**Definition :- The last but one coefficient of each transformed equation is called the Trial - Divisor .**It is advisable to use the Principal of Trial - Divisor generally after two or three transformations .

1) Before dimensioning the roots by a number , we must supply the missing terms with zero coeff. i.e. we should make the equation to be transformed complete .

2) If at any stage , the Trial - Divisor suggests the next figure in the decimal part of the root to be Zero in such a case , the roots should again be multiplied by 10 and Zero should be placed in the decimal part of the root .

The principal is based on " Newton's Method of Approximation "

Let we see that , If a root of f(x) = 0 differs from an approximate root a by a small quantity h then ,

h= -f(a)/f '(a)

Now in a Transformation process , when we diminish the roots of polynomials f(x)=0 , by a , and the least but one will be f '(a) . This can be shown as follows .

Let y=x-a , then x=a+y , Thus f(x)=0 => f(a=y)=0

=> f(a) + y f '(a) + y^2/2!f "(a) + ..... =0

Thus in the transformed equation will be f(a) and the coeff. of last one term i.e. the coeff. of y is f '(a) .

Similarly in Hoener's Process , when we diminish the roots of f(x)=0 by a , the last coeff. in the transformed equation will be f(a) and the last coeff. but one will be f '(a). Also since a is positive , f(a) and f'(a) should be the opposite sign .

Hence in Horner's Procss , after two or three stages have been completed and further transformation isnecessary , We may expct to get the next figure in the decimal part of the root

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