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### Distributive Property: (x+2)(x+4)

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Date: 09/18/2001 at 07:36:30
From: Lydia Flecha
Subject: Distributive Property

I'm confused about the distributive property.

Example: (x+2)(x+4). Would that be x+2 * x+4 = 2x + 6 ?
```

```
Date: 09/18/2001 at 11:23:49
From: Doctor Ian
Subject: Re: Distributive Property

Hi Lydia,

The distributive property says that

a(b + c) = ab + ac

It's easiest to understand as a picture:

3      4
----- -------
@ @ @ * * * * |
@ @ @ * * * * |
@ @ @ * * * * | 5    5(3 + 4) = 5(3) + 5(4)
@ @ @ * * * * |
@ @ @ * * * * |      The size of the whole rectangle is equal
-------------        to the sum of the sizes of the smaller
3 + 4            rectangles.

Extending this from particular cases to the general case is pretty
straightforward:

a          b
--------- -----------
@ @ ... @ * * ... * * |
@ @ ... @ * * ... * * |
.       .         .  | c    c(a + b) = ca + cb
.       .         .  |
@ @ ... @ * * ... * * |
@ @ ... @ * * ... * * |
---------------------
a + b

Note that c doesn't have to be a number. It can also be an expression,
which is the key to expanding something like (x+2)(x+4):

(x + 2)(x + 4) = (x + 2)(x) + (x + 2)(4)
\_____/ |   |    \_____/ |    \_____/ |
c    a   b       c    a        c   b

Then we use the same trick to expand each of the resulting products:

= (x + 2)(x) + (x + 2)(4)
|   |  |     |   |  |
a   b  c     A   B  C

= (x)(x) + (2)(x) + (x)(4) + (2)(4)
|  |     |  |     |  |     |  |
a  c     b  c     A  C     B  C

= x^2 + 2x + 4x + 8

Note that you can now use the distributive property in the other
direction to simplify this:

= x^2 + 2x + 4x + 8
||   ||
ac   bc

= x^2 + (2 + 4)x + 8

= x^2 + 6x + 8

There is a technique called FOIL that is often taught as a trick for
expanding expressions like this, but applying the distributive
property twice isn't any harder than applying FOIL, and the former
works in a lot of cases where the latter is useless:

When FOIL Fails
http://mathforum.org/dr.math/problems/ryan.03.22.01.html

In fact, once you move from using numbers to using variables, the
distributive property becomes one of your best friends, so it's good
that you want to become more familiar with it.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
Middle School Algebra

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