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Vieta's formulas can be proved by considering the equality + + + + = () (which is true since ,, …, are all the roots of this polynomial), expanding the products in the right-hand side, and equating the coefficients of each power of between the two members of the equation.
From the equation for H, one sees that 1 + x y ′ > 0. Since x > 0, it follows that y ′ ≥ 0. Hence the point (x, y ′) is in the first quadrant. By reflection, the point (y ′, x) is also a point in the first quadrant on H. Moreover from Vieta's formulas, yy ′ = x 2 - q, and y ′ = x 2 - q / y . Combining this equation with x ...
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 ...
Quadratic function#Upper bound on the magnitude of the roots; Real-root isolation – Methods for locating real roots of a polynomial; Root-finding of polynomials – Algorithms for finding zeros of polynomials; Square-free polynomial – Polynomial with no repeated root; Vieta's formulas – Relating coefficients and roots of a polynomial
Sturm's theorem (theory of equations) Sturm–Picone comparison theorem (differential equations) Subspace theorem (Diophantine approximation) Superrigidity theorem (algebraic groups) Supersymmetry nonrenormalization theorems ; Supporting hyperplane theorem (convex geometry) Švarc-Milnor lemma (geometric group theory) Swan's theorem (module theory)
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is The general form of a quartic equation is Graph of a polynomial function of degree 4, with its 4 roots and 3 critical points .
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
The four roots of the depressed quartic x 4 + px 2 + qx + r = 0 may also be expressed as the x coordinates of the intersections of the two quadratic equations y 2 + py + qx + r = 0 and y − x 2 = 0 i.e., using the substitution y = x 2 that two quadratics intersect in four points is an instance of Bézout's theorem.
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