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

ROOTS OF NUMERICAL EQUATIONS

63

Dividing the left member by 2 and multiplying the roots by 2, we
obtain - x2 -2 x - 12 = 0.
It is found that + 3 is the only commensurable root of this equation,
Hence, + 3 is the only commensurable root of the given equation.
Ex. 3. Find all the commensurable roots of
x3 + 4X2 + 6x + 3 0.
X4 - 3 X3 - 22 x2 - 39 x - 21 = 0.
5 - 10 X4 + 17 x2-x- 7 = 0.
x5 - 13x4 + 34 x3 - 26 2 - 18 x + 22 = 0.
6 X3 - 2 x2 + 3 x 4 =0.
4x3 + 20x2 - 23 x + 6 = 0.
56. Horner's Method. This method may be used advantageously for finding not only incommensurable roots, but also
commensurable roots when the process of ~ 55 is inconvenient.
In the application of Horner's method we must know the
first significant figure of the root, to start with. The first digit
may be found by the process indicated in ~ 42 or by Sturm's
Theorem.
Horner's method consists of successive transformations of an
equation. Each transformation diminishes the root by a certain
amount. If the required root is 2.24004, then the root is
diminished successively by 2,.2,.04,.00004. The mode of
effecting these transformations, by synthetic division, was
explained in ~ 32. The method will be readily understood by
the study of the following example:
Ex. 1. The equation x3 - x - 9 = 0, I
has a root between 2 and 3, for f(2) - 3 and f(3)= + 15. The first
figure of the root is therefore 2. Transforming the equation so that the
roots of the new equation will be smaller by 2, we obtain
1 +0 -1 -9 2
+2 +4 +6
+2 +3 -3
+2 +8
+ 4 +11
+2
- 0(