By substitution we have
a" + pa + pra"? +--+, 10 + Pn = O.
Transposing and dividing throughout by a, we obtain
in which it is evident that = must be an integer. Denoting = by Q and transposing — p,_1,
Q+ Pri=—-— poa" — pia? — a)
Dividing again by a gives
Q+ Dai
a =e — pa" — pa" — a.
Again, as before, the first member of the equation must be an integer. Denoting it by Q, and proceeding as before, we must after n divisions obtain a result
Q,-1 + Pi
=—1. a
Hence if a represents one of the integral divisors of the last term we have the following rule:
Divide the last term by a and add the coefficient of x to the quotient.
Divide this sum by a, and -if the quotient is an integer add to it the coefficient of x^2.
Proceed in this manner, and if a is a root of the equation each quotient will be an integer and the last quotient will be -1.
The advantage of Newton’s method is that the obtaining of a fractional quotient at any point of the division shows at once that the divisor is not a root of the equation.
Ex. Find the integral roots of «+423 «2—16%—12=0. By Descartes’ Rule the equation cannot have more than one positive root, nor more than three negative roots.
The integral divisors of —12 are +1, +2, +3, +4, +6. Substitution shows that — 1 is a root, and that + 1 is not a root.
