ILLUSTRATIONS OF USES OF SYMMETRIC FUNCTION TABLES. 
15 
ILLUSTRATIONS OF USES OF SYMMETRIC FUNCTION TABLES. 
To illustrate a use of symmetric functions we may solve the quartic 
equation 
x 4 —x 3 — 8x 2 + 2x + 12 = 0 ( 1 ) 
with roots say a 1} a 2 , a 3 , a 4 . We may begin by forming the cubic resolvent 
in Z, where Z has the three values of the function 
The equation is 
a k a h + a i 3 a u 11 
Z 3 -Z 2 Xa x 
a 2 + Z2 a\a 2 a 3 — (Sa?a 2 a 3 a 4 + 2ta?a|a|) = 0 
(2) 
By the use of tables of symmetric functions this may be written 
Z 3 + a 2 Z 2 + (a x a 3 — 4 a 4 ) Z + 4a 2 cc 4 — aj — ct?a 4 — 0 
(3) 
for the general quartic 
X 4 + a x x 3 + a 2 x 2 + a 3 x + a 4 = 0 
(4) 
or for (1) 
Z 3 + 8Z 2 — 50Z — 400 = 0 
(5) 
Solving the cubic (5) 
4 = a 4 a 2 + a 3 a 4 = —8 
(6) 
@2 ~ a l a 3 + a 2 a 4 “ 5^/ 2 
(7) 
(3 3 - oqoq + a 2 a 3 = — 5yj 2 
(8) 
and since 
a x + a 2 -f a 3 + a 4 = 1 
(9) 
we may solve the set (6), (7), ( 8 ), (9), for the a’s and get 
oq — 3 a 2 = — 2 a 3 = yj 2 a 4 = — -yj 2 
To illustrate, in particular, the use of the fifteenthic, let us find the 
resultant of a cubic equation and a quintic equation, say 
f 1 (x)=x 3 -Sx 2 -2x+l = 0 ( 10 ) 
and 
<pi(x) =x 5 — 2x + 3 = 0 ( 11 ) 
For the resultant of the cubic equation 
f(x) = x 3 + a x x 2 + a 2 x + a 3 = 0 , with roots a x , a 2 , a 3 ( 12 ) 
and the quintic equation 
$ (x) == x 5 + b x x 4 + b 2 x 3 + b 3 x 2 + b^x + b 5 = 0 
we may write 
= -qM -<pM 
(13) 
(14) 
11 To see that + a; 3 aj 4 is a three-valued function one may try various permutations of the sub¬ 
scripts and find that each leads to one of the forms 6, 7, 8. 
