Factoring vs. Simplifying

Date: 03/02/2003 at 19:33:05
From: Mike
Subject: Algebra (factoring)

Factor the expression 1+y(1+x)^2(1+xy).

It looks pretty factored in this form, and I can't figure out how to 
factor it any further.

Date: 03/03/2003 at 12:43:23
From: Doctor Ian
Subject: Re: Algebra (factoring)

Hi Mike,

It's pretty simplified, but that's not the same as factored. 

An expression is factored if it contains only factors. For example, 

  y(1 + x)^2(1 + xy)

is factored. Why do we make this distinction?  Often we're looking
for situations where we have 

  0 = some factored expression 

This is convenient, because we can then quickly see what the values of
the factors have to be.  If 

  0 = y(1 + x)^2(1 + xy)

then one of the following must be true:

  1) y = 0

  2) (1 + x) = 0, i.e., x = -1

  3) (1 + xy) = 0, i.e., xy = -1

So to factor your expression, you'd need to expand it out, gather up
like terms, and try to factor the result, using the methods you've
been learning over the years. Here's a simple example of how that
would look:

   -1 + (x+2)^2 = -1 + (x^2 + 4x + 4)

               = x^2 + 4x + (4 + -1)

               = x^2 + 4x + 3

               = (x+1)(x+3)

Yours will be a little trickier. 

Does this help? 

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

Date: 03/03/2003 at 21:10:18
From: Mike
Subject: Thank you (Algebra (factoring))

Your advice was appreciated, it gave me the help I needed 
to solve the problem.  The answer ended up being:
(x^2y+xy+1)(xy+y+1)