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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 quartic function is a function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
of degree four, called a quartic polynomial. A ''
quartic equation In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomi ...
'', or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form :ax^4+bx^3+cx^2+dx+e=0 , where . The
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of a quartic function is a
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 ...
. Sometimes the term biquadratic is used instead of ''quartic'', but, usually, biquadratic function refers to a
quadratic function 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 polynomia ...
of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form :f(x)=ax^4+cx^2+e. Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
. If ''a'' is positive, then the function increases to positive infinity at both ends; and thus the function has a
global minimum 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 ran ...
. Likewise, if ''a'' is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum. The degree four (''quartic'' case) is the highest degree such that every polynomial equation can be solved by radicals, according to 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 t ...
.


History

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 ...
is credited with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a
cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...
to be found, it could not be published immediately. The solution of the quartic was published together with that of the cubic by Ferrari's mentor
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, ...
in the book '' Ars Magna''. The Soviet historian I. Y. Depman ( ru) claimed that even earlier, in 1486, Spanish mathematician Valmes was
burned at the stake Death by burning (also known as immolation) is an execution and murder method involving combustion or exposure to extreme heat. It has a long history as a form of public capital punishment, and many societies have employed it as a punishment f ...
for claiming to have solved the quartic equation. Inquisitor General
Tomás de Torquemada Tomás de Torquemada (14 October 1420 – 16 September 1498), also anglicized as Thomas of Torquemada, was a Castilian Dominican friar and first Grand Inquisitor of the Tribunal of the Holy Office (otherwise known as the Spanish Inquisition). ...
allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding. However,
Petr Beckmann Petr Beckmann (November 13, 1924 – August 3, 1993) was a professor of electrical engineering who became a well-known advocate of libertarianism and nuclear power. Later in his life he disputed Albert Einstein's theory of relativity and oth ...
, who popularized this story of Depman in the West, said that it was unreliable and hinted that it may have been invented as Soviet antireligious propaganda. Beckmann's version of this story has been widely copied in several books and internet sites, usually without his reservations and sometimes with fanciful embellishments. Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed. The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in 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 t ...
in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
prior to dying in a duel in 1832 later led to an elegant
complete theory In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence \varphi, the theory T contains the sentence or its ...
of the roots of polynomials, of which this theorem was one result.


Applications

Each
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
of the intersection points of two
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 speci ...
s is a solution of a quartic equation. The same is true for the intersection of a line and a
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
. It follows that quartic equations often arise in
computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems a ...
and all related fields such as computer graphics,
computer-aided design Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
,
computer-aided manufacturing Computer-aided manufacturing (CAM) also known as computer-aided modeling or computer-aided machining is the use of software to control machine tools in the manufacturing of work pieces. This is not the only definition for CAM, but it is the most ...
and
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
. Here are examples of other geometric problems whose solution involves solving a quartic equation. In
computer-aided manufacturing Computer-aided manufacturing (CAM) also known as computer-aided modeling or computer-aided machining is the use of software to control machine tools in the manufacturing of work pieces. This is not the only definition for CAM, but it is the most ...
, the torus is a shape that is commonly associated with the
endmill An end mill is a type of milling cutter, a cutting tool used in industrial milling applications. It is distinguished from the drill bit in its application, geometry, and manufacture. While a drill bit can only cut in the axial direction, most m ...
cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the -axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated. A quartic equation arises also in the process of solving the
crossed ladders problem The crossed ladders problem is a puzzle of unknown origin that has appeared in various publications and regularly reappears in Web pages and Usenet discussions. The problem Two ladders of lengths ''a'' and ''b'' lie oppositely across an alley, a ...
, in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found. In optics,
Alhazen's problem Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD. It is named for the 11th-century Arab mathematician Alhazen (''Ibn al-Haytham'') who presented a ...
is "''Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.''" This leads to a quartic equation. Finding the distance of closest approach of two ellipses involves solving a quartic equation. The
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of a 4×4
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
are the roots of a quartic polynomial which is the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The char ...
of the matrix. The characteristic equation of a fourth-order linear
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
or
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending. Intersections between spheres, cylinders, or other
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is d ...
s can be found using quartic equations.


Inflection points and golden ratio

Letting and be the distinct
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case o ...
s of the graph of a quartic function, and letting be the intersection of the inflection
secant line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproc ...
and the quartic, nearer to than to , then divides into the
golden section In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ...
: :\frac=\frac= \varphi \; (\text). Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area.


Solution


Nature of the roots

Given the general quartic equation :ax^4 + bx^3 + cx^2 + dx + e = 0 with real coefficients and the nature of its roots is mainly determined by the sign of its
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
:\begin \Delta = &256 a^3 e^3 - 192 a^2 b d e^2 - 128 a^2 c^2 e^2 + 144 a^2 c d^2 e - 27 a^2 d^4 \\ &+ 144 a b^2 c e^2 - 6 a b^2 d^2 e - 80 a b c^2 d e + 18 a b c d^3 + 16 a c^4 e \\ &- 4 a c^3 d^2 - 27 b^4 e^2 + 18 b^3 c d e - 4 b^3 d^3 - 4 b^2 c^3 e + b^2 c^2 d^2 \end This may be refined by considering the signs of four other polynomials: :P = 8ac - 3b^2 such that is the second degree coefficient of the associated depressed quartic (see below); :R= b^3+8da^2-4abc, such that is the first degree coefficient of the associated depressed quartic; :\Delta_0 = c^2 - 3bd + 12ae, which is 0 if the quartic has a triple root; and :D = 64 a^3 e - 16 a^2 c^2 + 16 a b^2 c - 16 a^2 bd - 3 b^4 which is 0 if the quartic has two double roots. The possible cases for the nature of the roots are as follows: * If then the equation has two distinct real roots and two
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 - ...
non-real roots. * If then either the equation's four roots are all real or none is. ** If < 0 and < 0 then all four roots are real and distinct. ** If > 0 or > 0 then there are two pairs of non-real complex conjugate roots. * If then (and only then) the polynomial has a multiple root. Here are the different cases that can occur: ** If < 0 and < 0 and , there are a real double root and two real simple roots. ** If > 0 or ( > 0 and ( ≠ 0 or ≠ 0)), there are a real double root and two complex conjugate roots. ** If and ≠ 0, there are a triple root and a simple root, all real. ** If = 0, then: ***If < 0, there are two real double roots. ***If > 0 and = 0, there are two complex conjugate double roots. ***If , all four roots are equal to There are some cases that do not seem to be covered, and in fact they cannot occur. For example, , = 0 and ≤ 0 is not one of the cases. In fact, if and = 0 then > 0, since 16 a^2\Delta_0 = 3D + P^2; so this combination is not possible.


General formula for roots

The four roots , , , and for the general quartic equation :ax^4+bx^3+cx^2+dx+e=0 \, with ≠ 0 are given in the following formula, which is deduced from the one in the section on Ferrari's method by back changing the variables (see ) and using the formulas for the quadratic and
cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
s. :\begin x_\ &= -\frac - S \pm \frac12\sqrt\\ x_\ &= -\frac + S \pm \frac12\sqrt \end where and are the coefficients of the second and of the first degree respectively in the associated depressed quartic :\begin p &= \frac\\ q &= \frac \end : and where :\begin S &= \frac\sqrt\\ Q &= \sqrt \end (if or , see , below) with :\begin \Delta_0 &= c^2 - 3bd + 12ae\\ \Delta_1 &= 2c^3 - 9bcd + 27b^2 e + 27ad^2 - 72ace \end and :\Delta_1^2-4\Delta_0^3 = - 27 \Delta\ , where \Delta is the aforementioned
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
. For the cube root expression for ''Q'', any of the three cube roots in the complex plane can be used, although if one of them is real that is the natural and simplest one to choose. The mathematical expressions of these last four terms are very similar to those of their cubic counterparts.


Special cases of the formula

*If \Delta > 0, the value of Q is a non-real complex number. In this case, either all roots are non-real or they are all real. In the latter case, the value of S is also real, despite being expressed in terms of Q; this is casus irreducibilis of the cubic function extended to the present context of the quartic. One may prefer to express it in a purely real way, by using
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
, as follows: ::S = \frac \sqrt :where ::\varphi = \arccos\left(\frac\right). *If \Delta \neq 0 and \Delta_0 = 0, the sign of \sqrt=\sqrt has to be chosen to have Q \neq 0, that is one should define \sqrt as \Delta_1, maintaining the sign of \Delta_1. *If S = 0, then one must change the choice of the cube root in Q in order to have S \neq 0. This is always possible except if the quartic may be factored into \left(x+\tfrac\right)^4. The result is then correct, but misleading because it hides the fact that no cube root is needed in this case. In fact this case may occur only if the
numerator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
of q is zero, in which case the associated depressed quartic is biquadratic; it may thus be solved by the method described below. *If \Delta = 0 and \Delta_0 = 0, and thus also \Delta_1 = 0, at least three roots are equal to each other, and the roots are
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s of the coefficients. The triple root x_0 is a common root of the quartic and its second derivative 2(6ax^2+3bx+c); it is thus also the unique root of the remainder of the
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
of the quartic by its second derivative, which is a linear polynomial. The simple root x_1 can be deduced from x_1+3x_0=-b/a. *If \Delta=0 and \Delta_0 \neq 0, the above expression for the roots is correct but misleading, hiding the fact that the polynomial is reducible and no cube root is needed to represent the roots.


Simpler cases


Reducible quartics

Consider the general quartic :Q(x) = a_4x^4+a_3x^3+a_2x^2+a_1x+a_0. It is reducible if , where and are non-constant polynomials with
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
coefficients (or more generally with coefficients in the same
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 ...
as the coefficients of ). Such a factorization will take one of two forms: :Q(x) = (x-x_1)(b_3x^3+b_2x^2+b_1x+b_0) or :Q(x) = (c_2x^2+c_1x+c_0)(d_2x^2+d_1x+d_0). In either case, the roots of are the roots of the factors, which may be computed using the formulas for the roots of a
quadratic function 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 polynomia ...
or
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 ...
. Detecting the existence of such factorizations can be done using the resolvent cubic of . It turns out that: * if we are working over (that is, if coefficients are restricted to be real numbers) (or, more generally, over some
real closed field In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. D ...
) then there is always such a factorization; * if we are working over (that is, if coefficients are restricted to be rational numbers) then there is an algorithm to determine whether or not is reducible and, if it is, how to express it as a product of polynomials of smaller degree. In fact, several methods of solving quartic equations ( Ferrari's method, Descartes' method, and, to a lesser extent,
Euler's method In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit me ...
) are based upon finding such factorizations.


Biquadratic equation

If then the biquadratic function : Q(x) = a_4x^4+a_2x^2+a_0\,\! defines a biquadratic equation, which is easy to solve. Let the auxiliary variable . Then becomes a quadratic in : . Let and be the roots of . Then the roots of our quartic are : \begin x_1&=+\sqrt, \\ x_2&=-\sqrt, \\ x_3&=+\sqrt, \\ x_4&=-\sqrt. \end


Quasi-palindromic equation

The polynomial : P(x)=a_0x^4+a_1x^3+a_2x^2+a_1 m x+a_0 m^2 is almost
palindromic A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a plan, a canal – Panam ...
, as (it is palindromic if ). The change of variables in produces the
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quadr ...
. Since , the quartic equation may be solved by applying 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, gr ...
twice.


Solution methods


Converting to a depressed quartic

For solving purposes, it is generally better to convert the quartic into a depressed quartic by the following simple change of variable. All formulas are simpler and some methods work only in this case. The roots of the original quartic are easily recovered from that of the depressed quartic by the reverse change of variable. Let : a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 be the general quartic equation we want to solve. Dividing by , provides the equivalent equation , with , , , and . Substituting for gives, after regrouping the terms, the equation , where :\begin p&=\frac =\frac\\ q&=\frac =\frac\\ r&=\frac=\frac. \end If is a root of this depressed quartic, then (that is is a root of the original quartic and every root of the original quartic can be obtained by this process.


Ferrari's solution

As explained in the preceding section, we may start with the ''depressed quartic equation'' : y^4 + p y^2 + q y + r = 0. This depressed quartic can be solved by means of a method discovered by
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 ...
. The depressed equation may be rewritten (this is easily verified by expanding the square and regrouping all terms in the left-hand side) as : \left(y^2 + \frac p2\right)^2 = -q y - r + \frac4. Then, we introduce a variable into the factor on the left-hand side by adding to both sides. After regrouping the coefficients of the power of on the right-hand side, this gives the equation which is equivalent to the original equation, whichever value is given to . As the value of may be arbitrarily chosen, we will choose it in order to complete the square on the right-hand side. This implies that the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
in of this
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quadr ...
is zero, that is is a root of the equation : (-q)^2 - 4 (2m)\left(m^2 + p m + \frac4 - r\right) = 0,\, which may be rewritten as This is the
resolvent cubic In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: :P(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0. In each case: * The coefficients of the resolvent cubic can be obta ...
of the quartic equation. The value of may thus be obtained from Cardano's formula. When is a root of this equation, the right-hand side of equation (') is the square :\left(\sqrty-\frac q\right)^2. However, this induces a division by zero if . This implies , and thus that the depressed equation is bi-quadratic, and may be solved by an easier method (see above). This was not a problem at the time of Ferrari, when one solved only explicitly given equations with numeric coefficients. For a general formula that is always true, one thus needs to choose a root of the cubic equation such that . This is always possible except for the depressed equation . Now, if is a root of the cubic equation such that , equation (') becomes : \left(y^2 + \frac p2 + m\right)^2 = \left(y\sqrt-\frac\right)^2. This equation is of the form , which can be rearranged as or . Therefore, equation (') may be rewritten as : \left(y^2 + \frac p2 + m + \sqrty-\frac q\right) \left(y^2 + \frac p2 + m - \sqrty+\frac q\right)=0. This equation is easily solved by applying to each factor 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, gr ...
. Solving them we may write the four roots as :y=, where and denote either or . As the two occurrences of must denote the same sign, this leaves four possibilities, one for each root. Therefore, the solutions of the original quartic equation are :x=- + . A comparison with the general formula above shows that .


Descartes' solution

Descartes introduced in 1637 the method of finding the roots of a quartic polynomial by factoring it into two quadratic ones. Let : \begin x^4 + bx^3 + cx^2 + dx + e & = (x^2 + sx + t)(x^2 + ux + v) \\ & = x^4 + (s + u)x^3 + (t + v + su)x^2 + (sv + tu)x + tv \end By
equating coefficients In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters. It relies on the fact that two expressions are identical precisely when co ...
, this results in the following system of equations: : \left\{\begin{array}{l} b = s + u \\ c = t + v + su \\ d = sv + tu \\ e = tv \end{array}\right. This can be simplified by starting again with the depressed quartic , which can be obtained by substituting for . Since the coefficient of is , we get , and: : \left\{\begin{array}{l} p + u^2 = t + v \\ q = u (t - v) \\ r = tv \end{array}\right. One can now eliminate both and by doing the following: : \begin{align} u^2(p + u^2)^2 - q^2 & = u^2(t + v)^2 - u^2(t - v)^2 \\ & = u^2
t + v + (t - v))(t + v - (t - v)) T, or t, is the twentieth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''tee'' (pronounced ), plural ''tees''. It is de ...
\ & = u^2(2t)(2v) \\ & = 4u^2tv \\ & = 4u^2r \end{align} If we set , then solving this equation becomes finding the roots of the
resolvent cubic In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: :P(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0. In each case: * The coefficients of the resolvent cubic can be obta ...
which is done elsewhere. This resolvent cubic is equivalent to the resolvent cubic given above (equation (1a)), as can be seen by substituting U = 2m. If is a square root of a non-zero root of this resolvent (such a non-zero root exists except for the quartic , which is trivially factored), : \left\{\begin{array}{l} s = -u \\ 2t = p + u^2 + q/u \\ 2v = p + u^2 - q/u \end{array}\right. The symmetries in this solution are as follows. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of for the square root of merely exchanges the two quadratics with one another. The above solution shows that a quartic polynomial with rational coefficients and a zero coefficient on the cubic term is factorable into quadratics with rational coefficients if and only if either the resolvent cubic (') has a non-zero root which is the square of a rational, or is the square of rational and ; this can readily be checked using 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 ...
.


Euler's solution

A variant of the previous method is due to
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in m ...
. Unlike the previous methods, both of which use ''some'' root of the resolvent cubic, Euler's method uses all of them. Consider a depressed quartic . Observe that, if * , * and are the roots of , * and are the roots of , then * the roots of are , , , and , * , * . Therefore, . In other words, is one of the roots of the resolvent cubic (') and this suggests that the roots of that cubic are equal to , , and . This is indeed true and it follows 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 ...
. It also follows from Vieta's formulas, together with the fact that we are working with a depressed quartic, that . (Of course, this also follows from the fact that .) Therefore, if , , and are the roots of the resolvent cubic, then the numbers , , , and are such that :\left\{\begin{array}{l}r_1+r_2+r_3+r_4=0\\(r_1+r_2)(r_3+r_4)=-\alpha\\(r_1+r_3)(r_2+r_4)=-\beta\\(r_1+r_4)(r_2+r_3)=-\gamma\text{.}\end{array}\right. It is a consequence of the first two equations that is a square root of and that is the other square root of . For the same reason, * is a square root of , * is the other square root of , * is a square root of , * is the other square root of . Therefore, the numbers , , , and are such that :\left\{\begin{array}{l}r_1+r_2+r_3+r_4=0\\r_1+r_2=\sqrt{\alpha}\\r_1+r_3=\sqrt{\beta}\\r_1+r_4=\sqrt{\gamma}\text{;}\end{array}\right. the sign of the square roots will be dealt with below. The only solution of this system is: :\left\{\begin{array}{l}r_1=\frac{\sqrt{\alpha}+\sqrt{\beta}+\sqrt{\gamma2\\ mm_2=\frac{\sqrt{\alpha}-\sqrt{\beta}-\sqrt{\gamma2\\ mm_3=\frac{-\sqrt{\alpha}+\sqrt{\beta}-\sqrt{\gamma2\\ mm_4=\frac{-\sqrt{\alpha}-\sqrt{\beta}+\sqrt{\gamma2\text{.}\end{array}\right. Since, in general, there are two choices for each square root, it might look as if this provides choices for the set }, but, in fact, it provides no more than  such choices, because the consequence of replacing one of the square roots by the symmetric one is that the set } becomes the set }. In order to determine the right sign of the square roots, one simply chooses some square root for each of the numbers , , and and uses them to compute the numbers , , , and from the previous equalities. Then, one computes the number . Since , , and are the roots of ('), it is a consequence of Vieta's formulas that their product is equal to and therefore that . But a straightforward computation shows that : If this number is , then the choice of the square roots was a good one (again, by Vieta's formulas); otherwise, the roots of the polynomial will be , , , and , which are the numbers obtained if one of the square roots is replaced by the symmetric one (or, what amounts to the same thing, if each of the three square roots is replaced by the symmetric one). This argument suggests another way of choosing the square roots: * pick ''any'' square root of and ''any'' square root of ; * ''define'' as -\frac q{\sqrt{\alpha}\sqrt{\beta. Of course, this will make no sense if or is equal to , but is a root of (') only when , that is, only when we are dealing with a biquadratic equation, in which case there is a much simpler approach.


Solving by Lagrange resolvent

The
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \ ...
on four elements has the
Klein four-group In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one ...
as a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
. This suggests using a whose roots may be variously described as a discrete Fourier transform or a
Hadamard matrix In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in ...
transform of the roots; see
Lagrange resolvents In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rati ...
for the general method. Denote by , for from  to , the four roots of . If we set : \begin{align} s_0 &= \tfrac12(x_0 + x_1 + x_2 + x_3), \\ pts_1 &= \tfrac12(x_0 - x_1 + x_2 - x_3), \\ pts_2 &= \tfrac12(x_0 + x_1 - x_2 - x_3), \\ pts_3 &= \tfrac12(x_0 - x_1 - x_2 + x_3), \end{align} then since the transformation is an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
we may express the roots in terms of the four in exactly the same way. Since we know the value , we only need the values for , and . These are the roots of the polynomial :(s^2 - {s_1}^2)(s^2-{s_2}^2)(s^2-{s_3}^2). Substituting the by their values in term of the , this polynomial may be expanded in a polynomial in whose coefficients are
symmetric polynomial In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one ha ...
s in the . By the
fundamental theorem of symmetric polynomials In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sym ...
, these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If, for simplification, we suppose that the quartic is depressed, that is , this results in the polynomial This polynomial is of degree six, but only of degree three in , and so the corresponding equation is solvable by the method described in the article about
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 ...
. By substituting the roots in the expression of the in terms of the , we obtain expression for the roots. In fact we obtain, apparently, several expressions, depending on the numbering of the roots of the cubic polynomial and of the signs given to their square roots. All these different expressions may be deduced from one of them by simply changing the numbering of the . These expressions are unnecessarily complicated, involving the cubic roots of unity, which can be avoided as follows. If is any non-zero root of ('), and if we set :\begin{align} F_1(x) & = x^2 + sx + \frac{c}{2} + \frac{s^2}{2} - \frac{d}{2s} \\ F_2(x) & = x^2 - sx + \frac{c}{2} + \frac{s^2}{2} + \frac{d}{2s} \end{align} then :F_1(x)\times F_2(x) = x^4 + cx^2 + dx + e. We therefore can solve the quartic by solving for and then solving for the roots of the two factors 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, gr ...
. This gives exactly the same formula for the roots as the one provided by Descartes' method.


Solving with algebraic geometry

There is an alternative solution using algebraic geometry In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three reducible quadratic curves (pairs of lines) that pass through these points (this corresponds to the resolvent cubic, the pairs of lines being the Lagrange resolvents), and then use these linear equations to solve the quadratic. The four roots of the depressed quartic may also be expressed as the coordinates of the intersections of the two quadratic equations and i.e., using the substitution that two quadratics intersect in four points is an instance of
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
. Explicitly, the four points are for the four roots of the quartic. These four points are not collinear because they lie on the irreducible quadratic and thus there is a 1-parameter family of quadratics (a
pencil of curves In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane. Although the definiti ...
) passing through these points. Writing the projectivization of the two quadratics as
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s in three variables: :\begin{align} F_1(X,Y,Z) &:= Y^2 + pYZ + qXZ + rZ^2,\\ F_2(X,Y,Z) &:= YZ - X^2 \end{align} the pencil is given by the forms for any point in the projective line — in other words, where and are not both zero, and multiplying a quadratic form by a constant does not change its quadratic curve of zeros. This pencil contains three reducible quadratics, each corresponding to a pair of lines, each passing through two of the four points, which can be done \textstyle{\binom{4}{2 =  different ways. Denote these , , and . Given any two of these, their intersection has exactly the four points. The reducible quadratics, in turn, may be determined by expressing the quadratic form as a  matrix: reducible quadratics correspond to this matrix being singular, which is equivalent to its determinant being zero, and the determinant is a homogeneous degree three polynomial in and and corresponds to the resolvent cubic.


See also

* * * *


References


Further reading

* *


External links

*
Ferrari's achievement
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