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

CHAPTER III

LOCATION OF THE ROOTS OF AN EQUATION
37. In this chapter we shall deduce theorems giving limits
between which all the real roots of an equation with real coefficients lie. We shall also derive theorems which enable us to
separate from each other all the distinct real roots, and to
ascertain the exact number and location of the real roots.
38. An Upper Limit. If in the equation f(x) = 0 the coefficient
of xa is unity, then the numerically greatest negative coefficient,
increased by one, is an upper limit of the positive roots of the
equation.
Any positive value of x makes f(x) > 0, if it makes
xn _ p(x'-1 + X-2 +...2 + 1) > 0,
n,n _ 1
x -1
or, xZ-p _. >0,
where p is the numerical value of the greatest negative coefficient. All the more is f(x) > 0, if a positive value of x makes
n _ 1
(xn 1) -)- >O
x-1
or, ) ( z)
(ox — 1)1- x_ > 0.
But this last expression is always > 0, or positive, if p < x- 1;
that is, if x >p 1.
Since any real value of x, greater than p+l, makes f(x)>0,
every real value of x which makes f(x) equal to zero must be
equal to or less than p + 1. Hence p + 1 is an upper limit of
the real positive roots of f(x) = 0.
43