Cubic equation

In algebra, a cubic equation in one variable is an equation of the form
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients , , , and of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:
The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) complex numbers.
History.
Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians. Babylonian (20th to 16th centuries BC) cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did. The problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. In the 5th century BC, Hippocrates reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a compass and straightedge construction, a task which is now known to be impossible. Methods for solving cubic equations appear in "The Nine Chapters on the Mathematical Art", a Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui in the 3rd century. In the 3rd century AD, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations (Diophantine equations). Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections, though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. Some others like T. L. Heath, who translated all of Archimedes' works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two conics, but also discussed the conditions where the roots are 0, 1 or 2. 
In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form , 23 of them with , and two of them with .
In the 11th century, the Persian poet-mathematician, Omar Khayyam (1048–1131), made significant progress in the theory of cubic equations. In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution. 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.
In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation: . In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), wrote the "Al-Muʿādalāt" ("Treatise on Equations"), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also used the concepts of maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations.
In his book "Flos", Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation . Writing in Babylonian numerals he gave the result as 1,22,7,42,33,4,40 (equivalent to 1 + 22/60 + 7/602 + 42/603 + 33/604 + 4/605 + 40/606), which has a relative error of about 10−9.
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 . In fact, all cubic equations can be reduced to this form if one allows and to be negative, but negative numbers were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fior about it.
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 Fior, 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. Tartaglia received questions in the form , for which he had worked out a general method. Fior received questions in the form , 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 write a book about cubics, he would give Tartaglia time to publish. Some years later, Cardano learned about del Ferro's prior work and published del Ferro's method in his book "Ars Magna" in 1545, meaning Cardano gave Tartaglia six years to publish his results (with credit given to Tartaglia for an independent solution). Cardano's promise to Tartaglia said that he would 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 from 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 his income.
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.
François Viète (1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and René Descartes (1596–1650) extended the work of Viète.
Factorization.
If the coefficients of a cubic equation are rational numbers, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a common multiple of their denominators. Such an equation 
with integer coefficients, is said to be reducible if the polynomial of the left-hand side is the product of polynomials of lower degrees. By Gauss's lemma, if the equation is reducible, one can suppose that the factors have integer coefficients.
Finding the roots of a reducible cubic equation is easier than solving the general case. In fact, if the equation is reducible, one of the factors must have the degree one, and have thus the form
with and being coprime integers. The rational root test allows finding and by examining a finite number of cases (because must be a divisor of , and must be a divisor of ).
Thus, one root is formula_4 and the other roots are the roots of the other factor, which can be found by polynomial long division. This other factor is 
Then, the other roots are the roots of this quadratic polynomial and can be found by using the quadratic formula.
Depressed cubic.
Cubics of the form 
are said to be depressed. They are much simpler than general cubics, but are fundamental, because the study of any cubic may be reduced by a simple change of variable to that of a depressed cubic.
Let 
be a cubic equation. The change of variable
gives a cubic (in ) that has no term in . 
After dividing by one gets the "depressed cubic equation" 
with 
The roots formula_11 of the original equation are related to the roots formula_12 of the depressed equation by the relations 
for formula_14.
Discriminant and nature of the roots.
The nature (real or not, distinct or not) of the roots of a cubic can be determined without computing them explicitly, by using the discriminant.
Discriminant.
The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is square-free.
If are the three roots (not necessarily distinct nor real) of the cubic formula_15 then the discriminant is
The discriminant of the depressed cubic formula_6 is 
The discriminant of the general cubic formula_19 is 
It is the product of formula_21 and the discriminant of the corresponding depressed cubic. 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. In summary, the same information can be deduced from either one of these two discriminants.
To prove the preceding formulas, one can use Vieta's formulas to express everything as polynomials in , and . The proof then results in the verification of the equality of two polynomials.
Nature of the roots.
If the coefficients of a polynomial are real numbers, and its discriminant formula_23 is not zero, there are two cases:
This can be proved as follows. First, if is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.
As stated above, if are the three roots of the cubic formula_19, then the discriminant is 
If the three roots are real and distinct, the discriminant is a product of positive reals, that is formula_28
If only one root, say , is real, then and are complex conjugates, which implies that is a purely imaginary number, and thus that is real and negative. On the other hand, and are complex conjugates, and their product is real and positive. Thus the discriminant is the product of a single negative number and several positive ones. That is formula_29
Multiple root.
If the discriminant of a cubic is zero, the cubic has a multiple root. If furthermore its coefficients are real, then all of its roots are real.
The discriminant of the depressed cubic formula_30 is zero if formula_31 If is also zero, then , and 0 is a triple root of the cubic. If formula_32 and , then the cubic has a simple root
and a double root
In other words,
This result can be proved by expanding the latter product or retrieved by solving the rather simple system of equations resulting from Vieta's formulas.
By using the reduction of a depressed cubic, these results can be extended to the general cubic. This gives: If the discriminant of the cubic formula_36 is zero, then
Characteristic 2 and 3.
The above results are valid when the coefficients belong to a field of characteristic other than 2 or 3, but must be modified for characteristic 2 or 3, because of the involved divisions by 2 and 3.
The reduction to a depressed cubic works for characteristic 2, but not for characteristic 3. However, in both cases, it is simpler to establish and state the results for the general cubic. The main tool for that is the fact that a multiple root is a common root of the polynomial and its formal derivative. In these characteristics, if the derivative is not a constant, it has a single root, being linear in characteristic 3, or the square of a linear polynomial in characteristic 2. This allows computing the multiple root, and the third root can be deduced from the sum of the roots, which is provided by Vieta's formulas.
A difference with other characteristics is that, in characteristic 2, the formula for a double root involves a square root, and, in characteristic 3, the formula for a triple root involves a cube root.
Cardano's formula.
Gerolamo Cardano is credited with publishing the first formula for solving cubic equations, attributing it to Scipione del Ferro. The formula applies to depressed cubics, but, as shown in , it allows solving all cubic equations.
Cardano's result is that, if 
is a cubic equation such that and are real numbers such that formula_45 then the equation has the real root
See , below, for several methods for getting this result.
As shown in , the two other roots are non-real complex conjugate numbers, in this case. It was later shown (Cardano did not know complex numbers) that the two other roots are obtained by multiplying either of the cube roots by the primitive cube root of unity formula_47 and the other cube root by formula_48
If formula_49 there are three real roots, but Galois theory allows proving that, if there is no rational root, the roots cannot be expressed by an algebraic expression involving only real numbers. Therefore, the equation cannot be solved in this case with the knowledge of Cardano's time. This case has thus been called "casus irreducibilis", meaning "irreducible case" in Latin.
In "casus irreducibilis", Cardano's formula can still be used, but some care is needed in the use of cube roots. A first method is to define the symbols formula_50 and formula_51 as representing the principal values of the root function (that is the root that has the largest real part). With this convention Cardano's formula for the three roots remains valid, but is not purely algebraic, as the definition of a principal part is not purely algebraic, since it involves inequalities for comparing real parts. Also, the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers. Moreover, if the coefficients belong to another field, the principal cube root is not defined in general.
The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be . It results that a root of the equation is 
In this formula, the symbols formula_50 and formula_51 denote any square root and any cube root. The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is formula_55
This formula for the roots is always correct except when , with the proviso that if , the square root is chosen so that . However, the formula is useless in these cases as the roots can be expressed without any cube root. Similarly, the formula is also useless in the other cases where no cube root is needed, that is when formula_56 and when the cubic polynomial is not irreducible.
This formula is also correct when and belong to any field of characteristic other than 2 or 3.
General cubic formula.
A "cubic formula" for the roots of the general cubic equation (with )
can be deduced from every variant of Cardano's formula by reduction to a depressed cubic. The variant that is presented here is valid not only for real coefficients, but also for coefficients belonging to any field of characteristic different of 2 and 3.
The formula being rather complicated, it is worth splitting it in smaller formulas.
Let
and 
where the symbols formula_50 and formula_51 are interpreted as "any" square root and "any" cube root, respectively. The sign " before the square root is either " or ""; the choice is almost arbitrary, and changing it amounts to choosing a different square root. However, if a choice yields , then the other sign must be selected instead.
Then, one of the roots is
The other two roots can be obtained by changing the choice of the cube root in the definition of , or, equivalently by multiplying by a primitive cube root of unity, that is . In other words, the three roots are 
where .
As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots.
It's worth noting that both Δ₀ and Δ₁ can be found by taking resultants between the cubic at the top and its derivatives. Δ₁ is -1/(8a) times the resultant between the cubic and its second derivative. Δ₀ is -1/(12a) times the resultant between the first and second derivatives of the cubic polynomial.
Trigonometric and hyperbolic solutions.
Trigonometric solution for three real roots.
When a cubic equation with real coefficients has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. Galois theory allows proving that when the three roots are real, and none is rational ("casus irreducibilis"), one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using trigonometric functions, specifically in terms of cosines and arccosines. More precisely, the roots of the depressed cubic
are 
This formula is due to François Viète. It is purely real when the equation has three real roots (that is formula_66). Otherwise, it is still correct but involves complex cosines and arccosines when there is only one real root, and it is nonsensical (division by zero) when .
This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back substitution described in . It can be proved as follows:
Starting from the equation , let us set The idea is to choose to make the equation coincide with the identity
For this, choose formula_68 and divide the equation by formula_69 This gives 
Combining with the above identity, one gets 
and the roots are thus 
Hyperbolic solution for one real root.
When there is only one real root (and ), this root can be similarly represented using hyperbolic functions, as 
If and the inequalities on the right are not satisfied (the case of three real roots), the formulas remain valid but involve complex quantities.
When , the above values of are sometimes called the Chebyshev cube root. More precisely, the values involving cosines and hyperbolic cosines define, when , the same analytic function denoted , which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted , when .
Geometric solutions.
Omar Khayyám's solution.
For solving the cubic equation where , Omar Khayyám constructed the parabola , the circle that has as a diameter the line segment on the positive -axis, and a vertical line through the point where the circle and the parabola intersect above the -axis. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the -axis (see the figure).
A simple modern proof is as follows. Multiplying the equation by and regrouping the terms gives 
The left-hand side is the value of on the parabola. The equation of the circle being , the right hand side is the value of on the circle.
Solution with angle trisector.
A cubic equation with real coefficients can be solved geometrically using compass, straightedge, and an angle trisector if and only if it has three real roots.
A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root. This implies that the old problems of angle trisection and doubling the cube, set by ancient Greek mathematicians, cannot be solved by compass-and-straightedge construction.
Geometric interpretation of the roots.
Three real roots.
Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle. When the cubic is written in depressed form (), , as shown above, the solution can be expressed as
Here formula_76 is an angle in the unit circle; taking of that angle corresponds to taking a cube root of a complex number; adding for finds the other cube roots; and multiplying the cosines of these resulting angles by formula_77 corrects for scale.
For the non-depressed case () (shown in the accompanying graph), the depressed case as indicated previously is obtained by defining such that so . Graphically this corresponds to simply shifting the graph horizontally when changing between the variables and , without changing the angle relationships. This shift moves the point of inflection and the centre of the circle onto the -axis. Consequently, the roots of the equation in sum to zero.
One real root.
In the Cartesian plane.
When the graph of a cubic function is plotted in the Cartesian plane, if there is only one real root, it is the abscissa (-coordinate) of the horizontal intercept of the curve (point R on the figure). Further, if the complex conjugate roots are written as , then the real part is the abscissa of the tangency point H of the tangent line to cubic that passes through -intercept R of the cubic (that is the signed length RM, negative on the figure). The imaginary parts are the square roots of the tangent of the angle between this tangent line and the horizontal axis.
In the complex plane.
With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots.
The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. (The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.) Marden's theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse of the triangle—the unique ellipse that is tangent to the triangle at the midpoints of its sides. If the angle at the vertex on the real axis is less than then the major axis of the ellipse lies on the real axis, as do its foci and hence the roots of the derivative. If that angle is greater than , the major axis is vertical and its foci, the roots of the derivative, are complex conjugates. And if that angle is , the triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real roots.
Galois group.
Given a cubic irreducible polynomial over a field of characteristic different from 2 and 3, the Galois group over is the group of the field automorphisms that fix of the smallest extension of (splitting field). As these automorphisms must permute the roots of the polynomials, this group is either the group of all six permutations of the three roots, or the group of the three circular permutations.
The discriminant of the cubic is the square of 
where is the leading coefficient of the cubic, and , and are the three roots of the cubic. As formula_79 changes of sign if two roots are exchanged, formula_79 is fixed by the Galois group only if the Galois group is 
. In other words, the Galois group is if and only if the discriminant is the square of an element of .
As most integers are not squares, when working over the field of the rational numbers, the Galois group of most irreducible cubic polynomials is the group with six elements. An example of a Galois group with three elements is given by , whose discriminant is .
Derivation of the roots.
This section regroups several methods for deriving Cardano's formula.
Cardano's method.
This method is due to Scipione del Ferro and Tartaglia, but is named after Gerolamo Cardano who first published it in his book "Ars Magna" (1545).
This method applies to a depressed cubic . The idea is to introduce two variables and such that and to substitute this in the depressed cubic, giving
At this point Cardano imposed the condition . This removes the third term in previous equality, leading to the system of equations
Knowing the sum and the product of and , one deduces that they are the two solutions of the quadratic equation 
so
The discriminant of this equation is formula_85, and assuming it is positive, real solutions to this equations are (after folding division by 4 under the square root):
So (without loss of generality in choosing u or v):
As , the sum of the cube roots of these solutions is a root of the equation. That is 
is a root of the equation; this is Cardano's formula.
This works well when formula_90 but, if formula_49 the square root appearing in the formula is not real. As a complex number has three cube roots, using Cardano's formula without care would provide nine roots, while a cubic equation cannot have more than three roots. This was clarified first by Rafael Bombelli in his book "L'Algebra" (1572). The solution is to use the fact that , that is . This means that only one cube root needs to be computed, and leads to the second formula given in .
The other roots of the equation can be obtained by changing of cube root, or, equivalently, by multiplying the cube root by each of the two primitive cube roots of unity, which are formula_92
Vieta's substitution.
Vieta's substitution is a method introduced by François Viète (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of , and avoids the problem of computing two different cube roots.
Starting from the depressed cubic , Vieta's substitution is .
The substitution transforms the depressed cubic into
Multiplying by , one gets a quadratic equation in :
Let 
be any nonzero root of this quadratic equation. If , and are the three cube roots of , then the roots of the original depressed cubic are , , and . The other root of the quadratic equation is formula_96 This implies that changing the sign of the square root exchanges and for , and therefore does not change the roots. This method only fails when both roots of the quadratic equation are zero, that is when , in which case the only root of the depressed cubic is .
Lagrange's method.
In his paper "Réflexions sur la résolution algébrique des équations" ("Thoughts on the algebraic solving of equations"), Joseph Louis Lagrange introduced a new method to solve equations of low degree in a uniform way, with the hope that he could generalize it for higher degrees. This method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it requires solving a resolvent polynomial of degree at least six. 
Except that nobody succeeded before to solve the problem, this was the first indication of the non-existence of an algebraic formula for degrees 5 and higher. This has been proved later, and named Abel–Ruffini theorem. Nevertheless, the modern methods for solving solvable quintic equations are mainly based on Lagrange's method.
In the case of cubic equations, Lagrange's method gives the same solution as Cardano's. Lagrange's method can be applied directly to the general cubic equation , but the computation is simpler with the depressed cubic equation, .
Lagrange's main idea was to work with the discrete Fourier transform of the roots instead of with the roots themselves. More precisely, let be a primitive third root of unity, that is a number such that and (when working in the space of complex numbers, one has formula_97 but this complex interpretation is not used here). Denoting , and the three roots of the cubic equation to be solved, let
be the discrete Fourier transform of the roots. If , and are known, the roots may be recovered from them with the inverse Fourier transform consisting of inverting this linear transformation; that is,
By Vieta's formulas, is known to be zero in the case of a depressed cubic, and for the general cubic. So, only and need to be computed. They are not symmetric functions of the roots (exchanging and exchanges also and ), but some simple symmetric functions of and are also symmetric in the roots of the cubic equation to be solved. Thus these symmetric functions can be expressed in terms of the (known) coefficients of the original cubic, and this allows eventually expressing the as roots of a polynomial with known coefficients.
In the case of a cubic equation, , and are such symmetric polynomials (see below). It follows that and are the two roots of the quadratic equation . Thus the resolution of the equation may be finished exactly as with Cardano's method, with and in place of and .
In the case of the depressed cubic, one has and , while in Cardano's method we have set and . Thus we have, up to the exchange of and , and . In other words, in this case, Cardano's method and Lagrange's method compute exactly the same things, up to a factor of three in the auxiliary variables, the main difference being that Lagrange's method explains why these auxiliary variables appear in the problem.
Computation of and.
A straightforward computation using the relations and gives 
This shows that and are symmetric functions of the roots. Using Newton's identities, it is straightforward to express them in terms of the elementary symmetric functions of the roots, giving
with , and in the case of a depressed cubic, and , and , in the general case.
Applications.
Cubic equations arise in various other contexts.