'Un Distributing' Polynomials

Date: 01/11/2003 at 19:02:21
From: Mike
Subject: 'un distributing' polynomials

How can you factor:

   y^4 + 14y^2 + 49 - x^2 ?

I regrouped first:

   (y^4 + 14y^2) + (49 - x^2)

I factored that some more to get:

   y^2(y^2 + 14) + (7 + x)(7 - x)

but that just doesn't seem complete.

I tried another way and regrouped like this:

   (14y^2 + 49) + (y^4 - x^2)

factored that a little bit more to get:

   7(2y^2 + 7) + (y^2 + x)(y^2 - x)

I don't think I can get it down past that any more.

Which way is right?  Or are they both wrong?

Date: 01/12/2003 at 22:04:22
From: Doctor Kastner
Subject: Re: 'un distributing' polynomials

Hi Mike -

First of all, let me say that I really appreciate the fact that you
have worked hard on this problem and you showed me what you did.  That
helped me out since I would have tried the two things that you did.
However, I agree that they aren't quite right.

After staring at it for a bit, I think that I have found the key for
this one. How about trying this grouping:

(y^4 + 14y^2 + 49) - x^2

Here's how I came to this. Since you tried the obvious "two and two"
groupings and those didn't work, the next things to try were the 
"three and one groupings." However, it looks as if keeping the y terms 
together is a good idea because of the y^4 and the y^2, so that
means that either the 49 of the x^2 can be by itself. At that point I
noticed that the 14 and the 49 are part of a perfect square so the 49
should go with the y terms. That leaves the -x^2 to (we hope!) form
part of a difference of squares.

I hope this helps. Write back if you're still stuck, or if you have
other questions.

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

Date: 01/12/2003 at 22:42:56
From: Mike
Subject: 'un distributing' polynomials

That's right, I never thought of the three and one groupings... But 
I'm still a little stuck after that.

I broke it down and regrouped into the way you mentioned:

   (y^4 + 14y^2 + 49) - x^2

and factored that to

   (y^2 + 7)(y^2 +7) -x^2

but that's where I'm stuck.

I didn't quite follow what to do with the - x^2.

If you could show me or clear this up for me it'd be really great.

Date: 01/13/2003 at 09:34:00
From: Doctor Kastner
Subject: Re: 'un distributing' polynomials

Hi again Mike -

Things are looking good, but instead of writing the expression

   (y^2 + 7)(y^2 +7) - x^2

Collect the first two terms and write

   (y^2 + 7)^2 - x^2

This is now nothing more than the difference of squares: z^2-x^2. In
this case your z is actually a more complicated expression (y^2+7),
but that doesn't matter at all. The form is what is important, and
your problem fits the form exactly!

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