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

66

THEORY OF EQUATIONS

should be; for if the two terms had like signs, the value of x in
the transformed equation would be negative, showing that the
last digit in the root of the original equation had been taken too
large. For instance, if in Ex. 1 the first decimal had, by mistake, been taken as 3, instead of 2, then the second transformed equation would have been x3 + 6.9 x2 + 14.87 x +.867 = 0.
The approximate value of x in this equation is —.05, showing that in diminishing the roots by.3 we took away too
much.
If, by mistake, a digit is taken too small, the error will show
itself in the next step. Suppose that in Ex. 1 the first decimal had been taken to be.1, then the second transformed
equation would have been x3 + 6.3 x2 + 12.23 x - 1.839 = 0.
From 12.23 x - 1.839 = 0 we get approximately x =.15. This
changes.1 into.25, and thus discloses an error in the estimate
of the first decimal.
To find the value of a negative root by Horner's method, we
need only transform the given equation by writing -x for x
and then proceed as before.
Ex. 1. Find the real roots of:
(1) 4x5 - 3 - 2 x2+ 4 x - 10 = 0.
(2) 3 x5 +.X3 4 - x-4 x + 5 =0.
(3) 7X4 + 3 x-5 x2 + 4 x-6= 0.
(4) x7 - + x5 + X4 - 10 = 0.
(5) - 4x - 2 = 0.
57. Newton's Method of Approximation. This method is not
as convenient in the solution of numerical equations involving
algebraic functions as is the method of Horner, but it has the
advantage of being applicable to numerical equations involving transcendental functions. For instance, Newton's method
can be used in finding x in x - sin x = 2.
Let f(x) = 0 be the given equation. Suppose that we know a
quantity a which differs from one of the values of x by the small