4 
SYNOPSIS OF THE THEORY OF EQUATIONS. 
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In general, a is not a root of the equation f x (x) = 0; but it may be so, viz. 
f\ (x) may contain the factor x — a; when this is so, f(x) will contain the factor 
(x — a) 2 ; writing then f(x) = (x — a) 2 f 2 (x), and assuming that a is not a root of the 
equation f 2 (x) = 0, x = a is then said to be a double root of the equation. Similarly, 
f(x) may contain the factor {x — a) 3 and no higher power, and then x — a is said to 
be a triple root; and so on. 
Supposing, in general, that f(x)=(x — a) a F(x), where a is a positive integer which 
may be =1, and Fx is of the order n — a, then if & is a root different from a, we 
shall have x — b a factor (in general a simple one, but it may be a multiple one) of 
F(x), and f(x) will in this case become = (x — a) a (x — by <E> (x), where /3 is a positive 
integer which may be =1, and <I># is of the order n—a—fi. The original equation 
fx = 0 is in this case said to have a roots each = a, /3 roots each = b, and so on. 
We have the theorem, a numerical equation of the order n has in every case n 
roots, viz. there exist n numbers a, b,... (in general, all of them distinct, but they 
may arrange themselves in groups of equal values) such that 
f(x) = (x — a)(x — b) (x — c) ... identically. 
If an equation has equal roots, these can in general be determined; the case is at 
any rate a special one, which may be here omitted from consideration. It is there 
fore, in general, assumed that the equation f(x)= 0 under consideration has all its 
roots unequal. If the coefficients p 1} p 2 ,... are all or any one or more of them 
imaginary, then the equation f(x) = 0, separating the real and imaginary parts, may 
be written F (x) + i<i> (x) = 0, where F (x), <&(x) are each of them a function with real 
coefficients; and it thus appears that the equation f(x) = 0 with imaginary coefficients 
has not in general any real root; supposing it to have a real root a, this must be 
at once a root of each of the equations F (x) = 0 and <I> (x) — 0. 
But an equation with real coefficients may have as well imaginary as real roots; 
and we have further the theorem that for such an equation the imaginary roots enter 
in pairs, viz. a + (3i being a root, then will also a — ¡3i be a root. 
Considering an equation with real coefficients, the question arises as to the number 
and situation of its real roots; this is completely resolved by means of Sturm’s 
theorem, viz. we form a series of functions f(x), f (x), f 2 (x),.., f n (x) (a constant) of 
the degrees n, n— 1,.., 2, 1, 0 respectively; and substituting therein for x any two 
real values a and b, we find by means of the resulting signs of these functions how 
many real roots of f(x) lie between the limits a, b. 
The same thing can frequently be effected with greater facility by other means, 
but the only general method is the one just referred to. 
In the general case of an equation with imaginary (it may be real) coefficients, 
the like question arises as to the situation of the (real or imaginary) roots, viz. if 
for facility of conception we regard the constituents a, ¡3 of a root a + as the 
coordinates of a point in piano, and accordingly represent the root by such point; 
then drawing in the plane any closed curve or “contour,” the question is how many 
roots lie within such contour.