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Viète obtained his formula by comparing the areas of regular polygons with 2 n and 2 n + 1 sides inscribed in a circle. [ 1 ] [ 2 ] The first term in the product, 2 / 2 {\displaystyle {\sqrt {2}}/2} , is the ratio of areas of a square and an octagon , the second term is the ratio of areas of an octagon and a hexadecagon , etc.
A method similar to Vieta's formula can be found in the work of the 12th century Arabic mathematician Sharaf al-Din al-Tusi. It is plausible that the algebraic advancements made by Arabic mathematicians such as al-Khayyam, al-Tusi, and al-Kashi influenced 16th-century algebraists, with Vieta being the most prominent among them. [2] [3]
Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as (+). It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are real , the two discriminants have the same sign.
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 ...
If q = 2, then (a-b) 2 = 2 and there is no integral solution a, b. When q > 2, the equation x 2 + y 2 − qxy − q = 0 defines a hyperbola H and (a,b) represents an integral lattice point on H. If (x,x) is an integral lattice point on H with x > 0, then (since q is integral) one can see that x = 1. This proposition's statement is then true for ...
François Viète (French: [fʁɑ̃swa vjɛt]; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters in equations.
If b 2 – 3ac = 0, then there is only one critical point, which is an inflection point. If b 2 – 3ac < 0, then there are no (real) critical points. In the two latter cases, that is, if b 2 – 3ac is nonpositive, the cubic function is strictly monotonic. See the figure for an example of the case Δ 0 > 0.
Polynomial equations of degree two can be solved with the quadratic formula, which has been known since antiquity. Similarly the cubic formula for degree three, and the quartic formula for degree four, were found during the 16th century. At that time a fundamental problem was whether equations of higher degree could be solved in a similar way.