In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, a cubic equation in one variable is an
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
of the form
:
in which is nonzero.
The solutions of this equation are called
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of the
cubic function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
defined by the left-hand side of the equation. If all of the
coefficients
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
, , , and of the cubic equation are
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, 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:
*
algebraically, that is, they can be expressed by a cubic formula involving the four coefficients, the four basic
arithmetic operation
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th cen ...
s and
th roots (radicals). (This is also true of
quadratic (second-degree) and
quartic (fourth-degree) equations, but not of higher-degree equations, by the
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
.)
*
trigonometrically
*
numerical approximation
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
s of the roots can be found using
root-finding algorithm
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...
s such as
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
.
The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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 number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s.
History
Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians.
[ ]Babylonia
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...
n (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
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
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
Hippocrates of Kos (; grc-gre, Ἱπποκράτης ὁ Κῷος, Hippokrátēs ho Kôios; ), also known as Hippocrates II, was a Greek physician of the classical period who is considered one of the most outstanding figures in the history of ...
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
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
, a task which is now known to be impossible. Methods for solving cubic equations appear in ''The Nine Chapters on the Mathematical Art
''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'', a Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui
Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
in the 3rd century. In the 3rd century AD, the Greek mathematician Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
found integer or rational solutions for some bivariate cubic equations ( Diophantine equations).[Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ] Hippocrates, Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.''
Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, w ...
and Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
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
Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath transla ...
, who translated all of Archimedes' works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two conics
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
, but also discussed the conditions where the root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
s are 0, 1 or 2.
In the 7th century, the Tang dynasty
The Tang dynasty (, ; zh, t= ), or Tang Empire, was an Dynasties in Chinese history, imperial dynasty of China that ruled from 618 to 907 AD, with an Zhou dynasty (690–705), interregnum between 690 and 705. It was preceded by the Sui dyn ...
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
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...
(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 section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s.
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
Persian may refer to:
* People and things from Iran, historically called ''Persia'' in the English language
** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples
** Persian language, an Iranian language of the ...
mathematician, Sharaf al-Dīn al-Tūsī
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( fa, شرفالدین مظفر بن محمد بن مظفر توسی; 1135 – 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the ...
(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
Horner is an English and German surname that derives from the Middle English word for the occupation ''horner'', meaning horn-worker or horn-maker, or even horn-blower.
People
*Alison Horner (born 1966), British businesswoman
* Arthur Horner (dis ...
method" to numerically approximate the root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of a cubic equation. He also used the concepts of maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
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
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
(1170–1250), was able to closely approximate the positive solution to the cubic equation . Writing in Babylonian numerals
Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.
The Babylonians, who were fam ...
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
The approximation error in a data value is the discrepancy between an exact value and some '' approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute e ...
of about 10−9.
In the early 16th century, the Italian mathematician Scipione del Ferro
Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician who first discovered a method to solve the depressed cubic equation.
Life
Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filip ...
(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 number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
s 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 1535, Niccolò Tartaglia Niccolò is an Italian male given name, derived from the Greek Nikolaos meaning "Victor of people" or "People's champion".
There are several male variations of the name: Nicolò, Niccolò, Nicolas, and Nicola. The female equivalent is Nicole. The f ...
(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
Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
(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
Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian mathematician.
Biography
Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, and he began his ...
(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 number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s in ''Ars Magna'', but he did not really understand it. Rafael Bombelli
Rafael Bombelli (baptised
Baptism (from grc-x-koine, βάπτισμα, váptisma) is a form of ritual purification—a characteristic of many religions throughout time and geography. In Christianity, it is a Christian sacrament of initia ...
studied this issue in detail and is therefore often considered as the discoverer of complex numbers.
François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
(1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
(1596–1650) extended the work of Viète.[
]
Factorization
If the coefficients of a cubic equation are rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...
of their denominators. Such an equation
:
with integer coefficients, is said to be reducible if the polynomial on 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 degree one, and thus have the form
:
with and being coprime integers
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
. The rational root test
In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or theorem) states a constraint on rational solutions of a polynomial equation
:a_nx^n+a_x^+\cdots+a_0 = 0
with integer coefficients a_i\in\m ...
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 and the other roots are the roots of the other factor, which can be found by polynomial long division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, becaus ...
. This other factor is
:
(The coefficients seem not to be integers, but must be integers if is a root.)
Then, the other roots are the roots of this quadratic polynomial
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
and can be found by using the quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, g ...
.
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
Change or Changing may refer to:
Alteration
* Impermanence, a difference in a state of affairs at different points in time
* Menopause, also referred to as "the change", the permanent cessation of the menstrual period
* Metamorphosis, or change, ...
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
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of the original equation are related to the roots of the depressed equation by the relations
::
for .
Discriminant and nature of the roots
The nature (real or not, distinct or not) of the roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of a cubic can be determined without computing them explicitly, by using the discriminant.
Discriminant
The discriminant of a polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
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 {{no footnotes, date=December 2015
In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''.
A ...
.
If are the three roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
(not necessarily distinct nor real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
) of the cubic then the discriminant is
:
The discriminant of the depressed cubic is
:
The discriminant of the general cubic is
:
It is the product of and the discriminant of the corresponding depressed cubic. Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as
:
It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
, 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
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
Basic 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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, and its discriminant is not zero, there are two cases:
*If the cubic has three distinct real roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
*If the cubic has one real root and two non-real complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
roots.
This can be proved as follows. First, if is a root of a polynomial with real coefficients, then its complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
, at least one root must be real.
As stated above, if are the three roots of the cubic , then the discriminant is
:
If the three roots are real and distinct, the discriminant is a product of positive reals, that is
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
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 is zero if If is also zero, then , and 0 is a triple root of the cubic. If 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
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
Basic 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 is zero, then
*either, if the cubic has a triple root
::
:and
::
*or, if the cubic has a double root
::
:and a simple root,
::
:and thus
::
Characteristic 2 and 3
The above results are valid when the coefficients belong to a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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 is a linear polynomial in characteristic 3, and is the square of a linear polynomial in characteristic 2. Therefore, for either characteristic 2 or 3, the derivative has only one root. 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
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
Basic 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
Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
is credited with publishing the first formula for solving cubic equations, attributing it to Scipione del Ferro
Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician who first discovered a method to solve the depressed cubic equation.
Life
Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filip ...
and Niccolo Fontana Tartaglia. 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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s such that is positive, then the equation has the real root
:
where and are the two numbers and
See , below, for several methods for getting this result.
As shown in , the two other roots are non-real complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
numbers, in this case. It was later shown (Cardano did not know complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s) that the two other roots are obtained by multiplying one of the cube roots by the primitive cube root of unity and the other cube root by the other primutive cube root of the unity That is, the other roots of the equation are