + 2) = 7 + y(x) + y(2) = 7 + xy + 2y
If the problem above were written (7+x)(x+2) then we could distribute the 7 + x over both terms x and 2 in the parentheses.
Distribution of Division
Since division is the same as multiplying by the reciprocal of a number we can apply the distributive property on division as well.
