Theory of Equation

Symmetrical Functions of the Roots of an Equation

Notation: Let 𝑎, 𝑏, 𝑐, ⋯ be the roots of the equation 𝑓(𝑥)=0.
Let 𝑠_𝑚 denote 𝑎^𝑚+𝑏^𝑚+⋯, the sum of the 𝑚^th powers of the roots.
Let 𝑠_𝑚,𝑝 denote 𝑎^𝑚𝑏^𝑝+𝑏^𝑚𝑎^𝑝+𝑎^𝑚𝑐^𝑝+⋯, through all the permutations of the roots, two at a time.
Similarly, let 𝑠_{𝑚,𝑝,𝑞} denote 𝑎^𝑚𝑏^𝑝𝑐^𝑞+𝑎^𝑚𝑏^𝑝𝑑^𝑞+⋯, taking all the permutations of the roots three at a time; and on. 534

Sums of the powers of the roots

𝑠_𝑚+𝑝₁𝑠_𝑚−1+𝑝₂𝑠_𝑚−2+⋯+𝑝_𝑚−1𝑠₁+𝑚𝑝_𝑚=0 where 𝑚 is less than 𝑎, the degree of 𝑓(𝑥).
Obtained by expanding by division each term in the value of 𝑓′(𝑥) given at (432), arranging the whole in powers of 𝑥, and equating coefficients in the result and in the value of 𝑓′(𝑥), found by differentiation us in (424). 535 where 𝑚 is less than 𝑎, the formula will be 𝑠_𝑚+𝑝₁𝑠_𝑚−1+𝑝₂𝑠_𝑚−2+⋯+𝑝_𝑛𝑠_𝑚−𝑛=0 Obtained by multiplying 𝑓(𝑥)=0 by 𝑥^𝑚−𝑛, substituting for 𝑥 the roots 𝑎, 𝑏, 𝑐, ⋯ in succession and adding the results.
By these formula 𝑠₁, 𝑠₂, 𝑠₃, ⋯ may be calculated successively. 536 To find the sum of the negative powers of the roots, put 𝑚 equal to 𝑛−1, 𝑛−2, 𝑛−3, ⋯, successively in (535), in order to obtain 𝑠₋₁, 𝑠₋₂, 𝑠₋₃, ⋯ 537 To calculate 𝑠_𝑟 independently,
Rule: 𝑠_𝑟=−𝑟×coefficient of 𝑥^−𝑟 in the expansion of log>𝑓(𝑥)𝑥^𝑛
Proved by taking 𝑓(𝑥)=(𝑥−𝑎)(𝑥−𝑏)(𝑥−𝑐)⋯, dividing by 𝑥^𝑛, and expanding the logarithm of the right side of the equation by (456). 538

Symmetrical Functions which are not powers of the roots

These are expressed in terms of the sums of powers of the roots as under, and thence, by (534), in terms of the roots explicitly, 𝑠_𝑚,𝑝=𝑠_𝑚𝑠_𝑝−𝑠_𝑚+𝑝 539 𝑠_{𝑚,𝑝,𝑞}=𝑠_𝑚𝑠_𝑝𝑠_𝑞−𝑠_𝑚+𝑝𝑠_𝑞−𝑠_𝑚+𝑞𝑠_𝑝−𝑠_𝑝+𝑞𝑠_𝑚+2𝑠_{𝑚+𝑝+𝑞} The last equation may be proved by multiplying 𝑠_𝑚,𝑝 by 𝑠_𝑞; and expansions of other symmetrical functions may be obtained in a similar way. 540 If 𝜙(𝑥) be a rational integral function of 𝑥, then the symmetrical function of the roots of 𝑓(𝑥), denoted by 𝜙(𝑎)+𝜙(𝑏)+𝜙(𝑐)+⋯ is equal to the coefficient of 𝑥^𝑛−1 in the remainder obtained by dividing 𝜙(𝑥)𝑓′(𝑥) by 𝑓(𝑥).
Proved by multiplying the equation (432) by 𝜙(𝑥)𝑓(𝑥), and by theorem (401). 541 To find the equation whose roots are the squares of the differences of the roots of a given equation.
Let 𝐹(𝑥) be the given equation, and 𝑆_𝑟 the sum of the 𝑟^th powers of its roots. Let 𝑓(𝑥) and 𝑠_𝑟 have the same meaning with regard to the required equation.
The coefficients of the required equation can be calculated from those of the given one as follows: The coefficients of each equation may be connected with the sums of the powers of its roots by (534); and the sums of the powers of the roots of the two equations are connected by the formula. 542 2𝑠_𝑟=𝑛𝑆_2𝑟−2𝑟𝑆₁𝑆_2𝑟−1+2𝑟(2𝑟−1)1⋅2𝑆₂𝑆_2𝑟−2−⋯+𝑛𝑆_2𝑟 Rule: 2𝑠_𝑟 is equal to the formal expansion of (𝑆−𝑆)^2𝑟 by the Binomial Theorem, with the first and last terms each multiplied by 𝑛, and the indices all changed to suffixes. As the equi-distant terms are equal we can divide by 2, and take half the series.
Demonstration: Let 𝑎, 𝑏, 𝑐, ⋯ be the roots of 𝐹(𝑥) Let 𝜙(𝑥)=(𝑥−𝑎)^2𝑟+(𝑥−𝑏)^2𝑟+⋯i. Expand each term on the right by the Bin. Theor., and add, substituting 𝑆₁, 𝑆₂, ⋯. In the result change 𝑥 into 𝑎, 𝑏, 𝑐, ⋯ successively, and add the 𝑛 equations to obtain the formula, observing that, by [i.]. 𝜙(𝑎)+𝜙(𝑏)+⋯=2𝑠_𝑟 If 𝑛 be the degree of 𝐹(𝑥), then 12𝑛(𝑛−1) is the degree of 𝑓(𝑥). 543 The last term of the equation 𝑓(𝑥)=0 is equal to 𝑛^𝑛𝐹(𝛼)𝐹(𝛽)𝐹(𝛾)⋯ where 𝛼, 𝛽, 𝛾, ⋯, are the roots of 𝐹(𝑥). Proved by shewing that 𝐹′(𝛼)𝐹′(𝛽)⋯=𝑛^𝑛𝐹(𝛼)𝐹(𝛽)⋯ 544 If 𝐹(𝑥) has negative or imaginary roots, 𝑓(𝑥) must have imaginary roots. 545 The sum of the 𝑚^th powers of the roots of the quadratic equation 𝑥²+𝑝𝑥+𝑞=0 𝑠_𝑚=𝑝^𝑚−𝑚𝑝^𝑚−2𝑞+𝑚(𝑚−3)|2𝑝^𝑚−4𝑞²−⋯+(−1)^𝑟𝑚(𝑚−𝑟−1)⋯(𝑚−2𝑟+1)|𝑟𝑝^𝑚−2𝑟𝑞^𝑟+⋯ By (537) expanding the logarithm by (156) 546 The sum of the 𝑚^th powers of the roots of 𝑥^𝑛−1=0 is 𝑛 if 𝑚 be a multiple of 𝑛, and zero if it be not. By (537); expanding the logarithm by (156) 547 If 𝜙(𝑥)=𝑎₀+𝑎₁𝑥+𝑎₂𝑥²+⋯i. then the sum of the selected terms 𝑎_𝑚𝑥^𝑚+𝑎_𝑚+𝑛𝑥^𝑚+𝑛+𝑎_𝑚+2𝑛𝑥^𝑚+2𝑛+⋯ will be 𝑠=1𝑛{𝛼^𝑛−𝑚𝜙(𝛼𝑥)+𝛽^𝑛−𝑚𝜙(𝛽𝑥)+𝛾^𝑛−𝑚𝜙(𝛾𝑥)+⋯} where 𝛼, 𝛽, 𝛾, ⋯, are the 𝑛^th roots of unity.
For proof, multiply (i.) by 𝛼^𝑛−𝑚, and change 𝑥 into 𝛼𝑥; so with 𝛽, 𝛾, ⋯, and add the resulting equations. 548 To approximate to the root of an equation by means of the sums of the powers of the roots.
By taking 𝑚 large enough, the fraction 𝑠_𝑚+1𝑠_𝑚 will approximate to the value of the numberically greatest root, unless there be a modulus of imaginary roots greater than any real root, in which case the fraction has no limiting value. 549 Similarly the fraction 𝑠_𝑚𝑠_𝑚+2−𝑠²_𝑚𝑠_𝑚−1𝑠_𝑚+1−𝑠²_𝑚 approximates, as 𝑚 increases, to the greatest product of any pair of roots, real or imaginary; excepting in the case in which the product of the pair of imaginary roots, though less than the product of the two real roots, is greater than the square of the least of them, for then the fraction has no limiting value. 550 Similarly the fraction 𝑠_𝑚𝑠_𝑚+3−𝑠_𝑚+1𝑠_𝑚+2𝑠_𝑚𝑠_𝑚+2−𝑠²_𝑚+1 approximates, as 𝑚 increases, to the sum of the two numerically greatest roots, or to the sum of the two imaginary roots with the greatest modulus.

Sources and References

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