Factoring Polynomials

Date: 11/29/1999 at 00:05:20
From: (anonymous)
Subject: Factorizing the expressions

Dear Dr. Maths,

I'm having problems with some of these questions. Can you please 
enlighten me on these? 

1) I am not sure where I went wrong on this one. Can you please tell 
   me where I was wrong?

       4x^2 - 36 

     = a^2 - b^2 
     = (a+b)(a-b) 
     = (2x+6)(2x-6) 

2) I've learned how to use the Distributive law to show the following 
   special rules:

     (a+b)^2    = a^2 + 2ab + b^2 
     (a-b)^2    = a^2 - 2ab + b^2
     (a+b)(a-b) = a^2 - b^2 

But I'm very puzzled.

     x^2 + yz + xy + xz = ?

There are altogether 4 numbers. And none of the above rules can 
explain this expression. 

Thanks a lot.
Agnes

Date: 12/23/1999 at 22:31:54
From: Doctor Sandi
Subject: Re: Factorizing the expressions

Hi Agnes:

I can see nothing wrong with what you have done with expression (1) 
above. I do it a little differently and put them into their "squared" 
form as soon as I can, that is:

       4x^2 - 36
     = (2x)^2 - (6)^2
     = (2x-6)(2x+6)

Those other two rules (a+b)^2 and (a-b)^2 will help you a lot if you 
can memorize them.

As for the other one:

     x^2 + yz + xy + xz

this is a matter of factorization of four terms. If you collect terms 
that have the same sort of letters in them and put them together you 
can rearrange it like this:

     x^2 + xy + yz + xz

This is the same thing, just rearranged a little bit. Now you can 
factor the x out of the first pair of terms and the z out of the 
second pair of terms, like this:

     x(x+y) + z(y+x)

Multiply it out if you are unsure and you'll see that you end up with 
the original equation.

Then, because both x and z are being multiplied by exactly the same 
thing (x+y is the same as y+x), you can rewrite it as

     (x+z)(x+y)

What you're doing here is taking each of the terms that are 
multiplying (x+y) and putting them in a bracket of their own, because 
they each multiply (x+y).

With this factorization of four terms, you are looking for something 
that you can do that will cause you to finish with two brackets both 
containing the same thing, which you can then collect neatly.

Also, to check that you have arrived at the correct answer you could 
multiply your answer out; you should end up with the original 
expression. That way you know that you've found the right answer.

To give you a quick summary of factorization:

To factor a polynomial:

     1. Always remove common factors first

     2. If it has two terms look for:
         - difference of two squares
         - sum or difference of two cubes (if a cubic)

     3. If it has three terms:
         - it may be a perfect square (then use (a+b)^2 or (a-b)^2)
         - it may be factored by using the distributive law in reverse
         - completing the square
        (I don't know if you would have learned those last two yet)

     4. If it has four or more terms it may be factored by:
         - Grouping the terms in order to apply other methods
             (this is what we did)
         - applying the factor theorem
         - finding one factor using the factor theorem then finding 
             the other using long division.

You might also want to take a look at this page from the Dr. Math FAQ:

  Learning to Factor
  http://mathforum.org/dr.math/faq/faq.learn.factor.html   

I hope this has been a help to you. If you have any further questions, 
please don't hesitate to send them to Ask Dr. Math.

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