Theory of Equation

Commensurable Roots

502 To find the commensurable roots of an equation. First transform it by putting 𝑥=𝑦𝑘 into an equation of the form 𝑥^𝑛+𝑝₁𝑥^𝑛−1+𝑝₂𝑥^𝑛−2+⋯+𝑝_𝑛=0 having 𝑝₀=1, and the remaining coefficients integers. 431 503 This equation cannot have a rational fractional root, and the integral roots may be found by Newton's method of Divisors (459).
These roots, divided each by 𝑘, will furnish the commensurable roots of the original equation. 504 Example: To find the commensurable roots of the equation 81𝑥⁵−207𝑥⁴−9𝑥³+89𝑥²+2𝑥−8=0 Dividing by 81, and proceeding as in (431), we find the requisite substitution to be 𝑥=𝑦9 The transformed equation is 𝑦⁵−23𝑦⁴−9𝑦³+801𝑦²+162𝑦−5832=0 The roots all lie between 24 and −34, by (451).
The method of divisors gives the integral roots 6, −4, and 3. Therefore, dividing each by 9, we find the commensurable roots of the original equation to be 23, −49, and 13, 505 To obtain the remaining roots; diminish the transformed equation by the roots 6, −4, and 3, in the following manner (see 427):   1−23−9+801+162−5832 6   6−102+666+810−5832   1−17−111+135+972 -4  −4+84+108−972   1−21−27+243 -3   3−54−243   1−18−31 The depressed equation is therefore 𝑦²−18𝑦−81=0 The roots of which are 9(1+2) and 9(1−2); and, consequently, the incommensurable roots of the proposed equation are 1+2 and 1−2.

Sources and References

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