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From: nglasser@yale-com.UUCP (Nathan Glasser)
Newsgroups: net.math
Subject: Solution to high school problem
Message-ID: <2225@yale-com.UUCP>
Date: Sat, 22-Oct-83 08:18:41 EDT
Article-I.D.: yale-com.2225
Posted: Sat Oct 22 08:18:41 1983
Date-Received: Sun, 23-Oct-83 07:55:05 EDT
Lines: 32

The following is a solution to the problem I posted earlier. The problem
was to find the sum of the seventeenth powers of the roots of the equation
   17     2
  x   + 3x  + 2x - 1 = 0.

Let the roots of this equation be X , i = 1,2,...,17.
				   i
Also let S  denote the sum of the kth powers of the roots of the given equation.
	  k
Then since each X  is a root of the equation,
		 i
 17    2
X  + 3X  + 2X - 1 = 0           for each i. If we add all 17 such equations
 i     i     i                  we get

S  + 3S  + 2S  - 17 = 0.
 17    2     1

>From the coefficients of the polynomial, it is clear that S = 0. Also,
					   2               1
S , the sum of the squares of the roots = S  - 2(X X  + ... + X  X  ).
 2                                         1      1 2          16 17

But the sum of the products of the roots taken two at a time = the coeff.
of the x^15 term, which is 0. So S = 0. Hence S  = 17.
				  2            17


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