An introduction to the modern theory of equations, by Florian Cajori.

86

THEORY OF EQUATIONS

The values of Sm may be expressed in determinant form
as follows:
a1 1 0 0
82= a1 1 1a, 0a a, 1 1 0
S2a2, - 2 a2 cI 1, s4 = a., etc.
2a2al aac a2 1 30a3 a2 ai 1l
4a4 a 3 a 2- a
69. Coefficients expressed in Terms of Sm. From the formulae
of ~ 68 one readily obtains
s 1 0 0
a-? I s,Ia3= a.= s2 8s 2,a4= 2 0 etc.
2 S2 83 I3 sIS 4 S3 S2 s, 3
84 8 3 8 2 S1
Ex. 1. Find the sums of positive powers of the roots of
a4 + x3 + 2 + X + 1 0.
We have sI - l = - 1,
2 = - asl - 2 a - 1,
S3 - al2 - a2l - 3 03 = -1,
S4 = als3 - a23 2 - a31 - 4 4 - 1,
and so on. The roots are the primitive fifth roots of unity. Verify our
result for s2 by actually squaring the roots given in Ex. 2, ~ 67.
Ex. 2. Find the sums of positive and negative powers of the roots of
x - 2 x + 5 x - 4 = 0
si = 2, - 6,s = -10, S4 = + 18, and so on. To get s_-, put x
y
and the equation becomes x3 - 5 X2 + 1 x- = 0. Then s_- =,
S-3, and so on.
Ex. 3. Find the sums of positive and negative powers of the roots of
xa + 1 = 0.
Ex. 4. Show that if the sum of an even power of the roots is zero or
negative, the equation has at least two complex roots.
* Ex. 5. Show that, for xn -1= 0, sm= n or 0, according as m is
divisible or not divisible by n.
Substitute for al,..., a, their values in I and II, ~ 68.