Theory of Equation

Newton's Method of Divisors

459 To discover the integral roots of an equation.

Example

To ascertain if 5 be a root of 𝑥⁴−6𝑥³+86𝑥²−176𝑥+105=0 If 5 be a root it will divide 105. Add the quotient to the next coefficient. Result, −155. If 5 be a root it will divide −155. Add the quotient to the next coefficient; and so on. If the number tried be a root, the divisions will be effectible to the end, and the last quotient will be −1, or −𝑝₀, if 𝑝₀ be not unity.

 5)105
    21
  −176
5)−155
   −31
    86
  5)55
    11
    −6
  5)−5
    −1

460 In employing this method, limits of the roots may first be found, and divisors chosen between those limits. 461 Also, to lessen the number of trial divisors, take any integer 𝑚; then any divisor 𝑎 of the last term can be rejected if 𝑎−𝑚 does not divide 𝑓(𝑚).
In practice take 𝑚=+1 and −1.
To find whether any of the roots determined as above are repeated, divide 𝑓(𝑥) by the factors corresponding to them, and then apply the method of divisors to the resulting equation.

Example

Take the equation 𝑥⁶+2𝑥⁵−17𝑥⁴−26𝑥³+88𝑥²+72𝑥−144=0 Putting 𝑥=1, we find 𝑓(1)=−24. The divisors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 24, ⋯ The values of 𝑎−𝑚 (since 𝑚=1) are therefore 0, 1, 2, 3, 5, 7, 8, 11, 15, 23, ⋯ Of these last numbers only 1, 2, 3, and 8 will divide 24. Hence 2, 3, 4, and 9 are the only divisors of 144 which it is of use to try. The only integral roots of the equation will be found to be ±2 and ±3. 462 If 𝑓(𝑥) and 𝐹(𝑋) have common roots, they are contained in the greatest common measure of 𝑓(𝑥) and 𝐹(𝑋). 463 If 𝑓(𝑥) has for its roots 𝑎, 𝜙(𝑎), 𝑏, 𝜙(𝑏) amongst others; then the equations 𝑓(𝑥)=0 and 𝑓{𝜙(𝑥)}=0 have the common roots 𝑎 and 𝑏. 464 But, if all the roots occur in pairs in this way, these equations coincide. For example, suppose that each pair of roots, 𝑎 and 𝑏, satisfies the equation 𝑎 + 𝑏=2𝑟. We may then assume 𝑎 − 𝑏=2𝑧. Therefore 𝑓(𝑧+𝑟)=0. This equation involves only even powers of 𝑧, and may be solved for 𝑧². 465 Otherwise, Let 𝑎𝑏=𝑧; then 𝑓(𝑥) is divisible by (𝑥−𝑎)(𝑥−𝑏)=𝑥² −2𝑟𝑥+𝑧. Perform the division until a remainder is obtained of the form 𝑃𝑥+𝑄. where 𝑃, 𝑄 only involve 𝑧.
The equations 𝑃=0, 𝑄=0 determine 𝑧, by (462); and 𝑎 and 𝑏 are found from 𝑎 + 𝑏=2𝑟, 𝑎𝑏=𝑧.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive