Simultaneous Equations

Date: 11/28/95 at 13:26:8
From: Anonymous
Subject: Simultaneous equations

     Hi!

     x+y+z=1
     x^2+y^2+z^2=2
     x^3+y^3+z^3=3

     The problem is to find the value of:

     x^4+y^4+z^4=?

     Thanks very much!

              Danny Bromberg
              bc3dnky@bell-atl.com

Date: 11/28/95 at 16:6:57
From: Doctor Ken
Subject: Re: Simultaneous equations

Hello!

I'll tell you right now, I think this is going to be a messy problem.  
What you DON'T want to do is try to solve for x, y, and z and then plug 
in.  But nonetheless, I think it's kind of messy.  

Hm, how can I give you a hint ... 

Okay, try this: square the first equation.  That gives us
x^2 + y^2 + z^2 + 2(xy + xz + yz) = 1.  Since the first three terms add 
to 2, we get

              2 + 2(xy + xz + yz) = 1
                    xy + xz + yz  = -1/2

Notice that we found the value of xy + xz + yz without ever finding the
value of x, y, or z.  That's pretty neat.  See if you can do something 
similar to find out what x^4 + y^4 + z^4 is without finding x, y, and z.

Good luck!

-Doctor Ken,  The Geometry Forum