Principle of Trial - Divisor

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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