Quartic function - Wikipedia 22/09/2019, 1003 Quartic function In algebra, a quartic function is a function of the form where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form where a ≠ 0. The derivative of a quartic function is a cubic function. Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum. The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals. https://en.wikipedia.org/wiki/Quartic_function Page 1 of 21 Quartic function - Wikipedia 22/09/2019, 1003 Contents History Applications Inflection points and golden ratio Solution Nature of the roots General formula for roots Special cases of the formula Simpler cases Reducible quartics Biquadratic equation Quasi-palindromic equation Graph of a polynomial of degree 4, with Solution methods 3 critical points and four real roots Converting to a depressed quartic (crossings of the x axis) (and thus no Ferrari's solution complex roots). If one or the other of Descartes' solution the local minima were above the x axis, Euler's solution or if the local maximum were below it, Solving by Lagrange resolvent or if there were no local maximum and Solving with algebraic geometry one minimum below the x axis, there See also would only be two real roots (and two complex roots). If all three local References extrema were above the x axis, or if Further reading there were no local maximum and one External links minimum above the x axis, there would be no real root (and four complex roots). History Lodovico Ferrari is credited with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately.[1] The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book Ars Magna.[2] https://en.wikipedia.org/wiki/Quartic_function Page 2 of 21 Quartic function - Wikipedia 22/09/2019, 1003 The Soviet historian I. Y. Depman claimed that even earlier, in 1486, Spanish mathematician Valmes was burned at the stake for claiming to have solved the quartic equation.[3] Inquisitor General Tomás de Torquemada allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding.[4] However Beckmann, who popularized this story of Depman in the West, said that it was unreliable and hinted that it may have been invented as Soviet antireligious propaganda.[5] Beckmann's version of this story has been widely copied in several books and internet sites, usually without his reservations and sometimes with fanciful embellishments. Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed.[6] The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois prior to dying in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result.[7] Applications Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus. It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics. Here are examples of other geometric problems whose solution involves solving a quartic equation. In computer-aided manufacturing, the torus is a shape that is commonly associated with the endmill cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the z-axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated.[8] A quartic equation arises also in the process of solving the crossed ladders problem, in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found. In optics, Alhazen's problem is "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer." This leads to a quartic equation.[9][10][11] Finding the distance of closest approach of two ellipses involves solving a quartic equation. The eigenvalues of a 4×4 matrix are the roots of a quartic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a fourth-order linear difference equation or differential equation is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending. https://en.wikipedia.org/wiki/Quartic_function Page 3 of 21 Quartic function - Wikipedia 22/09/2019, 1003 Intersections between spheres, cylinders, or other quadrics can be found using quartic equations. Inflection points and golden ratio Letting F and G be the distinct inflection points of a quartic, and letting H be the intersection of the inflection secant line FG and the quartic, nearer to G than to F, then G divides FH into the golden section:[12] Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area. Solution Nature of the roots Given the general quartic equation with real coefficients and a ≠ 0 the nature of its roots is mainly determined by the sign of its discriminant This may be refined by considering the signs of four other polynomials: https://en.wikipedia.org/wiki/Quartic_function Page 4 of 21 Quartic function - Wikipedia 22/09/2019, 1003 P such that is the second degree coefficient of the associated depressed quartic (see below); 8a2 R such that is the first degree coefficient of the associated depressed quartic; 8a3 which is 0 if the quartic has a triple root; and which is 0 if the quartic has two double roots. The possible cases for the nature of the roots are as follows:[13] If ∆ < 0 then the equation has two distinct real roots and two complex conjugate non-real roots. If ∆ > 0 then either the equation's four roots are all real or none is. If P < 0 and D < 0 then all four roots are real and distinct. If P > 0 or D > 0 then there are two pairs of non-real complex conjugate roots.[14] If ∆ = 0 then (and only then) the polynomial has a multiple root. Here are the different cases that can occur: If P < 0 and D < 0 and ∆0 ≠ 0, there are a real double root and two real simple roots. If D > 0 or (P > 0 and (D ≠ 0 or R ≠ 0)), there are a real double root and two complex conjugate roots. If ∆0 = 0 and D ≠ 0, there are a triple root and a simple root, all real. If D = 0, then: If P < 0, there are two real double roots. If P > 0 and R = 0, there are two complex conjugate double roots. b If , all four roots are equal to ∆0 = 0 −4a https://en.wikipedia.org/wiki/Quartic_function Page 5 of 21 Quartic function - Wikipedia 22/09/2019, 1003 There are some cases that do not seem to be covered, but they cannot occur. For example, ∆0 > 0, P = 0 and D ≤ 0 is not one of the cases. However, if ∆0 > 0 and P = 0 then D > 0 so this combination is not possible. General formula for roots The four roots x1, x2, x3, and x4 for the general quartic equation with a ≠ 0 are given in the following formula, which is deduced Solution of written out in full. This formula is too unwieldy for general use; hence other methods, or simpler formulas for special cases, are generally used.[15] from the one in the section on Ferrari's method by back changing the variables (see § Converting to a depressed quartic) and using the formulas for the quadratic and cubic equations. where p and q are the coefficients of the second and of the first degree respectively in the associated depressed quartic and where https://en.wikipedia.org/wiki/Quartic_function Page 6 of 21 Quartic function - Wikipedia 22/09/2019, 1003 (if S = 0 or Q = 0, see § Special cases of the formula, below) with and where is the aforementioned discriminant.