This article is about cubic equations in one variable. For cubic equations in
two variables, see
elliptic curve.
Graph of a cubic function with 3
real roots (where the curve crosses the horizontal axis—where
y = 0). It has 2
critical points. Here the function is
Failed to parse (Missing texvc executable; please see math/README to configure.): y=(x^3+3x^2-6x-8)/4 .
In mathematics, a cubic function is a function of the form
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=ax^3+bx^2+cx+d,\,
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.
Setting ƒ(x) = 0 produces a cubic equation of the form:
- Failed to parse (Missing texvc executable; please see math/README to configure.): ax^3+bx^2+cx+d=0.\,
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 2 or 3. To solve a cubic equation is to find the roots (zeros) of a cubic function. There are various ways to solve a cubic equation. The roots of a cubic, like those of a quadratic or quartic (fourth degree) function but no higher degree function, can always be found algebraically (as a formula involving simple functions like the square root and cube root functions). The roots can also be found trigonometrically. Alternatively, one can find a numerical approximation of the roots in the field of the real or complex numbers. This may be obtained by any root-finding algorithm, like Newton's method.
Solving cubic equations is a necessary part of solving the general quartic equation, since solving the latter requires solving its resolvent cubic equation.
Cubic equations were known to ancient Greek mathematician Diophantus;[1] even earlier to ancient Babylonians who were able to solve certain cubic equations;[2] and also to the ancient Egyptians. Doubling the cube is the simplest and oldest studied cubic equation, and one which the ancient Egyptians considered to be impossible.[3] 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,[4] a task which is now known to be impossible. Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections,[4] 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 Archimedes' works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two cones, but also discussed the conditions where the roots are 0, 1 or 2.[5]
In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved 25 cubic equations of the form Failed to parse (Missing texvc executable; please see math/README to configure.): x^3+px^2+qx=N , 23 of them with Failed to parse (Missing texvc executable; please see math/README to configure.): p,q \ne 0 , and two of them with Failed to parse (Missing texvc executable; please see math/README to configure.): q = 0 .[6]
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 and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution.[7][8] 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.[9][10]
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:[11]
Failed to parse (Missing texvc executable; please see math/README to configure.): x^3+12x=6x^2+35
In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), wrote the Al-Mu'adalat (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 developed the concepts of a derivative function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions.[12] He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations.[13]
Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to find the positive solution to the cubic equation x3+2x2+10x = 20, using the Babylonian numerals. He gave the result as 1,22,7,42,33,4,40 which is equivalent to: 1+22/60+7/602+42/603+33/604+4/605+40/606.[14]
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 to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fiore about it.
Niccolò Fontana Tartaglia
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. 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.[15]
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.[16]
Through the quadratic formula the roots of the derivative f ′(x) = 3ax2 + 2bx + c are given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): x=\frac{-b \pm \sqrt {b^2-3ac}}{3a}
and provide the critical points where the slope of the cubic function is zero. If b2 − 3ac > 0, then the cubic function has a local maximum and a local minimum. If b2 − 3ac = 0, then the cubic's inflection point is the only critical point. If b2 − 3ac < 0, then there are no critical points. In the cases where b2 − 3ac ≤ 0, the cubic function is strictly monotonic.
The general cubic equation has the form
- Failed to parse (Missing texvc executable; please see math/README to configure.): ax^3+bx^2+cx+d=0 \qquad(1)
with Failed to parse (Missing texvc executable; please see math/README to configure.): a\neq 0\,.
This section describes how the roots of such an equation may be computed. The coefficients a, b, c, d are generally assumed to be real numbers, but most of the results apply when they belong to any field of characteristic not 2 or 3.
Every cubic equation (1) 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,
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta = 18abcd -4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. \,
The following cases need to be considered: [17]
- If Δ > 0, then the equation has three distinct real roots.
- If Δ = 0, then the equation has a multiple root and all its roots are real.
- If Δ < 0, then the equation has one real root and two nonreal complex conjugate roots.
For the general cubic equation (1) with real coefficients, the general formula for the roots, in terms of the coefficients, is as follows. Note that the expression under the square root sign in what follows is Failed to parse (Missing texvc executable; please see math/README to configure.): \ (2 b^3-9 a b c+27 a^2 d)^2-4 (b^2-3 a c)^3=-27 a^2 \Delta , where Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta
is the above-mentioned discriminant.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} x_1 = &-\frac{b}{3 a}\\ &-\frac{1}{3 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d+\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\ &-\frac{1}{3 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d-\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\ x_2 = &-\frac{b}{3 a}\\ &+\frac{1+i \sqrt{3}}{6 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d+\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\ &+\frac{1-i \sqrt{3}}{6 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d-\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\ x_3 = &-\frac{b}{3 a}\\ &+\frac{1-i \sqrt{3}}{6 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d+\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\ &+\frac{1+i \sqrt{3}}{6 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d-\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]} \end{align}
However, this formula is applicable without further explanation only when the operand of the square root is non-negative and a,b,c,d are real coefficients. When this operand is real and non-negative, the square root refers to the principal (positive) square root and the cube roots in the formula are to be interpreted as the real ones. Otherwise, there is no real square root and one can arbitrarily choose one of the imaginary square roots (the same one in both parts of the solution for each xi). For extracting the complex cube roots of the resulting complex expression, we have also to choose among three cube roots in each part of each solution, giving nine possible combinations of one of three cube roots for the first part of the expression and one of three for the second. The correct combination is such that the two cube roots chosen for the two terms in a given solution expression are complex conjugates of each other (whereby the two imaginary terms in each solution cancel out).
Another way of writing the solution may be obtained by noting that the proof of above formula shows that the product of the two cube roots is rational. This gives the following formula in which Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{ }
or Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt[3]{ }
stands for any choice of the square or cube root, if Failed to parse (Missing texvc executable; please see math/README to configure.): b^2-3ac \text{ } \neq \text{ } 0.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} Q = &\sqrt{(2 b^3-9 a b c+27 a^2 d)^2-4 (b^2-3 a c)^3}\\ C = &\sqrt[3]{\tfrac12 (Q + 2 b^3-9 a b c+27 a^2 d)}\\ x_1 = &-\frac{b}{3 a}-\frac{C}{3 a}-\frac{b^2-3 a c}{3 a C}\\ x_2 = &-\frac{b}{3 a}+\frac{C(1+i \sqrt{3})}{6 a} +\frac{(1-i \sqrt{3}) (b^2-3 a c)}{6 a C}\\ x_3 = &-\frac{b}{3 a}+\frac{C(1-i \sqrt{3})}{6 a} +\frac{(1+i \sqrt{3}) (b^2-3 a c)}{6 a C} \end{align}
If Failed to parse (Missing texvc executable; please see math/README to configure.): Q \neq 0
and Failed to parse (Missing texvc executable; please see math/README to configure.): \ b^2-3ac = 0
, the sign of Failed to parse (Missing texvc executable; please see math/README to configure.): \ Q
has to be chosen to have Failed to parse (Missing texvc executable; please see math/README to configure.): C \mbox{ } \neq \mbox{ } 0
.
If Failed to parse (Missing texvc executable; please see math/README to configure.): \ Q = 0
and Failed to parse (Missing texvc executable; please see math/README to configure.): \ b^2-3ac = 0
, the three roots are equal:
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_1=x_2=x_3=-\frac{b}{3a}.
If Failed to parse (Missing texvc executable; please see math/README to configure.): Q = 0
and Failed to parse (Missing texvc executable; please see math/README to configure.): b^2-3ac \mbox{ } \neq \mbox{ } 0
, the above expression for the roots is correct but misleading, hiding the fact that no radical is needed to represent the roots. In fact, in this case, there is a double root,
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_1=x_2=\frac{bc-9ad}{2(3ac-b^2)},
and a simple root
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_3=\frac{9a^2d-4abc+b^3}{a(3ac-b^2)}.
The next sections describe how these formulas may be obtained.
Dividing Equation (1) by Failed to parse (Missing texvc executable; please see math/README to configure.): a
and substituting Failed to parse (Missing texvc executable; please see math/README to configure.): x
by Failed to parse (Missing texvc executable; please see math/README to configure.): t-\frac{b}{3a}
(the Tschirnhaus transformation) we get the equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): t^3+pt+q=0 \qquad(2)
where
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} p=&\frac{3ac-b^2}{3a^2}\\ q=&\frac{2b^3-9abc+27a^2d}{27a^3}. \end{align}
Any formula for the roots of Equation (2) may be transformed into a formula for the roots of Equation (1) by substituting the above values for Failed to parse (Missing texvc executable; please see math/README to configure.): p
and Failed to parse (Missing texvc executable; please see math/README to configure.): q
and using the relation Failed to parse (Missing texvc executable; please see math/README to configure.): x=t-\frac{b}{3a}
.
Therefore, only Equation (2) is considered in the following.
The solutions can be found with the following method due to Scipione del Ferro and Tartaglia, published by Gerolamo Cardano in 1545.[18]
We first apply preceding reduction, giving the so-called depressed cubic
- Failed to parse (Missing texvc executable; please see math/README to configure.): t^3 + pt + q = 0\,. \qquad (2)
We introduce two variables u and v linked by the condition
- Failed to parse (Missing texvc executable; please see math/README to configure.): u+v=t\,
and substitute this in the depressed cubic (2), giving
- Failed to parse (Missing texvc executable; please see math/README to configure.): u^3+v^3+(3uv+p)(u+v)+q=0 \qquad (3)\,
.
At this point Cardano imposed a second condition for the variables u and v:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 3uv+p=0\,
. As the first parenthesis vanishes in (3), we get Failed to parse (Missing texvc executable; please see math/README to configure.): u^3+v^3=-q
and Failed to parse (Missing texvc executable; please see math/README to configure.): u^3v^3=-p^3/27
. Thus Failed to parse (Missing texvc executable; please see math/README to configure.): u^3
and Failed to parse (Missing texvc executable; please see math/README to configure.): v^3
are the two roots of the equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): z^2 + qz - {p^3\over 27} = 0\,.
At this point, Cardano, who did not know complex numbers, supposed that the roots of this equation were real, that is that Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{q^2}{4}+\frac{p^3}{27} >0\,.
Solving this equation and using the fact that Failed to parse (Missing texvc executable; please see math/README to configure.): u
and Failed to parse (Missing texvc executable; please see math/README to configure.): v
may be exchanged, we find
- Failed to parse (Missing texvc executable; please see math/README to configure.): u^{3}=-{q\over 2} + \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}
and Failed to parse (Missing texvc executable; please see math/README to configure.): v^{3}=-{q\over 2} - \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}
. As these expressions are real, their cube roots are well defined and, like Cardano, we get
- Failed to parse (Missing texvc executable; please see math/README to configure.): t_1=u+v=\sqrt[3]{-{q\over 2}+ \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}} +\sqrt[3]{-{q\over 2}- \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}
The two complex roots are obtained by considering the complex cubic roots; the fact Failed to parse (Missing texvc executable; please see math/README to configure.): uv
is real implies that they are obtained by multiplying one of the above cubic roots by Failed to parse (Missing texvc executable; please see math/README to configure.): \,\tfrac{-1}{2} + i\tfrac{\sqrt{3}}{2}\,
and the other by Failed to parse (Missing texvc executable; please see math/README to configure.): \,\tfrac{-1}{2} - i\tfrac{\sqrt{3}}{2}\,
.
If Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{q^2}{4}+\frac{p^3}{27}\,
is not necessarily positive, we have to choose a cube root of Failed to parse (Missing texvc executable; please see math/README to configure.): u^3
. As there is no direct way to choose the corresponding cube root of Failed to parse (Missing texvc executable; please see math/README to configure.): v^3 , one has to use the relation Failed to parse (Missing texvc executable; please see math/README to configure.): v=-\frac{p}{3u} , which gives
- Failed to parse (Missing texvc executable; please see math/README to configure.): u=\sqrt[3]{-{q\over 2}- \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}} \qquad (4)
and
- Failed to parse (Missing texvc executable; please see math/README to configure.): t=u-\frac{p}{3u}\,.
Note that the sign of the square root does not affect the resulting Failed to parse (Missing texvc executable; please see math/README to configure.): t , because changing it amounts to exchanging Failed to parse (Missing texvc executable; please see math/README to configure.): u
and Failed to parse (Missing texvc executable; please see math/README to configure.): v
. We have chosen the minus sign to have Failed to parse (Missing texvc executable; please see math/README to configure.): u\ne 0
when Failed to parse (Missing texvc executable; please see math/README to configure.): p = 0
and Failed to parse (Missing texvc executable; please see math/README to configure.): q\ne 0
, in order to avoid a division by zero. With this choice, the above expression for Failed to parse (Missing texvc executable; please see math/README to configure.): t
always works, except when Failed to parse (Missing texvc executable; please see math/README to configure.): p = q=0
, where the second term becomes 0/0. In this case there is a triple root Failed to parse (Missing texvc executable; please see math/README to configure.): t=0 .
Note also that in several cases the solutions are expressed with fewer square or cube roots
- If Failed to parse (Missing texvc executable; please see math/README to configure.): p=q=0
then we have the triple real root
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): t=0.\,
- If Failed to parse (Missing texvc executable; please see math/README to configure.): p=0
and Failed to parse (Missing texvc executable; please see math/README to configure.): q\ne 0
then
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): u=-\sqrt[3]{q} \text{ and } v = 0
- and the three roots are the three cube roots of Failed to parse (Missing texvc executable; please see math/README to configure.): -q
.
- If Failed to parse (Missing texvc executable; please see math/README to configure.): p\ne 0
and Failed to parse (Missing texvc executable; please see math/README to configure.): q=0
then
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): u=\sqrt{{p\over 3}} \qquad \text{and} \qquad v=-\sqrt{{p\over 3}},
- in which case the three roots are
- Failed to parse (Missing texvc executable; please see math/README to configure.): t=u+v=0 , \qquad t=\omega_1u-{p\over 3\omega_1u}=\sqrt{-p} , \qquad t={u\over \omega_1}-{\omega_1p\over 3u}=-\sqrt{-p} ,
- where
- Failed to parse (Missing texvc executable; please see math/README to configure.): \omega_1=e^{i\frac{2\pi}{3}}=-\tfrac{1}{2} + \tfrac{\sqrt{3}}{2}i.
- Finally if Failed to parse (Missing texvc executable; please see math/README to configure.): 4p^3+27q^2=0 \text{ and } p\ne 0
, there is a double root and a simple root which may be expressed rationally in term of Failed to parse (Missing texvc executable; please see math/README to configure.): p \text{ and } q , but this expression may not be immediately deduced from the general expression of the roots:
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): t_1=t_2= -\frac{3q}{2p}\quad \text{and} \quad t_3=\frac{3q}{p}\,.
To pass from these roots of Failed to parse (Missing texvc executable; please see math/README to configure.): t
in Equation (2) to the general formulas for roots of Failed to parse (Missing texvc executable; please see math/README to configure.): x
in Equation (1), subtract Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{b}{3a}
and replace Failed to parse (Missing texvc executable; please see math/README to configure.): p
and Failed to parse (Missing texvc executable; please see math/README to configure.): q
by their expressions in terms of Failed to parse (Missing texvc executable; please see math/README to configure.): a,b,c,d
.
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.
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.[19][20][21] This is explained by the Abel–Ruffini theorem, which proves that such polynomials cannot be solved by radicals. Nevertheless the modern methods for solving solvable quintic equations are mainly based on Lagrange's method.[21]
In the case of cubic equations, Lagrange's method gives the same solution as Cardano's, where the latter may seem almost magical to the modern reader. But Cardano explains in his book Ars Magna how he arrived at the idea of considering the unknown of the cubic equation as a sum of two other quantities, by drawing attention to a geometrical problem that involves two cubes of different size. Lagrange's method may also be applied directly to the general cubic equation (1) without using the reduction to the trinomial equation (2). Nevertheless the computation is much easier with this reduced equation.
Suppose that x0, x1 and x2 are the roots of equation (1) or (2), and define Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta = -\tfrac{1}{2} + \tfrac{\sqrt{3}}{2}i , so that ζ is a primitive third root of unity which satisfies the relation Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta^2+\zeta+1=0 . We now set
- Failed to parse (Missing texvc executable; please see math/README to configure.): s_0 = x_0 + x_1 + x_2,\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): s_1 = x_0 + \zeta x_1 + \zeta^2 x_2,\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): s_2 = x_0 + \zeta^2 x_1 + \zeta x_2.\,
This is the discrete Fourier transform of the roots: observe that while the coefficients of the polynomial are symmetric in the roots, in this formula an order has been chosen on the roots, so these are not symmetric in the roots. The roots may then be recovered from the three si by inverting the above linear transformation via the inverse discrete Fourier transform, giving
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_0 = \tfrac13(s_0 + s_1 + s_2),\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_1 = \tfrac13(s_0 + \zeta^2 s_1 + \zeta s_2),\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_2 = \tfrac13(s_0 + \zeta s_1 + \zeta^2 s_2).\,
The polynomial Failed to parse (Missing texvc executable; please see math/README to configure.): s_0
is an elementary symmetric polynomial and is thus equal to Failed to parse (Missing texvc executable; please see math/README to configure.): -b/a
in case of Equation (1) and to zero in case of Equation (2), so we only need to seek values for the other two.
The polynomials Failed to parse (Missing texvc executable; please see math/README to configure.): s_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_2
are not symmetric functions of the roots: Failed to parse (Missing texvc executable; please see math/README to configure.): s_0
is invariant, while the two non-trivial cyclic permutations of the roots send Failed to parse (Missing texvc executable; please see math/README to configure.): s_1
to Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta s_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_2
to Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta^2 s_2
, or Failed to parse (Missing texvc executable; please see math/README to configure.): s_1
to Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta^2 s_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_2
to Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta s_2
(depending on which permutation), while transposing Failed to parse (Missing texvc executable; please see math/README to configure.): x_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): x_2
switches Failed to parse (Missing texvc executable; please see math/README to configure.): s_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_2
- other transpositions switch these roots and multiply them by a power of Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta.
Thus, Failed to parse (Missing texvc executable; please see math/README to configure.): s_1^3 , Failed to parse (Missing texvc executable; please see math/README to configure.): s_2^3
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_1 s_2
are left invariant by the cyclic permutations of the roots, which multiply them by Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta^3=1
. Also Failed to parse (Missing texvc executable; please see math/README to configure.): s_1 s_2
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_1^3+s_2^3
are left invariant by the transposition of Failed to parse (Missing texvc executable; please see math/README to configure.): x_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): x_2
which exchanges Failed to parse (Missing texvc executable; please see math/README to configure.): s_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_2
. As the permutation group Failed to parse (Missing texvc executable; please see math/README to configure.): S_3
of the roots is generated by these permutations, it follows that Failed to parse (Missing texvc executable; please see math/README to configure.): s_1^3+s_2^3
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_1 s_2
are symmetric functions of the roots and may thus be written as polynomials in the elementary symmetric polynomials and thus as rational functions of the coefficients of the equation. Let Failed to parse (Missing texvc executable; please see math/README to configure.): s_1^3+s_2^3=A
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_1 s_2=B
in these expressions, which will be explicitly computed below.
We have that Failed to parse (Missing texvc executable; please see math/README to configure.): s_1^3
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_2^3
are the two roots of the quadratic equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): z^2-Az+B^3 = 0 \,.
Thus the resolution of the equation may be finished exactly as described for Cardano's method, with Failed to parse (Missing texvc executable; please see math/README to configure.): s_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_2
in place of Failed to parse (Missing texvc executable; please see math/README to configure.): u
and Failed to parse (Missing texvc executable; please see math/README to configure.): v
.
[edit] Computation of A and B
Setting Failed to parse (Missing texvc executable; please see math/README to configure.): E_1=x_0+x_1+x_2 , Failed to parse (Missing texvc executable; please see math/README to configure.): E_2=x_0x_1+x_1x_2+x_2x_0
and Failed to parse (Missing texvc executable; please see math/README to configure.): E_3=x_0x_1x_2
, the elementary symmetric polynomials, we have, using that Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta^3=1
- Failed to parse (Missing texvc executable; please see math/README to configure.): s_1^3=x_0^3+x_1^3+x_2^3+3\zeta (x_0^2x_1+x_1^2x_2+x_2^2x_0) +3\zeta^2 (x_0x_1^2+x_1x_2^2+x_2x_0^2) +6x_0x_1x_2\,.
The expression for Failed to parse (Missing texvc executable; please see math/README to configure.): s_2^3
is the same with Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta
and Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta^2
exchanged. Thus, using Failed to parse (Missing texvc executable; please see math/README to configure.): \zeta^2+\zeta=-1
we get
- Failed to parse (Missing texvc executable; please see math/README to configure.): A=s_1^3+s_2^3=2(x_0^3+x_1^3+x_2^3)-3(x_0^2x_1+x_1^2x_2+x_2^2x_0+x_0x_1^2+x_1x_2^2+x_2x_0^2)+12x_0x_1x_2\,,
and a straightforward computation gives
- Failed to parse (Missing texvc executable; please see math/README to configure.): A=s_1^3+s_2^3=2E_1^3-9E_1E_2+27E_3\,.
Similarly we have
- Failed to parse (Missing texvc executable; please see math/README to configure.): B=s_1s_2=x_0^2+x_1^2+x_2^2+(\zeta+\zeta^2)(x_0x_1+x_1x_2+x_2x_0)=E_1^2-3E_2\,.
When solving Equation (1) we have
- Failed to parse (Missing texvc executable; please see math/README to configure.): E_1=-b/a
, Failed to parse (Missing texvc executable; please see math/README to configure.): E_2=c/a
and Failed to parse (Missing texvc executable; please see math/README to configure.): E_3=-d/a
With Equation (2), we have Failed to parse (Missing texvc executable; please see math/README to configure.): E_1=0 , Failed to parse (Missing texvc executable; please see math/README to configure.): E_2=p
and Failed to parse (Missing texvc executable; please see math/README to configure.): E_3=-q
and thus:
- Failed to parse (Missing texvc executable; please see math/README to configure.): A=-27q
and Failed to parse (Missing texvc executable; please see math/README to configure.): B=-3p
.
Note that with Equation (2), we have Failed to parse (Missing texvc executable; please see math/README to configure.): x_0 = \tfrac13(s_1 + s_2)
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_1s_2=-3p
, while in Cardano's method we have set Failed to parse (Missing texvc executable; please see math/README to configure.): x_0 = u+v
and Failed to parse (Missing texvc executable; please see math/README to configure.): uv=-\frac13p\,.
Thus we have, up to the exchange of Failed to parse (Missing texvc executable; please see math/README to configure.): u
and Failed to parse (Missing texvc executable; please see math/README to configure.): v
- Failed to parse (Missing texvc executable; please see math/README to configure.): s_1=3u
and Failed to parse (Missing texvc executable; please see math/README to configure.): s_2=3v
. In other words, in this case, Cardano's 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.
When a cubic equation has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. It has been proved that when none of the three real roots is rational—the casus irreducibilis— one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using hypergeometric functions,[22] or more elementarily in terms of trigonometric functions, specifically in terms of the cosine and arccosine functions.
The formulas which follow, due to François Viète,[16] are true in general (except when p = 0), are purely real when the equation has three real roots, but involve complex cosines and arccosines when there is only one real root.
Starting from Equation (2), Failed to parse (Missing texvc executable; please see math/README to configure.): t^3+pt+q=0 , let us set Failed to parse (Missing texvc executable; please see math/README to configure.): t=u\cos\theta\,.
The idea is to choose Failed to parse (Missing texvc executable; please see math/README to configure.): u
to make Equation (2) coincide with the identity
- Failed to parse (Missing texvc executable; please see math/README to configure.): 4\cos^3\theta-3\cos\theta-\cos(3\theta)=0\,.
In fact, choosing Failed to parse (Missing texvc executable; please see math/README to configure.): u=2\sqrt{-\frac{p}{3}}
and dividing Equation (2) by Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{u^3}{4}
we get
- Failed to parse (Missing texvc executable; please see math/README to configure.): 4\cos^3\theta-3\cos\theta-\frac{3q}{2p}\sqrt{\frac{-3}{p}}=0\,.
Combining with the above identity, we get
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(3\theta)=\frac{3q}{2p}\sqrt{\frac{-3}{p}}
and thus the roots are[23]
- Failed to parse (Missing texvc executable; please see math/README to configure.): t_k=2\sqrt{-\frac{p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-k\frac{2\pi}{3}\right) \quad \text{for} \quad k=0,1,2 \,.
This formula involves only real terms if Failed to parse (Missing texvc executable; please see math/README to configure.): p<0
and the argument of the arccosine is between −1 and 1. The last condition is equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): 4p^3+27q^2\leq 0\,,
which implies also Failed to parse (Missing texvc executable; please see math/README to configure.): p<0
. Thus the above formula for the roots involves only real terms if and only if the three roots are real.
Denoting by Failed to parse (Missing texvc executable; please see math/README to configure.): C(p,q)
the above value of t0, and using the inequality Failed to parse (Missing texvc executable; please see math/README to configure.): -\pi\le \arccos(u) \le \pi
for a real number u such that Failed to parse (Missing texvc executable; please see math/README to configure.): -1\le u\le 1\,,
the three roots may also be expressed as
- Failed to parse (Missing texvc executable; please see math/README to configure.): t_0=C(p,q),\qquad t_2=-C(p,-q), \qquad t_1=-t_0-t_2\,.
If the three roots are real, we have
- Failed to parse (Missing texvc executable; please see math/README to configure.): t_0\ge t_1\ge t_2\,.
All these formulas may be straightforwardly transformed into formulas for the roots of the general cubic equation (1), using the back substitution described in Section Reduction to a monic trinomial.
When there is only one real root (and p ≠ 0), it may be similarly represented using hyperbolic functions, as[24][25]
- Failed to parse (Missing texvc executable; please see math/README to configure.): t_0=-2\frac{|q|}{q}\sqrt{-\frac{p}{3}}\cosh\left(\frac{1}{3}\operatorname{arcosh}\left(\frac{-3|q|}{2p}\sqrt{\frac{-3}{p}}\right)\right) \quad \text{if } \quad 4p^3+27q^2>0 \text{ and } p<0\,,
- Failed to parse (Missing texvc executable; please see math/README to configure.): t_0=-2\sqrt{\frac{p}{3}}\sinh\left(\frac{1}{3}\operatorname{arsinh}\left(\frac{3q}{2p}\sqrt{\frac{3}{p}}\right)\right) \quad \text{if } \quad p>0\,.
If p ≠ 0 and the inequalities on the right are not satisfied the formulas remain valid but involve complex quantities.
When Failed to parse (Missing texvc executable; please see math/README to configure.): p=\pm 3 , the above values of Failed to parse (Missing texvc executable; please see math/README to configure.): t_0
are sometimes called the Chebyshev cube root.[26] More precisely, the values involving cosines and hyperbolic cosines define, when Failed to parse (Missing texvc executable; please see math/README to configure.): p=-3
, the same analytic function denoted Failed to parse (Missing texvc executable; please see math/README to configure.): C_{\frac13}(q) , which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted Failed to parse (Missing texvc executable; please see math/README to configure.): S_{\frac13}(q),
when Failed to parse (Missing texvc executable; please see math/README to configure.): p=3
.
If the cubic equation Failed to parse (Missing texvc executable; please see math/README to configure.): ax^3 + bx^2 + cx +d=0
with integer coefficients has a rational real root, it can be found using the rational root test: If the root is r = m / n fully reduced, then m is a factor of d and n is a factor of a, so all possible combinations of values for m and n can be checked for whether they satisfy the cubic equation.
The rational root test may also be used for a cubic equation with rational coefficients: by multiplication by the lowest common denominator) of the coefficients, one gets an equation with integer coefficients which has exactly the same roots.
The rational root test is particularly useful when there are three real roots because the algebraic solution unhelpfully expresses the real roots in terms of complex entities. The rational root test is also helpful in the presence of one real and two complex roots because it allows all of the roots to be written without the use of cube roots.
If r is any root of the cubic, then we may factor out (x–r ) using polynomial long division to obtain
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left (x-r\right )\left (ax^2+(b+ar)x+c+br+ar^2 \right ) = ax^3+bx^2+cx+d\,.
Hence if we know one root we can find the other two by using the quadratic formula to solve the quadratic Failed to parse (Missing texvc executable; please see math/README to configure.): ax^2+(b+ar)x+c+br+ar^2 , giving
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{-b-ra \pm \sqrt{b^2-4ac-2abr-3a^2r^2}}{2a}
for the other two roots.
If there are three real roots and none of them is rational, we have the so-called casus irreducibilis in which the cubic cannot be factored into the product of a linear polynomial and a quadratic polynomial each with real coefficients.
Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle.[16][27] When the cubic is written in depressed form as above as Failed to parse (Missing texvc executable; please see math/README to configure.): t^3+pt+q=0 , as shown above the solution can be expressed as
- Failed to parse (Missing texvc executable; please see math/README to configure.): t_k=2\sqrt{-\frac{p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-k\frac{2\pi}{3}\right) \quad \text{for} \quad k=0,1,2 \,.
Here Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)
is an angle in the unit circle; taking Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{1}{3}
of that angle corresponds to taking a cube root of a complex number; adding Failed to parse (Missing texvc executable; please see math/README to configure.): -k\frac{2\pi}{3}
for k = 1, 2 finds the other cube roots; and multiplying the cosines of these resulting angles by Failed to parse (Missing texvc executable; please see math/README to configure.): 2\sqrt{-\frac{p}{3}}
corrects for scale.
For the non-depressed case Failed to parse (Missing texvc executable; please see math/README to configure.): x^3+bx^2+cx+d=0
(shown in the accompanying graph), the depressed case as indicated previously is obtained by defining t such that Failed to parse (Missing texvc executable; please see math/README to configure.): x=t-\tfrac{b}{3}
so Failed to parse (Missing texvc executable; please see math/README to configure.): t=x+\tfrac{b}{3}
. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables t and x, without changing the angle relationships.
The slope of line RA is twice that of RH. Denoting the complex roots of the cubic as
g±
hi,
g =
Failed to parse (Missing texvc executable; please see math/README to configure.): \overline{OM}
(negative here) and h = Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{\tan ORH} = \sqrt{\text{slope of line RH}} = \overline{BE} = \overline{DA}
.
If a cubic is plotted in the Cartesian plane, the real root can be seen graphically as the horizontal intercept of the curve. But further,[28][29][30] if the complex conjugate roots are written as g+hi, then g is the abscissa (the positive or negative horizontal distance from the origin) of the tangency point of a line that is tangent to the cubic curve and intersects the horizontal axis at the same place as does the cubic curve; and |h| is the square root of the tangent of the angle between this line and the horizontal axis.
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{\pi}{3}
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{\pi}{3}
, the major axis is vertical and its foci, the roots of the derivative, are complex. And if that angle is Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{\pi}{3} , 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.
Omar Khayyám's geometric solution of a cubic equation.
As shown in this graph, to solve the third-degree equation Failed to parse (Missing texvc executable; please see math/README to configure.): x^3 + a^2x = b
Omar Khayyám constructed the parabola Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 = ay,
a circle with diameter Failed to parse (Missing texvc executable; please see math/README to configure.): b/a^2,
and a vertical line through an intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis.
- ^ Van de Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ISBN 0-387-12159-5
- ^ British Museum BM 85200
- ^ Guilbeau (1930, p. 8) states, "The Egyptians considered the solution impossible, but the Greeks came nearer to a solution."
- ^ a b Guilbeau (1930, pp. 8–9)
- ^ The works of Archimedes, translation by T. L. Heath
- ^ Mikami, Yoshio (1974) [1913], "Chapter 8 Wang Hsiao-Tung and Cubic Equations", The Development of Mathematics in China and Japan (2nd ed.), New York: Chelsea Publishing Co., pp. 53–56, ISBN 978-0-8284-0149-4
- ^ A paper of Omar Khayyam, Scripta Math. 26 (1963), pages 323-337
- ^ In O'Connor, John J.; Robertson, Edmund F., "Omar Khayyam", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html . one may read This problem in turn led Khayyam to solve the cubic equation x^3 + 200x = 20x^2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. The then in the last assertion is erroneous and should, at least, be replaced by also. The geometric construction was perfectly suitable for Omar Khayyam, as it occurs for solving a problem of geometric construction. At the end of his article he says only that, for this geometrical problem, if approximations are sufficient, then a simpler solution may be obtained by consulting trigonometric tables. Textually: If the seeker is satisfied with an estimate, it is up to him to look into the table of chords of Almagest, or the table of sines and versed sines of Mothmed Observatory. This is followed by a short description of this alternate method (seven lines).
- ^ J. J. O'Connor and E. F. Robertson (1999), Omar Khayyam, MacTutor History of Mathematics archive, states, "Khayyam himself seems to have been the first to conceive a general theory of cubic equations."
- ^ Guilbeau (1930, p. 9) states, "Omar Al Hay of Chorassan, about 1079 AD did most to elevate to a method the solution of the algebraic equations by intersecting conics."
- ^ Datta and Singh, History of Hindu Mathematics, p76,Equation of Higher Degree; Bharattya Kala Prakashan, Delhi, India 2004 ISBN 81-86050-86-8
- ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf.html .
- ^ J. L. Berggren (1990), "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2): 304–9
- ^ "The life and numbers of Fibonacci" [1], Plus Magazine
- ^ Katz, Victor. A History of Mathematics. pp. 220. Boston: Addison Wesley, 2004.
- ^ a b c Nickalls, R. W. D. (July 2006), "Viète, Descartes and the cubic equation", Mathematical Gazette 90: 203–208, http://www.nickalls.org/dick/papers/maths/descartes2006.pdf
- ^ Irving, Ronald S. (2004), Integers, polynomials, and rings, Springer-Verlag New York, Inc., ISBN 0-387-40397-3, http://books.google.com/?id=B4k6ltaxm5YC , Chapter 10 ex 10.14.4 and 10.17.4, pp. 154–156
- ^ Jacobson (2009), p. 210.
- ^ Prasolov, Viktor; Solovyev, Yuri (1997), Elliptic functions and elliptic integrals, AMS Bookstore, ISBN 978-0-8218-0587-9, http://books.google.com/?id=fcp9IiZd3tQC , §6.2, p. 134
- ^ Kline, Morris (1990), Mathematical Thought from Ancient to Modern Times, Oxford University Press US, ISBN 978-0-19-506136-9, http://books.google.com/?id=aO-v3gvY-I8C , Algebra in the Eighteenth Century: The Theory of Equations
- ^ a b Daniel Lazard, "Solving quintics in radicals", in Olav Arnfinn Laudal, Ragni Piene, The Legacy of Niels Henrik Abel, pp. 207–225, Berlin, 2004,. ISBN 3-540-43826-2
- ^ Zucker, I. J., "The cubic equation — a new look at the irreducible case", Mathematical Gazette 92, July 2008, 264–268.
- ^ Shelbey, Samuel (1975), CRC Standard Mathematical Tables, CRC Press, ISBN 0-87819-622-6
- ^ These are Formulas (80) and (83) of Weisstein, Eric W. 'Cubic Formula'. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CubicFormula.html, rewritten for having a coherent notation.
- ^ Holmes, G. C., "The use of hyperbolic cosines in solving cubic polynomials", Mathematical Gazette 86. November 2002, 473–477.
- ^ Abramowitz, Milton; Stegun, Irene A., eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover (1965), chap. 22 p. 773
- ^ Nickalls, R. W. D. (November 1993), "A new approach to solving the cubic: Cardan's solution revealed", The Mathematical Gazette 77 (480): 354–359, DOI:10.2307/3619777, ISSN 0025-5572, JSTOR 3619777, http://www.nickalls.org/dick/papers/maths/cubic1993.pdf See esp. Fig. 2.
- ^ Henriquez, G., "The graphical interpretation of the complex roots of cubic equations," American Mathematical Monthly 42, June–July 1935, 383–384.
- ^ Barr, C. F., American Mathematical Monthly 25, 1918, p. 268
- ^ Barr, C. F., Annals of Mathematics 19, 1917, p. 157.
- Anglin, W. S.; Lambek, Joachim (1995), "Mathematics in the Renaissance", The Heritage of Thales, Springers, pp. 125–131, ISBN 978-0-387-94544-6, http://books.google.com/?id=mZfXHRgJpmQC&pg=PA125&lpg=PA125&dq=%22mathematics+in+the+renaissance%22+heritage+thales&q Ch. 24.
- Guilbeau, Lucye (1930), "The History of the Solution of the Cubic Equation", Mathematics News Letter 5 (4): 8–12, DOI:10.2307/3027812, JSTOR 3027812
- Jacobson, Nathan (2009), Basic algebra, 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1
- Dunnett, R., "Newton–Raphson and the cubic," Mathematical Gazette 78, November 1994, 347–348.
- Dence, T., "Cubics, chaos and Newton's method," Mathematical Gazette 81, November 1997, 403–408.
- Mitchell, D. W., "Solving cubics by solving triangles," Mathematical Gazette 91, November 2007, 514–516.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.6 Quadratic and Cubic Equations", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, http://apps.nrbook.com/empanel/index.html?pg=227
- Zucker, I. J., "The cubic equation—A new look at the irreducible case," Mathematical Gazette 92, July 2008, 264–268.
- Rechtschaffen, E., "Real roots of cubics: Explicit formula for quasi-solutions," Mathematical Gazette 92, July 2008, 268–276.
- Mitchell, D. W., "Powers of Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
as roots of cubics," Mathematical Gazette 93, November 2009.
