Graph of a cubic function; the roots are where the curve crosses the x-axis (y = 0). It has 2 critical points.

In mathematics, a cubic function is a function of the form Polynomial of degree 3: y = x3/5+4x2/5-7x/5-2=1/5 (x+5)(x+1)(x-2) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Polynomial of degree 3: y = x3/5+4x2/5-7x/5-2=1/5 (x+5)(x+1)(x-2) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In mathematics, a critical point (or critical number) is a point on the domain of a function where: one dimension: the derivative is equal to zero or does not exist: it is points that are either stationary points or non-differentiable points. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about functions in mathematics. ...

where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function. In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... This article is about the term degree as used in mathematics. ... For other uses, see Derivative (disambiguation). ... f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where . ... Polynomial of degree 4: f(x) = (x+4)(x+1)(x-1)(x-3)/14+0. ...

If you set f(x) = 0, you get a cubic equation of the form: An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...

where

(if a = 0, then the equation becomes a quadratic equation). In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

Usually, the coefficients a, b, c, d are real numbers. However, most of the theory is also valid if they belong to a field of characteristic other than two or three. In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ...

Solving a cubic equation amounts to finding the roots of a cubic function. In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

History

Cubic equations were known to the ancient Indians and ancient Greeks since the 5th century BC, and even earlier to the ancient Egyptians, who dealt with the problem of doubling the cube, and attempted to solve it using compass and straightedge constructions.^[1] Hippocrates, Menaechmus and Archimedes are believed to have come close to solving this problem using intersecting conic sections,^[1] though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. This article is under construction. ... Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... This article or section is in need of attention from an expert on the subject. ... Doubling the cube (also known as The Delian Problem) is one of the three most famous geometric problems unsolvable by compass and straightedge construction. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... For other uses, see Hippocrates (disambiguation). ... Greek mathematician and geometer said to have been the tutor of Alexander the Great. ... For other uses, see Archimedes (disambiguation). ... In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. ...

Two-dimensional graph of a cubic, the polynomial f(x) = 2x³ − 3x² − 3x + 2

In the 11th century, the Persian poet-mathematician Omar Khayyám (1048–1131) made significant progress in the theory of cubic equations. In an early paper he wrote regarding cubic equations, he discovered that a cubic equation can have more than one solution, that it cannot be solved using earlier compass and straightedge constructions, and found a geometric solution which could be used to get a numerical answer by consulting trigonometric tables. In his later work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.^[2][3] In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... This article is about the Persian people, an ethnic group found mainly in Iran. ... Kelileh va Demneh Persian manuscript copy dated 1429, from Herat, depicts the Jackal trying to lead the Lion astray. ... In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... Events The city of Oslo is founded by Harald Hardråde of Norway. ... Events May 9 - Tintern Abbey is founded. ... For other uses, see Geometry (disambiguation). ... Tables of trigonometric functions are useful in a number of areas. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...

In the early 16th century, the Italian mathematician Scipione del Ferro (1465-1526) found a method for solving a class of cubic equations, namely those of the form x³ + mx = n. In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fiore about it. (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Scipione del Ferro (Bologna 1465–1526) was an Italian mathemtatician who first discovered a means to solve cubic equations. ... Events July 13 - Battle of Montlhéry Troops of King Louis XI of France fight inconclusively against an army of the great nobles organized as the League of the Public Weal. ... January 14 - Treaty of Madrid. ... A negative number is a number that is less than zero, such as −2. ...

In 1530, Niccolò Tartaglia (1500-1557) received two problems in cubic equations from Zuanne da Coi and announced that he could solve them. He was soon challenged by Fiore, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the money. Niccolò Fontana Tartaglia. ... 1500 was a common year starting on Monday (see link for calendar) of the Gregorian calendar. ... Events Spain is effectively bankrupt. ...

Tartaglia received questions in the form x³ + mx = n, for which he had worked out a general method. Fiore received questions in the form x³ + mx² = n, which proved to be too difficult for him to solve, and Tartaglia won the contest.

Later, Tartaglia was persuaded by Gerolamo Cardano (1501-1576) to reveal his secret for solving cubic equations. In 1539, Tartaglia did so only on the condition that Cardano would never reveal it and that if he did reveal a book about cubics, that he would give Tartaglia time to publish. Some years later, Cardano learned about Ferro's prior work and published Ferro's method in his book Ars Magna in 1545, meaning Cardano gave Tartaglia 6 years to publish his results (with credit given to Tartaglia for an independent solution). Cardano's promise with Tartaglia stated that he not publish Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as to get around the promise. Nevertheless, this led to a challenge to Cardano by Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's student Lodovico Ferrari (1522-1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income ^[4]. Gerolamo Cardano. ... 1501 was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar. ... Events May 5 - Peace of Beaulieu or Peace of Monsieur (after Monsieur, the Duc dAnjou, brother of the King, who negotiated it). ... Events May 30 - In Florida, Hernando de Soto lands at Tampa Bay with 600 soldiers with the goal to find gold. ... The Ars Magna (Latin: Great Work) is an important book on mathematics written by Gerolamo Cardano. ... Events February 27 - Battle of Ancrum Moor - Scots victory over superior English forces December 13 - Official opening of the Council of Trent (closed 1563) Battle of Kawagoe - between two branches of Uesugi families and the late Hojo clan in Japan. ... Lodovico Ferrari (February 2, 1522 - October 5, 1565) was an Italian mathematician. ... Events January 9 - Adrian Dedens becomes Pope Adrian VI. February 26 - Execution by hanging of Cuauhtémoc, Aztec ruler of Tenochtitlan under orders of conquistador Hernán Cortés. ... // Events March 1 - the city of Rio de Janeiro is founded. ...

Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these complex numbers in Ars Magna, but he did not really understand it. Rafael Bombelli studied this issue in detail and is therefore often considered as the discoverer of complex numbers. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Raphael Bombelli (1526-1572) was an Italian mathematician. ...

Roots of a cubic function

The nature of the roots

Every cubic equation with real coefficients has at least one solution x among the real numbers; this is a consequence of the intermediate value theorem. We can distinguish several possible cases using the discriminant, In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In Mathematical analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ...

The following cases need to be considered.

• If Δ < 0, then the equation has three distinct real roots.
• If Δ > 0, then the equation has one real root and a pair of complex conjugate roots.
• If Δ = 0, then (at least) two roots coincide. It may be that the equation has a double real root and another distinct single real root; alternatively, all three roots coincide yielding a triple real root. A possible way to decide between these subcases is to compute the resultant of the cubic and its second derivative: a triple root exists if and only if this resultant vanishes.

See also: multiplicity of a root of a polynomial In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... For the technique in organ building, see Resultant (organ). ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ...

Cardano's method

The solutions can be found with the following method due to Scipione del Ferro and Tartaglia, published by Gerolamo Cardano in 1545. Scipione del Ferro (Bologna 1465–1526) was an Italian mathemtatician who first discovered a means to solve cubic equations. ... Niccolo Fontana Tartaglia. ... Gerolamo Cardano. ... Events February 27 - Battle of Ancrum Moor - Scots victory over superior English forces December 13 - Official opening of the Council of Trent (closed 1563) Battle of Kawagoe - between two branches of Uesugi families and the late Hojo clan in Japan. ...

We first divide the standard equation by the leading coefficient to arrive at an equation of the form

The substitution x = t − a / 3 eliminates the quadratic term; in fact, we get the equation

This is called the depressed cubic. A simple and elegant way of solving the depressed cubic is due to Thomas Harriot (1560 – 1621): substituting into it and multiplying both sides by y³ yields, after much cancellation, . Described below is the original, somewhat long-winded method of Cardano and Tartaglia, which still dominates the textbooks today. Thomas Harriot (ca. ... Gerolamo Cardano. ... Niccolo Fontana Tartaglia. ...

Suppose that we can find numbers u and v such that

A solution to our equation is then given by

as can be checked by directly substituting this value for t in (2), as a consequence of the third order binomial identity In elementary algebra, a binomial is a polynomial with two terms: the sum of two monomials. ...

The system (3) can be solved by solving the second equation for v, which gives

Substituting this into the first equation in (3) yields

Moving all to the q side and multiplying by 27u³ yields

This can be seen as a quadratic equation for u³. If we solve this equation, we find that In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

Since t = v + u, t = x + a/3, and v = −p/3u, we find

Note that there are six possibilities in computing u with (4), since there are two solutions to the square root (), and three complex solutions to the cubic root — the principal root and the principal root multiplied by . However, the sign of the square root (plus or minus) does not affect the final resulting x, although care must be taken in two special cases to avoid divisions by zero. First, if p = 0, then one should choose the negative square root so that u does not equal zero, i.e., . Second, if p = q = 0, then we have the triple real root x = −a/3.

In summary, for the cubic equation

the solutions for x are given by

where

Although this method is simple and elegant, it fails for the case of three real roots, e.g. when:

For this case a different method (e.g. goniometrical) has to be used.

Lagrange resolvents

The symmetric group S₃ has the cyclic group of order three as a normal subgroup, which suggests making use of the discrete Fourier transform of the roots, an idea due to Lagrange. Suppose that r₀, r₁ and r₂ are the roots of equation (1), and define , so that ζ is a primitive third root of unity. We now set In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ... Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. ...

The roots may then be recovered from the three s_i by inverting the above linear transformation, giving

We already know the value s₀ = −a, so we only need to seek values for the other two. However, if we take the cubes, a cyclic permutation leaves the cubes invariant, and a transposition of two roots exchanges s₁³ and s₂³, hence the polynomial

is invariant under permutations of the roots, and so has coefficients expressible in terms of (1). Using calculations involving symmetric functions or alternatively field extensions, we can calculate (5) to be

The roots of this quadratic equation are

where D is the discriminant. Taking cube roots give us s₁ and s₂, from which we can recover the roots r_i of (1).

Factorization

If r is any root of (1), then we may factor using r to obtain

Hence if we know one root we can find the other two by solving a quadratic equation, giving

for the other two roots.

Root-finding formula

The formula for finding the roots of a cubic function is fairly complicated. Therefore, it is common to use the rational root test or a numerical solution instead. In algebra, the rational root theorem (or rational root test to find the zeros) states a constraint on solutions (or roots) to the polynomial equation an xn + an−1 xn −1 + ... + a1 x + a0 = 0 with integer coefficients. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...

If we have

let

and

Now, let

and

The solutions are

q³ + r² is the discriminant, if it is less than 0 then all three solutions are real and unequal and the roots can then be expressed by trigonometric functions (trigonometric cosine). In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ...

Solution in terms of Chebyshev radicals

If we have a cubic equation which is already in depressed form, we may write it as . Substituting we obtain or equivalently

From this we obtain solutions to our original equation in terms of the Chebyshev cube root as

If now we start from a general equation

and reduce it to the depressed form under the substitution x = t − a/3, we have and , leading to

This gives us the solutions to (1) as

The case of a cubic equation with real coefficients

Suppose the coefficients of (1) are real. If s is the quantity q/r from the section on real roots, then s = t²; hence 0 < s < 4 is equivalent to −2 < t < 2, and in this case we have a polynomial with three distinct real roots, expressed in terms of a real function of a real variable, quite unlike the situation when using cube roots. If s > 4 then either t > 2 and is the sole real root, or t < −2 and is the sole real root. If s < 0 then the reduction to Chebyshev polynomial form has given a t which is a pure imaginary number; in this case is the sole real root. We are now evaluating a real root by means of a function of a purely imaginary argument; however we can avoid this by using the function

which is a real function of a real variable with no singularities along the real axis. If a polynomial can be reduced to the form x³ + 3x − t with real t, this is a convenient way to solve for its roots.

Derivative

The derivative will yield when . Bearing its resemblance to the quadratic formula, this formula can be used to find the critical points of a cubic function. It turns out that, if , then the cubic function will have two critical points — a local maximum and a local minimum; if , then there is one critical point, and it will yield the inflection point; if , then there are no critical points. For other uses, see Derivative (disambiguation). ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... In mathematics, a critical point (or critical number) is a point on the domain of a function where: one dimension: the derivative is equal to zero or does not exist: it is points that are either stationary points or non-differentiable points. ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ... Plot of y = x3 with inflection point of (0,0). ...

Bipartite cubics

The graph of

where 0 < a < b is called a bipartite cubic. This is from the theory of elliptic curves. A catalog of elliptic curves. ...

You can graph a bipartite cubic on a graphing device by graphing the function

corresponding to the upper half of the bipartite cubic. It is defined on

See also

• Linear equation
• Quartic equation
• Quintic equation
• Newton's method
• Spline (mathematics)

Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... Graph of a polynomial of degree 5, with 4 critical points. ... In numerical analysis, Newtons method (also known as the Newton–Raphson method or the Newton–Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... One type of spline, a bézier curve In the mathematical subfield of numerical analysis, a spline is a special function defined piecewise by polynomials. ...

Notes

References

• W. S. Anglin; & J. Lambek (1995). "Mathematics in the Renaissance", in The heritage of Thales, Ch. 24. Springers.
• Lucye Guilbeau (1930). "The History of the Solution of the Cubic Equation", Mathematics News Letter 5 (4), p. 8-12.
• R.W.D. Nickalls (1993). A new approach to solving the cubic: Cardan's solution revealed, The Mathematical Gazette, 77:354–359.

External links

• Tartaglia's work (and poetry) on the solution of the Cubic Equation at Convergence
• Cubic Equation Solver.
• Quadratic, cubic and quartic equations on MacTutor archive.
• Cubic Formula on PlanetMath
• Cardano solution calculator as java applet at some local site. Only takes natural coefficients.
• Graphic explorer for cubic functions With interactive animation, slider controls for coefficients
• On Solution of Cubic Equations at Holistic Numerical Methods Institute
• Dave Auckly, Solving the quartic with a pencil American Math Monthly 114:1 (2007) 29--39
• "Cubic Equation" by Eric W. Weisstein, The Wolfram Demonstrations Project, 2007.