SOLUTION TO PROBLEM H-8

Let $x_1, x_2, \dots , x_{2002}$ be the roots of p(x) repeated according to multiplicity. Since the constant term is 1, according to Viete ' s relations between roots and coefficients one has that

$$S_1 := x_1 + x_2 + \dots + x_{2002} = \frac{-0}{1} = 0 \qquad \hbox{and}$$

$$S_2 := x_1x_2 + \dots + x_1x_{2002} + x_2x_3 + \dots +x_2x_{2002} + \dots + x_{2001}x_{2002} = \frac{1}{1} = 1$$

Therefore

$$S_1^2 - 2S_2 = -1 \qquad \hbox{but also}$$

$$S_1^2-2S_2 = x_1^2 + x_2^2 + \dots + x_{2002}^2$$

Hence

$$x_1^2 + x_2^2 + \dots + x_{2002}^2 = -1 < 0$$

for which reason some of the numbers $x_1, x_2, \dots , x_{2002}$must be complex and non-real.

Questions and/or comments should be directed to Judy Downey or Griff Elder

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Last modified:   Sat Mar 9 16:35:46 CST 2002