NationMaster - Encyclopedia: Cubic equation

In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. The standard form of a cubic equation is Polynomial of degree 3: y = x3/5+4x2/5-7x/5-2=1/5 (x+5)(x+1)(x-2) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Polynomial of degree 3: y = x3/5+4x2/5-7x/5-2=1/5 (x+5)(x+1)(x-2) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

$ax^3+bx^2+cx+d=0 ,$ .

Usually, the coefficients a,..., d are real numbers. However, most of the theory is also valid if they belong to a field of characteristic other than two or three. We will always assume that a is non-zero (otherwise it is a quadratic equation). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

Solving a cubic equation amounts to finding the roots of a cubic function. In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... Polynomial of degree 3 In mathematics, a cubic function is a function of the form where a is nonzero; or in other words, a polynomial of degree three. ...

1 History
2 The nature of the roots
3 Cardano's method
4 Lagrange resolvents
5 Factorization
6 Chebyshev radicals
7 See also
8 External links
9 References

History

Cubic equations were first discovered by Jaina mathematicians in ancient India sometime between 400 BC and 200 CE. The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BCE) and Vedic civilization (1500-500 BCE) to modern India (21st century CE). ... The history of India begins with the archaeological record of Homo sapiens ca. ... The Celtics claim Vienna, Austria. ... For other uses, see number 200. ...

The Persian mathematician Omar Khayyám (1048–1123) constructed solutions of cubic equations by intersecting a conic section with a circle. He showed how this geometric solution could be used to get a numerical answer by consulting trigonometric tables. Islamic mathematics is the profession of Muslim Mathematicians. ... Omar KhayyÃ¡m, Persian Ø¹Ù…Ø± Ø®ÛŒØ§Ù… (born: May 31, 1048 in Nishapur, Iran (Persia) â€“ died: December 4, 1131), was a Persian poet, mathematician and astronomer. ... Events The city of Oslo is founded by Harald Hardråde of Norway. ... Events First Council of the Lateran confirms Concordat of Worms and demands that priests remain celibate End of the reign of Emperor Toba of Japan. ...

In the early 16th century, the Italian mathematician Scipione del Ferro (1465-1526) found a method for solving a class of cubic equations, namely those of the form x³ + mx = n. In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fiore about it. (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Scipione del Ferro (Bologna 1465â€“1526) was an Italian mathemtatician who first discovered a means to solve cubic equations. ... Events July 13 - Battle of MontlhÃ©ry Troops of King Louis XI of France fight inconclusively against an army of the great nobles organized as the League of the Public Weal. ... Events January 14 - Treaty of Madrid. ... A negative number is a number that is less than zero, such as âˆ’3. ...

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. NiccolÃ² Fontana Tartaglia. ... 1500 was a common year starting on Monday (see link for calendar) of the Gregorian calendar. ... Events Spain is effectively bankrupt. ...

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

Later, Tartaglia was persuaded by Gerolamo Cardano (1501-1576) to reveal his secret for solving cubic equations. Tartaglia did so only on the condition that Cardano would never reveal it. A few years later, Cardano learned about Ferro's prior work and broke the promise by publishing Tartaglia's method in his book Ars Magna (1545) with credit given to Tartaglia. This led to another competition between Tartaglia and Cardano, for which the latter did not show up but was represented by his student Lodovico Ferrari (1522-1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income. Gerolamo Cardano or Jerome Cardan or Girolamo Cardan (September 24, 1501 - September 21, 1576) was a celebrated Italian Renaissance mathematician, physician, astrologer, and gambler. ... 1501 was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar. ... Events May 5 - Peace of Beaulieu or Peace of Monsieur (after Monsieur, the Duc dAnjou, brother of the King, who negotiated it). ... The Ars Magna (Latin: Great Work) is an important book on mathematics written by Gerolamo Cardano. ... Events February 27 - Battle of Ancrum Moor - Scots victory over superior English forces December 13 - Official opening of the Council of Trent (closed 1563) Battle of Kawagoe - between two branches of Uesugi families and the late Hojo clan in Japan. ... Lodovico Ferrari (February 2, 1522 - October 5, 1565) was an Italian mathematician. ... Events January 9 - Adrian Dedens becomes Pope Adrian VI. February 26 - Execution by hanging of CuauhtÃ©moc, Aztec ruler of Tenochtitlan under orders of conquistador HernÃ¡n CortÃ©s. ... // Events March 1 - the city of Rio de Janeiro is founded. ...

Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these complex numbers in Ars Magna, but he did not really understand it. Rafael Bombelli studied this issue in detail and is therefore often considered as the discoverer of complex numbers. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... Raphael Bombelli (1526-1572) was an Italian mathematician. ...

The nature of the roots

Every cubic equation with real coefficients has at least one solution x among the real numbers; this is a consequence of the intermediate value theorem. We can distinguish several possible cases using the discriminant, In mathematics, the real numbers may be described informally in several different ways. ... In analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... In mathematics, a discriminant is an expression which discriminates qualities of algebraic structures. ...

$Delta = 4alpha_1^3alpha_3 - alpha_1^2alpha_2^2 + 4alpha_0alpha_2^3 - 18alpha_0alpha_1alpha_2alpha_3 + 27alpha_0^2alpha_3^2.$

The following cases need to be considered.

If Δ < 0, then the equation has three distinct real roots.
If Δ > 0, then the equation has one real root and a pair of complex conjugate roots.
If Δ = 0, then (at least) two roots coincide. To decide how many distinct roots there are, we define

$Delta_2 = 2alpha_2^3 - 9alpha_1alpha_2alpha_3 + 27alpha_0alpha_3^2,$

and consider two further cases.

If Δ₂ = 0, then all three roots coincide and we have a triple real root.
Otherwise, the equation has a double real root and another distinct single real root.

The number Δ₂ is the resultant of the cubic and its second derivative.

See also: multiplicity of a root of a polynomial

In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, the resultant of two monic polynomials and over a field is defined as the product of the differences of their roots, where and take on values in the algebraic closure of . ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ...

Cardano's method

The solutions can be found with the following method due to Scipione del Ferro and Tartaglia, published by Gerolamo Cardano in 1545. Scipione del Ferro (Bologna 1465â€“1526) was an Italian mathemtatician who first discovered a means to solve cubic equations. ... Niccolo Fontana Tartaglia. ... Gerolamo Cardano or Jerome Cardan or Girolamo Cardan (September 24, 1501 - September 21, 1576) was a celebrated Italian Renaissance mathematician, physician, astrologer, and gambler. ... Events February 27 - Battle of Ancrum Moor - Scots victory over superior English forces December 13 - Official opening of the Council of Trent (closed 1563) Battle of Kawagoe - between two branches of Uesugi families and the late Hojo clan in Japan. ...

We first divide the given equation by α₃ to arrive at an equation of the form

$x^3 + ax^2 + bx +c = 0. qquad (1)$

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

$t^3 + pt + q = 0, quadmbox{where } p = b - frac{a^2}3 quadmbox{and}quad q = c + frac{2a^3-9ab}{27}. qquad (2)$

This is called the depressed cubic.

Suppose that we can find numbers u and v such that

$u^3-v^3 = q quadmbox{and}quad uv = frac{p}{3}. qquad (3)$

A solution to our equation is then given by

$t = v - u, ,$

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

$(v-u)^3+3uv(v-u)+(u^3-v^3)=0 .$

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

$v = frac{p}{3u}.$

Substituting this in the first equation in (3) yields

$u^3 - frac{p^3}{27u^3} = q.$

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

$u=sqrt[3]{{qover 2}pm sqrt{{q^{2}over 4}+{p^{3}over 27}}}. qquad (4)$

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

$x=frac{p}{3u}-u-{aover 3}.$

Note that there are six possibilities in computing u with (4), since there are two solutions to the square root ( $pm$ ), and three complex solutions to the cubic root (the principal root and the principal root multiplied by $-1/2 pm isqrt{3}/2$ ). However, which sign of the square root is chosen does not affect the final resulting x, although care must be taken in two special cases to avoid divisions by zero. First, if p = 0, then one should choose the sign of the square root that gives a nonzero value for u, i.e. $u = sqrt[3]{q}$ . Second, if p = q = 0, then we have the triple real root x = −a/3.

Lagrange resolvents

The symmetric group S₃ of order three has the cyclic group of order three as a normal subgroup, which suggests making use of the discrete Fourier transform of the roots, an idea due to Lagrange. Suppose that r₀, r₁ and r₂ are the roots of equation (1), and define $zeta = (-1+isqrt{3})/2$ , so that ζ is a primitive third root of unity. We now set In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ... Joseph-Louis Lagrange Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ...

$s_0 = r_0 + r_1 + r_2,,$

$s_1 = r_0 + zeta r_1 + zeta^2 r_2,,$

$s_2 = r_0 + zeta^2 r_1 + zeta r_2.,$

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

$r_0 = (s_0 + s_1 + s_2)/3,,$

$r_1 = (s_0 + zeta^2 s_1 + zeta s_2)/3,,$

$r_2 = (s_0 + zeta s_1 + zeta^2 s_2)/3.,$

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

$(z-s_1^3)(z-s_2^3) qquad (5)$

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

${z}^{2}+ left( -9,ba+2,{a}^{3}+27,c right) z+ left( {a}^{2}-3,bright)^{3}.$

The roots of this quadratic equation are

$frac92,ab-{a}^{3}- frac{27}{2},c pm frac32,sqrt{3Delta},$

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

Factorization

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

(x - r)(x 2 + (a + r) x + b + a r + r 2) = x 3 + a x 2 + b x + c .

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

$frac12 left(-a-r pm sqrt{-3r^2-2ar+a^2-4b}right)$

for the other two roots.

Chebyshev radicals

The cube root function is in some respects not a well-behaved function, or one convenient for the purposes of finding the roots of a cubic equation. While cube roots are well-known and traditional, it is possible to use other algebraic functions to determine the roots, and avoid some of the problems of cube roots. The cube root function has a branch singularity at zero, as a result of which the real cube root function does not extend nicely to a complex cube root function. Moreover, when using cube roots to find the roots of a polynomial with three real roots we must take the roots of complex numbers, which introduces complex numbers into a situation which does not, in fact, require them. Plot of y = In mathematics, the cube root ( ) of a number is the number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. ...

We can get around these problems by using Chebyshev cube roots in place of ordinary cube roots. The polynomial $C 3 = x 3 - 3 x$ is the third Chebyshev polynomial normalized to obtain a monic polynomial. The Chebyshev cube root is then defined as a (suitably chosen) root (depending on t) of the polynomial equation Pafnuty Lvovich Chebyshev Pafnuty Lvovich Chebyshev (Russian: ) ( May 16 [O.S. May 4] 1821 â€“ December 8 [O.S. November 26] 1894) was a Russian mathematician. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. ...

$x^3 - 3x = t .$

The polynomial $C 3 (x)$ satisfies the third order addition relations

$, 2,cos(3x)= C_3(2 cos x)$

and (as $, 2cos(ix)=2cosh(x)$ )

$2,cosh(3x)=C_3(2cosh x) .$

If t is represented as $t = 2cos y$ , then the polynomial equation $x 3 - 3 x = t$ can now be transformed into

$t=2,cos y=C_3(2cos (y/3)) .$

The function $C_{1over3}(t)$ is then defined as (a branch of) the algebraic function of the third order which transforms $2cos(x)$ into $2cos(x / 3)$ . It is given (inverting the relation $t = 2cos(x)$ to $x = arccos(t / 2)$ ) as This article or section does not cite its references or sources. ...

$C_{1over3}(t) = 2 ,operatorname{cos}left(operatorname{arccos}left({tover2}right)/3right) ,$

if t lies in the real interval [−2, 2]. If t lies in the interval $[2,infty]$ , then the Chebyshev root is given as

$C_{1over3}(t) = 2 ,operatorname{cosh}left(operatorname{arccosh}left({tover2}right)/3right) .$

The branch is uniquely defined by the value at $t = 0$ , which is $2, operatorname{cos}left(operatorname{arccos}(0)/3right)=2, operatorname{cos}(pi/6)=sqrt{3}$ , corresponding to the positive solution of $x 3 - 3 x = x (x 2 - 3) = 0$ .

This procedure is precisely analogous to the definition of the cube root in terms of logarithms and exponentials, with arccosh(x/2) resp. arccos(x/2) in the place of ln(x), and 2cosh(x) resp. 2cos(x) in the place of exp(x). The Chebyshev cube root can be constructed as an analytic function on the cut plane $mathbb{C}setminus [-infty,-2]$ and is the unique branch of the algebraic function $C_{1over3}(t)$ with this property. In the domain $D_1 := {z in mathbb{C}, | , Re{z}>2}$ it can be defined as In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, the domain of a function is the set of all input values to the function. ...

$C_{1over3}(t)= 2,operatorname{cosh}left(operatorname{arccosh}left({tover2}right)/3right)$

where $operatorname{arccosh}(z/2)=ln{{z+sqrt{z^2-4}}over 2},$ using the branch of the logarithm which is real on the positive real line and the branch of the square root which is positive on the real axis. On the domain $D_2 :=mathbb{C}setminus{{[-infty,-2] cup [2,infty]}}$ it can be defined as

$C_{1over3}(t)= 2 ,operatorname{cos}left(operatorname{arccos}left({tover2}right)/3right)$ ,

where $operatorname{arccos}(z/2)={pi over 2}+iln{{iz+sqrt{4-z^2}}over 2} .$ Both D₁ and D₂ are simply-connected domains in $mathbb{C}$ on which the functions $operatorname{arccos}(z)$ and $operatorname{arccosh}(z)$ are well-defined analytic functions (because the square roots $sqrt{pm (z^2-4)}$ exist as analytic functions on D₁ resp. D₂ and the argument functions ${z+sqrt{z^2-4}}over 2$ and ${iz+sqrt{4-z^2}}over 2$ of the logarithm do not vanish on each domain). Both (partially overlapping) definitions of the Chebyshev cube root on the domains D₁ and D₂ can be put together to define the Chebyshev cube root unambiguously as an analytic function on the larger domain $D= mathbb{C}setminus [-infty,-2]$ . In fact, if one approaches the critical value $t = 2$ from either the left or the right on the real axis the value of each representative will tend to 2. Because $x = 2$ is a simple root of the polynomial $x 3 - 3 x - 2$ the branch of the Chebyshev root (defined as the algebraic function F(t)=2+G(t) satisfying See also Simple Lie group. ...

$, F(t)^3-3F(t)-2=9G(t)+6G(t)^2+G(t)^3=0$

and $F (2) = 2$ exists locally as an analytic function in a (sufficiently small) neighbourhood U of $t = 2$ (according to the (complex-analytic ) inverse function theorem) and takes real values if $t=2pm epsilon, ,epsilon >0$ . Then it must coincide (on the intersection $U cap D_1$ and $U cap D_2$ ) with each of the two representatives (in terms of arccos z resp. arccosh z) constructed above. Therefore the Chebyshev cube root is in fact an analytic function on the whole of the domain D. In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ...

An alternative construction of the Chebyshev cube root in terms of hypergeometric functions is sketched in the next subsection.

The Chebyshev cube root as a hypergeometric function

The expression

$2 ,operatorname{cos}left(operatorname{arccos}left({tover2}right)/3right)=2,operatorname{cos}left({piover 6}-operatorname{arcsin}left({tover2}right)/3right)$

can be transformed (using the difference-to-product trigonometric identity for the cosine) into the representation In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

$sqrt{3} ,operatorname{cos}left(operatorname{arcsin}left({tover2}right)/3right)+ operatorname{sin }left(operatorname{arcsin}left({tover2}right)/3right) .$

For general complex parameter $lambda ne 0$ the functions $2,operatorname{cos},(lambda, operatorname{arcsin}(x/2))$ and $2,operatorname{sin},(lambda ,operatorname{arcsin}(x/2))$ are two linearly independent solutions of the second-order linear differential equation In mathematics, a linear differential equation is a differential equation Lf = g, where the differential operator L is a linear operator. ...

$, (4-x^2)y''-xy'+lambda^2 y=0$

which can be obtained by differentiating the functional relations $, f(2sin x)=2sin(lambda x)$ resp. $, f(2sin x)=2cos(lambda x)$ twice with respect to x. The differential equation

$, (4-x^2)y''-xy'+lambda^2 y=0$

is equivalent (under the affine substitution $x mapsto (2-4x)$ ) to the hypergeometric differential equation In mathematics, the hypergeometric differential equation is a second-order linear ordinary differential equation (ODE) whose solutions are given by the hypergeometric series. ...

$x(1-x) ,y''+{{1-2x}over 2},y'+lambda^2 y=0$

with parameters $c={1over 2},, a=lambda,, b=-lambda$ . According to the general theory of the hypergeometric equation it has (unless c is zero or a negative integer) a uniquely defined solution g which is analytic in x=0 and satisfies $,g(0)=1$ . It is given by the hypergeometric series (see hypergeometric function) In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ...

$,F(a,b,c;z):=,_2F_1 (a,b;c;z) = sum_{n=0}^infty frac{(a)_n(b)_n}{(c)_n} , frac {z^n} {n!} .$

Transforming back to the original differential equation one finds a solution $g(x)=F(lambda,-lambda,{1over2} ;{{2-x}over 4})$ of the differential equation

$, (4-x^2)y''-xy'+lambda^2 y=0$

which is analytic at $x = 2$ (unique up to scalar multiple). The representation

$C_{{1over 3}}(t)= sqrt{3} ,operatorname{cos}left(operatorname{arcsin}left({tover2}right)/3right)+ operatorname{sin}left(operatorname{arcsin}left({tover2}right)/3right)$

obtained above shows that the Chebyshev cube root is a solution of the differential equation

$, (4-x^2)y''-xy'+lambda^2 y=0$

for $lambda={1over 3}$ which is analytic at $x = 2$ . It must be proportional to the argument-shifted hypergeometric series and thus

$C_{{1over 3}}(t)=2F({1over 3},-{1over 3},{1over2} ;{{2-t}over 4}) = sum_{n=0}^infty frac{2}{1-3n} {3n choose n}left(frac{2-t}{27}right)^n ,$

where the last series converges if $| t - 2 | < 4$ . All three roots $r 1, r 2, r 3$ of the equation $x 3 - 3 x - t = 0$ are linear combinations of the two functions $f_1(t)=2operatorname{sin},left({1over 3} operatorname{arcsin}{tover2}right)$ and $f_2(t)=2operatorname{cos},left({1over 3} operatorname{arcsin}{tover2}right) .$ By construction

$r_1=C_{{1over 3}}(t)=sqrt{3} ,operatorname{cos}left(operatorname{arcsin}left({tover2}right)/3right)+ operatorname{sin},left(operatorname{arcsin}left({tover2}right)/3right), ,$

the other two roots are

$r_2=-C_{{1over 3}}(-t)=-sqrt{3} ,operatorname{cos}left(operatorname{arcsin}left({tover2}right)/3right)+ operatorname{sin},left(operatorname{arcsin}left({tover2}right)/3right)$

and

$r_3=-r_1-r_2= -2 ,operatorname{sin}left(operatorname{arcsin}left({tover2}right)/3right) .$

One derives the further relations

$r_1={sqrt{3}over 2}sqrt{4-r_3^2}-{r_3over 2} , qquad r_2=-{sqrt{3}over 2} sqrt{4-r_3^2}- {r_3over 2}$

which can be verified independently by calculating the other two roots ( here $r 1, r 2$ ) given one root (here $r 3$ ) by means of the relation

$t=x^3-3x=y^3-3y Longrightarrow (y-x)(y^2+xy+x^2-3)=0,$

solving the quadratic equation $, y^2+xy+(x^2-3)=0$ for y, given x. In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

Solving a general cubic equation using Chebyshev cube roots

If we have a cubic equation which is already in depressed form, we may write it as $,x^3 - 3px - q = 0$ . Substituting $x = sqrt{p} z$ we obtain $z^3 - 3z - p^{-frac{3}{2}}q = 0$ or equivalently

$z^3 - 3z = p^{-frac{3}{2}}q .$

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

$r_0 = sqrt{p},C_{1over3}(p^{-frac{3}{2}}q),,$

$r_1 = -sqrt{p},C_{1over3}(-p^{-frac{3}{2}}q),,$

$r_2 = -r_0 - r_1 .$

If now we start from a general equation

$x^3 + ax^2 + bx +c = 0 qquad (1)$

and reduce it to the depressed form under the substitution x = t − a/3, we have $, p = (a^2-3b)/9$ and $, q = -(2a^3-9ab+27c)/27$ , leading to

$t_{a;b;c} = p^{-frac{3}{2}}q = -frac{2a^3-9ab+27c}{(a^2-3b)^{3/2}}.$

This gives us the solutions to (1) as

$r_0 = sqrt{p},C_{1over3}(t_{a;b;c})-{aover 3} ,,$

$r_1 = -sqrt{p},C_{1over3}(-t_{a;b;c})-{aover 3},,$

$r_2 = -r_0 - r_1 - a .$

The case of a cubic equation with real coefficients

Suppose the coefficients of (1) are real. If s is the quantity q/r from the section on real roots, then s = t²; hence 0 < s < 4 is equivalent to −2 < t < 2, and in this case we have a polynomial with three distinct real roots, expressed in terms of a real function of a real variable, quite unlike the situation when using cube roots. If s > 4 then either t > 2 and $C_{1over3}(t)$ is the sole real root, or t < −2 and $-C_{1over3}(-t)$ is the sole real root. If s < 0 then the reduction to Chebyshev polynomial form has given a t which is a pure imaginary number; in this case $iC_{1over3}(-it)-iC_{1over3}(it)$ is the sole real root. We are now evaluating a real root by means of a function of a purely imaginary argument; however we can avoid this by using the function

$S_{1over3}(t) = iC_{1over3}(-it)-iC_{1over3}(it) = 2 operatorname{sinh}left(operatorname{arcsinh}left({tover2}right)/3right),,$

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

External links

Cubic Equation Solver.
Quadratic, cubic and quartic equations on MacTutor archive.
Cubic Formula on PlanetMath
Cardano solution calculator as java applet at some local site. Only takes natural coefficients.
Graphic explorer for cubic functions With interactive animation, slider controls for coefficients

BITCH!111 ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

References

W. S. Anglin; & J. Lambek (1995). Mathematics in the Renaissance. In The heritage of Thales, Ch. 24. Springers.
R.W.D. Nickalls (1993). A new approach to solving the cubic: Cardan's solution revealed, The Mathematical Gazette, 77:354–359.

Categories: Elementary algebra | Equations | Polynomials

Results from FactBites:

Math Forum: Ask Dr. Math FAQ: Cubic and Quartic Equations (1200 words)
	To try to go backward and come up with a closed form for the Cubic Formula in terms of the original a, b, c, d would be a real pain.
	The roots of the original equation are then x = -a/4 and the roots of that cubic with a/4 subtracted from each.
	The Math Forum is a research and educational enterprise of the Drexel School of Education.

cubic equation (264 words)
	A polynomial equation of the third degree, the general form of which is
	Early studies of cubics helped legitimize negative numbers, give a deeper insight into equations in general, and stimulate work that eventually led to the discovery and acceptance of complex numbers.
	He also noted an important fact connecting solutions of a cubic equation to its coefficients, namely, that the sum of the solutions is the negation of b, the coefficient of the x

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Contents

The nature of the roots

Cardano's method

Lagrange resolvents

Factorization

Chebyshev radicals

See also

External links

References