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

44 THEORLY OF EQUATIONS
39. Another Upper Limit. If the numerical value of each
negative coefficient is divided by the sum of all the positive coegficients which precede it, the greatest of the fractions thus formed,
increased by one, is an upper limit of the positive roots off(x) = 0.
Let f(x) - o ax + alxn-1 - a2x-2 + CaL-3 - Cx-4 -+.. + a,
in which the coefficients of x"-2 and x"-4 are negative. Since
( 1 - l) = ( -- 1) (XI.-' + a'-2 +...* + 1),
we have x"s (x - 1) (a-1 m-2+ x-2+ + x + 1) +1.
If we transform all the positive terms in f(x) by means of
this formula, we obtain f(x)=
ao(x - 1)x -1 + Co(X2 - 1)x^ 2 + 0(x- _ 1)X1- + Co(x- )X-4 +... + ao
+al(x -l)Xn-2+al(x-l)1x -3+a(x- 1)l-) + -4- — +a
-a2n
- a -2
+ a,(x- ) x ) '-4 + aQ
a4n-4
If in this expression x is assigned a positive value large
enough to make the sum of the coefficients in each colmlnl
of terms positive, then f(x) will be positive for that value
of x. The coefficients in the first and third column are positive, if x>1. The same is true of all other columns which
are free of negative coefficients.
The sum of the coefficients in the second column, containing the negative coefficient - a, is positive if x is large enough
to make a0 (x-) + a,(x- 1)-a2 > 0.
Whence x > — _ +1.
ao + aI
Similarly, we obtain from the fourth column, if
o (x - 1) + 0 (x -1 ) +,, (ax - 1) - a4 > 0,
the inequality x > - - 1.
ao + ia + a3