Theory of Equation

Binomial Equations

472 If 𝛼 be a root of 𝑥^𝑛−1=0, then 𝛼^𝑚 is likewise a root where 𝑚 is any positive or negative integer. 473 If 𝛼 be a root of 𝑥^𝑛+1=0, then 𝛼^2𝑚+1 is likewise a root. 474 If 𝑚 and 𝑛 be prime to each other, 𝑥^𝑚−1 and 𝑥^𝑛−1 have no common root but unity. Take 𝑝𝑚−𝑞𝑛=1 for an indirect proof. 475 If 𝑛 be a prime number, and if 𝛼 be a root of 𝑥^𝑛−1=0, the other roots are 𝛼, 𝛼², 𝛼³, ⋯, 𝛼^𝑛. These are all roots, by (472). Prove, by (474), that no two can be equal. 476 If 𝑛 be not a prime number, other roots besides these may exist. The successive powers, however, of some root will furnish all the rest. 477 If 𝑥^𝑛−1=0 has the index 𝑛=𝑚𝑝𝑞; 𝑚, 𝑝, 𝑞 being prime factors; then the roots are the terms of the product (1+𝛼+𝛼²+⋯+𝛼^𝑚−1)(1+𝛽+𝛽²+⋯+𝛽^𝑝−1)×(1+𝛾+𝛾²+⋯+𝛾^𝑞−1) where 𝛼 is a root of 𝑥^𝑚−1 𝛽 is a root of 𝑥^𝑝−1 𝛾 is a root of 𝑥^𝑞−1 but neither 𝛼, 𝛽, nor 𝛾=1 Proof as in (475) 478 If 𝑛=𝑚³, and 𝛼 be a root of 𝑥^𝑚−1=0 𝛽 be a root of 𝑥^𝑚−𝛼=0 𝛾 be a root of 𝑥^𝑚−𝛽=0 then the roots of 𝑥^𝑛−1=0 will be the terms of the product (1+𝛼+𝛼²+⋯+𝛼^𝑚−1)(1+𝛽+𝛽²+⋯+𝛽^𝑚−1)×(1+𝛾+𝛾²+⋯+𝛾^𝑚−1) 479 𝑥^𝑛±1=0 may be treated as a reciprocal equation, and depressed in degree after the manner of (468). 480 The complete solution of the equation 𝑥^𝑛−1=0 is obtained by De Moivre's Theorem. (757) The 𝑛 different roots are given by the formula 𝑥=cos2𝑟𝜋𝑛±-1sin2𝑟𝜋𝑛 in which 𝑟 must have the successive values 0, 1, 2, 3, ⋯, concluding with 𝑛2, if 𝑛 be even; and with 𝑛−12, if 𝑛 be odd. 481 Similarly the 𝑛 roots of the equation 𝑥^𝑛+1=0 are given by the formula 𝑥=cos(2𝑟+1)𝜋𝑛±-1sin(2𝑟+1)𝜋𝑛 𝑟 taking the successive values 0, 1, 2, 3, ⋯, up to 𝑛−22, if 𝑛 be even; and up to 𝑛−32, if 𝑛 be odd. 482 The number of different values of the product 𝐴^1𝑚𝐵^1𝑚 is equal to the least common multiple of 𝑚 and 𝑛, when 𝑚 and 𝑛 are integers.

Sources and References

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