Theory of Equation

Reciprocal Equations

466 A reciprocal equation has its roots in pairs of the form 𝑎, 1𝑎; also the relation between the coefficients is 𝑝_𝑟=𝑝_𝑛−𝑟, or else 𝑝_𝑟=−𝑝_𝑛−𝑟 467 A reciprocal equation of an even degree, with its last term positive, may be made to depend upon the solution of an equation of half the same degree. 468

Example

4𝑥⁶−24𝑥⁵+57𝑥⁴−73𝑥³+57𝑥²−24𝑥+4=0 is a reciprocal equation of an even degree, with its last term positive.
Any reciprocal equation which is not of this form may be reduced to it by dividing by 𝑥+1 if the last term be positive; and, if the last term be negative, by dividing by 𝑥−1 or 𝑥²−1, so as to bring the equation to an even degree. Then proceed in the following manner:- 469 First bring together equdistant terms, and divide the equation by 𝑥³; thus 4𝑥³+1𝑥³−24𝑥²+1𝑥²+57𝑥+1𝑥−73=0 By putting 𝑥+1𝑥=𝑦, and by making repeated use of the relation 𝑥²+1𝑥²=𝑥+1𝑥² the equation is reduced to a cubic in 𝑦, the degree being one-half that of the original equation. 470 Put 𝑝 for 𝑥+1𝑥, and 𝑝_𝑚 for 𝑥_𝑚+1𝑥_𝑚. The relation between the successive factors of the form 𝑝_𝑚 may be expressed by the equation 𝑝_𝑚=𝑝𝑝_𝑚−1−𝑝_𝑚−2 471 The equation for 𝑝_𝑚, in terms of 𝑝, is 𝑝_𝑚=𝑝^𝑚−𝑚𝑝^𝑚−2+𝑚(𝑚−3)1⋅2𝑝^𝑚−4−⋯+(−1)^𝑟𝑚(𝑚−𝑟−1)⋯(𝑚−2𝑟+1)|𝑟𝑝^𝑚−2𝑟+⋯ By (54), putting 𝑞=1

Sources and References

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