Thursday, April 10, 2008

Distributive Property

Credit is due to: Elizabeth Stapel at

Distributive Property
The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition".

Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, for example, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation uses the Distributive Property.

So, for instance:

Why is the following true? 2(x + y) = 2x + 2y

Since they distributed through the parentheses, this is true by the Distributive Property.
Use the Distributive Property to rearrange: 4x – 8

The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go into, you must need to factor. Then the answer is "By the Distributive Property, 4x – 8 = 4(x – 2)"

"But wait!" you say. "The Distributive Property says multiplication distributes over addition, not subtraction! What gives?" You make a good point. This is one of those times when it's best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number ("x – 2") or else as the addition of a negative number ("x + (–2)"). In the latter case, it's easy to see that the Distributive Property applies, because you're still adding; you're just adding a negative.

The other two properties come in two versions each: one for addition and the other for multiplication. (Note that the Distributive Property refers to both addition and multiplication, too, but both in just one rule.)

When simplifying expressions with parentheses, you will be applying the Distributive Property. That is, you will be distributing over (multiplying through) the parentheses in order to simplify a given expression. I will walk you through examples of increasing difficulty, and you should note, as this lesson progresses, the importance of simplifying as you go and of doing each step neatly, completely, and exactly.

Simplify 3(x + 4).
To "simplify" this, I have to get rid of the parentheses. The Distributive Property says to multiply the 3 onto everything inside the parentheses. I sometimes draw arrows to emphasize this:

3(x) + 3(4)
3x + 12

Written all in one line, this would look like:
3(x+4) = (3 * x) + (3 * 4) = 3x + 12

The most common error at this stage is to take the 3 through the parentheses but only onto the x, forgetting to carry it through onto the 4 as well. If you need to draw arrows to help you remember to carry through onto everything inside the parentheses, then use them!
Simplify –2(x – 4)

I have to take the –2 through the parentheses.
This gives me:
–2(x – 4) –2(x) – 2(–4) –2x + 8

The common mistake to make with this type of problem is to lose a "minus" sign somewhere, such as doing "–2(x – 4) = –2(x) – 2(4) = –2x – 8". (Did you notice how the "–4" somehow turned into a "4" when the –2 went through the parentheses? That's why the answer ended up being wrong.)

Be careful with the "minus" signs! Until you are confident in your skills, take the time to write out the distribution, complete with the signs, as I did.

–2(x – 4) –2(x) – 2(–4) –2x + 8

If you have difficulty with the subtraction, try converting it to addition of a negative:
–2(x – 4) –2(x + [–4]) –2(x) + (–2)(–4) –2x + 8

Do as many steps as you need to, in order consistently to get the correct answer.
Simplify –(x – 3)

I have to take the "minus" through the parentheses. Many students find it helpful to write in the little understood "1" before the parentheses:
–1(x – 3)

So I need to take the –1 through the parentheses:
–(x – 3) –1(x – 3) –1(x) – 1(–3) –1x + 3 –x + 3

Note that, technically, "–1x + 3" and "–x + 3" are the same thing and, in my classes, either would be a perfectly acceptable answer. However, some teachers will accept only "–x + 3" and would count "–1x + 3" as not fully simplified.