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

8

THEORY OF EQUATIONS

The ~ denotes an ambiguity; that is, the sign of a term sc
affected is here undetermlined. We see that the dots which
follow ~ are ambiguities; that is, each permanence of sign in
f(x) is here replaced in (x - ().f(x) by an ambiguity. We
see also that to every variation of sign in f(x) there corresponds
a variation in (x - ) ff(x). In the product there is, in addition, a variation introduced at the end. Hence the product
contains at least one more variation than does f(x). It may
contain more; for, successive permanences like + - + or
- - -, occurring in f(x) and replaced in (x a) f(c) ) by ambiguities, may in reality be replaced by the signs + — + or
---—. But such changes in sign always increase the variations by an even number. Hence in (x - a) - f(x) the total number of variations exceeds that in f(x) by the odd number 1 or
14+2.
The same conclusion is reached when the last term in f(x)
is negative.
Descartes' Pule follows now easily. Suppose the product of
all the factors, corresponding to negative and complex roots of
f(x) = 0, to be already formed. Designate this product by
F(x). Since F(x) =0 has no positive roots, the first and last
terms in F(x) have like signs. Hence the number of variations in F(x) is an even number, 2 k, where k is zero or a positive integer. Now, if F(x) is multiplied by the factor x - a,,
where ar is a positive root, we get in the product 2 kI + 1 variations, where km kI. In the same way a second factor x - a2
gives rise to 2 c2+ 2 variations, and so on. Thus, the introduction of v positive roots results in 2 kv + v variations, where
kV is zero or a positive integer. Hence, the theorem is
established.
12. Negative Roots. To apply Descartes' Rule to negative
roots of f(x)= 0 we write down an equation whose roots are
those off(x) = 0 with their signms changed. The new equation
can be derived by substituting in f(x) = 0, - x for x. The