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

THEORY OF EQUATIONS

7 y4 + 23 y3 + 19 y2 - 1 = 0. The upper limits of the roots of this equation
are a and.9; hence the lower limits of the positive roots of the given
equation are 7 and 9.
Writing - y for x, we obtain y4 - 19 y2 + 23 y - 7 = 0. We obtain 20
as a superior limit and 7~ as an inferior limit of the positive values of y.
Hence the negative roots of the given equation lie between — 5 and -20,
and all the roots lie between 24 and -20.
To convey an idea of how the limits compare with the actual values of
x, we give the roots: 4.8977..., - 3.6331..., -.7124..., -.5522....
Ex. 2. Between what limits do the real roots of x5 + 5 x4 + x3 - 16 x2
-20 x- 16 = 0 lie?
By ~ 38 and ~ 41, the roots lie between 21 and -21. By ~ 39 and ~ 41,
the roots lie between 27 and -6. The roots are 2, -2, — 4, ~(-1 v-3).
Ex. 3. Between what limits are the real roots of
(1) x4 44x3- X2 -16 X -12 = 0,
(2) x -3x + 3x-1=0,
(3) X5 - 11x4 + 17x3 + 17x2 - 11 X+ 1= 0?
42. Change of Sign of f(x). If two real numbers a and b, when
substituted for x in f(x), give to f(x) contrary signs, an odd number of roots of the equation f(x) = 0 must lie between a and b; if
they give to f(x) the same sign, either no root or an even number
of roots must lie between a and b.
Since f(x) varies continuo'usly with x (~ 25), and f(x) changes
sign in going from f(a) t f(f(b), passing through all the intermediate values, it follows that f(x) must pass through the value
zero. That is, there is some real value of x, between a and b,
which causes f(x) to vanish and is a root of the equation f(x) =0.
But f(x), in passing from f(a) to f(b), may go through zero
nore than once. When f(a) andf(b) have opposite signs, f(x)
must pass through zero an odd number of times. Since a real
root corresponds to a point where the graph of f(x) crosses the
axis of x, the statement just made simply means that, to pass
from a point on one side of the axis to a point on the other
side of it, we must cross the axis an odd number of times.