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

SYMMETRIC FUNCTIONS OF THE ROOTS

89

71. By the aid of the theorem of ~ 70 we can calculate the
value, in terms of the coefficients, of any rational symmetric
function. But this method is laborious, and usually other
methods are preferable. For convenience of reference we state
here some of the results obtained in ~ 15, viz.,
For the cubic x3 + ax2 + bx + c = 0,
ja2' = 3 c-ab,
>:~2/ = b2 _ 2 ac,:a 3 = ab - 2 b2 - ac,
(+ ~ )(j + ~)(/ + )( = c - ab.
For the quartic x4 + axC + bx2 + cx + d = 0,
Sa23 =3 c - ab,
cCa2 = b2 - 2 ac + 2 d.
Ex. 1. For the cubic find the value, expressed in terms of the
coefficients, of A za22
aa3/ - za2
Ex. 2. For the quartic find the value of the irrational symmetric
function -x/Za3P.
Ex. 3. For f(x) 0 calculate Zai2ac2cc, where ca, a2, a*, an are
the roots.
Multiply aC1 = — al
and Zlaca2as =- a3.
In the product the term a2ca2cca occurs only once, the term aa102a3c4
occurs 4 times. Hence,
Za12ccc23 + 4 ccala2a3a4 = aa3,
and ZcC12a(X23 = aCa,- 4 a4.
If the calculation is carried on by ~ 70, II, we have, since p=q=l and
m ~2, 22 ac12Ca2a3 = S2S12 - 2 S3 - S22 2 S4.
Substituting for sl, S2, S3, S4 their values, ~ 68, and carrying out the
indicated operations, we get the same answer.