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Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R.Then, the quotients / belong to the field of fractions of R (and possibly are in R itself if happens to be invertible in R) and the roots are taken in an algebraically closed extension.
The other root, x 2 is determined using Vieta's formulas. For all (a, b) above a certain base case, show that 0 < x 2 < b < a and that x 2 is an integer. Thus, while maintaining the same k, we may replace (a, b) with (b, x 2) and repeat this process until we arrive at the base case.
The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle formula from trigonometry leads to a generalized formula, discovered by Leonhard Euler, that has Viète's formula as a special case. Many similar formulas involving nested ...
By Vieta's formulas, s 0 is known to be zero in the case of a depressed cubic, and − b / a for the general cubic. So, only s 1 and s 2 need to be computed. They are not symmetric functions of the roots (exchanging x 1 and x 2 exchanges also s 1 and s 2 ), but some simple symmetric functions of s 1 and s 2 are also symmetric in the ...
The characteristic polynomial of a square matrix is an example of application of Vieta's formulas. The roots of this polynomial are the eigenvalues of the matrix . When we substitute these eigenvalues into the elementary symmetric polynomials, we obtain – up to their sign – the coefficients of the characteristic polynomial, which are ...
A lesser known quadratic formula, as used in Muller's method, provides the same roots via the equation =. This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is c/a.
If this number is −q, then the choice of the square roots was a good one (again, by Vieta's formulas); otherwise, the roots of the polynomial will be −r 1, −r 2, −r 3, and −r 4, which are the numbers obtained if one of the square roots is replaced by the symmetric one (or, what amounts to the same thing, if each of the three square ...
It follows by Vieta's formulas that x and y must be roots of the quadratic equation + = ; its = = > (≠ 0, otherwise c would be the square of a), hence x and y must be + and . Thus x and y are rational if and only if d = a 2 − c {\displaystyle d={\sqrt {a^{2}-c}}~} is a rational number.