[631
SYNOPSIS OF THE THEORY OF EQUATIONS.
9
631]
C. X.
2
a assignable order)
f a given equation,
is possible to find
a sextic equation ;
the coefficients of
d, and therefore
ic equation.
of values (obtained
roots ; for instance,
e function ab + cd\
is 3, viz. ab + cd,
tain functions of a
In the case of a quadric equation x 2 +px + q = 0, we can find for x, by the
assistance of the sign V( ) or ( )l, an expression for x as a two-valued function
of the coefficients p, q, such that, substituting this value in the equation, the equation
is thereby identically satisfied, viz. we have
x=- : kp±V(lp 2 -q),
giving
0C 2 = %p 2 -q + p^/(ip 2 - q)
+px = -^p 2 ± p\/(ip 2 - q)
+ q = +q
af+px + q =0,
the value of any
letermine rationally
jans of a numerical
2, 3, 4; then if it
2 (viz. a — 1, 6 = 2,
c = 4, d = 3); and
md not any other
,s regards particular
nines a to be =1
a 2 b = 16 does not
, or else 2 and 4,
i = (a + b) 3 + (c + d) s ,
there are only the
% + 4 2 2/ 2 +1 3 % will
Ld therefore rational
then the equations
of known functions
rown. But observe
. as regards t 2 , t 3 ;
of tn ~f“ ¿3, t 2 t 3 1 01',
but these last will
onally in terms of
a of t x and of the
n of equations, or,
and the equation is on this account said to be algebraically solvable, or, more accurately,
to be solvable by radicals. Or we may, by writing x = — ^p+ z, reduce the equation
to z 2 = \p 2 — q, viz. to an equation of the form z 2 = a, and, in virtue of its being thus
reducible, we may say that the equation is solvable by radicals. And the question for
an equation of any higher order is, say of the order n, can we by means of radicals,
that is, by aid of the sign ^/( ) or ( ) m , using as many as we please of such
signs and with any values of m, find an ^-valued function (or any function) of the
coefficients, which substituted for x in the equation shall satisfy it identically.
It will be observed that the coefficients p, q, ... are not explicitly considered as
numbers, but that even if they do denote numbers, the question whether a numerical
equation admits of solution by radicals is wholly unconnected with the before-mentioned
theorem of the existence of the n roots of such an equation. It does not even follow
that, in the case of a numerical equation solvable by radicals, the algebraical expression
of x gives the numerical solution; but this requires explanation. Consider, first, a
numerical quadric equation with imaginary coefficients; in the formula x = — %p + \J(\p 2 — q),
substituting for p, q their given numerical values we obtain for x an expression of the
form x = a. + /3i ± V(7 + Si), where a, /3, 7, 8 are real numbers; this value substituted
in the numerical equation would satisfy it identically and it is thus an algebraical
solution; but there is no obvious d priori reason why the expression V(y + Si) should
have a value = c + di, where c and d are real numbers calculable by the extraction
of a root or roots of real numbers; it appears upon investigation that + 8i) has
such a value calculable by means of the radical expression \/{V(7 2 + S 2 ) ± 7}; and hence
that the algebraical solution of a quadric equation does in every case give the
numerical solution of a numerical quadric. The case of a numerical cubic will be
considered presently.
A cubic equation can be solved by radicals, viz. taking for greater simplicity the
cubic in the reduced form a? - qx — r = 0, and writing x = a + b, this will be a solution
if only Sab = q, and a 3 + b 3 = r, or say % (a 3 + b 3 ) = \r; whence
1 (a 3 - V) = ± V(i?’ 2 - 2T? 3 )>