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

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.  

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 
                 
- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/