'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/
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