Expanding Polynomials

Date: 06/16/99 at 20:08:14
From: Chelsea
Subject: Expanding polynomials

Help! I have a math exam on Friday and I don't understand how to 
expand three binomials multiplied together.

My example question is

     (s+4)(2s-1)(s-2)

I have the answer sheet but I can't get the answer. I am also having 
problems with this question

     (z^2-2z+3)(2z^2+z-1)

I have tried the "foil" method for both of them, but I got both wrong. 
Please help me. Thanks!

Date: 06/17/99 at 18:02:20
From: Doctor Rick
Subject: Re: Expanding polynomials

Hi, Chelsea.

The FOIL method (First, Outside, Inside, Last) only applies to a 
product of 2 binomials.

     (s+4)(2s-1) = (s)(2s) + (s)(-1) + (4)(2s) + (4)(-1)
                    First     Outside   Inside    Last

FOIL is a particular example of using the Distributive Property more 
than once. I'll show you how the Distributive Property gives you FOIL.

First keep the second binomial intact, but break up the first binomial 
using the Distributive Property:

     (a+b)  c    = a  c    + b  c
     (s+4)(2s-1) = s(2s-1) + 4(2s-1)

Next, in each of the two terms that you got from the first application 
of the Distributive Property, use the property again:

     a(b +c) = a b   + a c
     s(2s-1) = s(2s) + s(-1)

     4(2s-1) = 4(2s) + 4(-1)

Now you have 4 terms, and they are the same as those that we got using 
FOIL. Does this give you ideas? The Distributive Property is much more 
flexible than FOIL; we can apply it to trinomials or more. I'll show 
you:

  ( a + b+c)    d     =   a     d      +   b      d      + c    d
  (z^2-2z+3)(2z^2+z-1) = z^2(2z^2+z-1) + (-2z)(2z^2+z-1) + 3(2z^2+z-1)

   a ( b   + c + d) =  a   b    +  a  c  +  a  d
  z^2(2z^2 + z - 1) = z^2(2z^2) + z^2(z) + z^2(-1)

I'll let you expand the other 2 terms. You will end up with 9 terms. 
You can "collect terms" by grouping all those with the same power of 
z, and using the Distributive Property in the other direction to "pull 
out" the power of z. For instance, you will have terms

     Z^2(z) + (-2z)(2z^2) = z^3 - 4z^3
                          = 1z^3 - 4z^3
                          = (1-4)z^3
                          = -3z^3

Do you see how to expand products of trinomials now? You can use the 
same method to expand products of any two polynomials. But you can 
also expand products of more than two polynomials, like your example

     (s+4)(2s-1)(s-2)

Just expand the product of the first two factors; it will become

     (s+4)(2s-1) = 2s^2 - s + 8s - 4
                 = 2s^2 +   7s   - 4

Replace the first two factors with this expansion:

     (2s^2 + 7s - 4)(s-2)

Expand this product of polynomials, and you're done.

The hardest part of this is making sure you don't lose a term. Just 
keep it organized as I did, and you won't have much trouble. When I 
regroup terms according to the power of the variable, I cross out a 
term as I use it, so I can tell when I have used them all.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/