### Horner's Method

Before we explain this method , we would like to point out that the scope of Horner's Method is restricted to Algebraic Equations only and that it can be conveniently used for finding both the commensurable (rational) and incommensurable roots of a Polynomial Equation . If the root is commensurable then , then it can be found exactly but if it is incommensurable , it can be found correctly to any specified number of decimal places .
By this method ,

1)   We first of all determine two consecutive integers between which the roots of equation lies . Suppose we find by trial that the root lies between the interval (a, a+1) where "a" is a positive integer . Thus we find the interval part  "a" of the root and then the decimal part is evolved figure by figure . This is done as follows .

2)  We diminish the roots of the given equation by "a" . The roots of the transformed equation will then lie in the interval (0, 1) .

3)  We now multiply the roots of the transformed equation by 10 so that the roots of the new transformed equation lie in the interval (0 , 10) .

4)  We then find by trial as in (1) an interval of the form (b, b+1) within the interval (0, 10) , such that a root of the last transformed equation lies in the interval  (i.e. we find two consecutive integers in the interval (0, 10)  between which the root of transformed equation lies )

Then "b" is the first figure of the decimal part of the root of the given equation i.e. the part of the root found so far is    a.b..........

We then repeat the entire step from (1) to (4)  as many times we like to have the numbers of figures in the decimal part of the root . Thus we again diminish the roots of the last transformed equation by "b" so that the resulting equation will have a root of the interval (0, 1) . We then multiply the roots of the last resulting equation by 10 so that the root lies in the interval (0, 10). Let this root be in the interval (c, c+1) , So that "c" is the second figure of the decimal part of the root of the given equation . Next we diminish the roots of the last transformed equation by "c" and proceed as before . Continuing this process we can obtain the root of equations to any desire number of decimal place .