In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
quartic function, or equation of the fourth degree, is a
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of the form
where
a is nonzero; or in other words, a
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
of
degreeThe degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
four. Such a function is sometimes called a
biquadratic function, but the latter term can occasionally also refer to a quadratic function of a square, having the form
or a product of two quadratic factors, having the form
Setting
results in a quartic
equationAn equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
of the form:
where
a ≠ 0.
The
derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of a quartic function is a
cubic functionIn mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
.
Since a quartic function is a polynomial of even degree, it has the same limit when the argument goes to positive or negative
infinityInfinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
. If
a is positive, then the function increases to positive infinity at both sides; and thus the function has a
global minimumIn mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...
. Likewise, if
a is negative, it decreases to negative infinity and has a global maximum.
The
quartic is the highest order polynomial equation that can be solved by
radicalsIn mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
in the general case (i.e., one where the coefficients can take any value).
History
Lodovico FerrariLodovico Ferrari was an Italian mathematician.Born in Milan, Italy, grandfather, Bartholomew Ferrari was forced out of Milan to Bologna. He settled in Bologna, Italy and he began his career as the servant of Gerolamo Cardano. He was extremely bright, so Cardano started teaching him mathematics...
is attributed 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 to be found, it couldn't be published immediately. The solution of the quartic was published together with that of the cubic by Ferrari's mentor
Gerolamo CardanoGerolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler...
in the book
Ars MagnaThe Ars Magna is an important book on Algebra written by Gerolamo Cardano. It was first published in 1545 under the title Artis Magnæ, Sive de Regulis Algebraicis Liber Unus . There was a second edition in Cardano's lifetime, published in 1570...
(1545).
It is reported that even earlier, in 1486, Spanish mathematician Paolo Valmes was
burned at the stakeBurned at the Stake is a 1981 film directed by Bert I. Gordon. It stars Susan Swift and Albert Salmi.-Cast:*Susan Swift as Loreen Graham / Ann Putnam*Albert Salmi as Captaiin Billingham*Guy Stockwell as Dr. Grossinger*Tisha Sterling as Karen Graham...
for claiming to have solved the quartic equation. Inquisitor General
Tomás de TorquemadaTomás de Torquemada, O.P. was a fifteenth century Spanish Dominican friar, first Inquisitor General of Spain, and confessor to Isabella I of Castile. He was described by the Spanish chronicler Sebastián de Olmedo as "The hammer of heretics, the light of Spain, the saviour of his country, the...
allegedly told him that
it was the will of God that such a solution be inaccessible to human understanding. However, attempts to find corroborating evidence for this story, or for the existence of Paolo Valmes, have not succeeded.
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 theoremIn algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...
in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by
Évariste GaloisÉvariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
prior to dying in a duel in 1832 later led to an elegant
complete theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
of the roots of polynomials, of which this theorem was one result.
Applications
Polynomials of high degrees often appear in problems involving
optimizationIn mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
, and sometimes these polynomials happen to be quartics, but this is a coincidence.
Quartics often arise in
computer graphicsComputer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
and during ray-tracing against surfaces such as
quadricIn mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
or
toriIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
surfaces, which are the next level beyond the
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
and
developable surfaceIn mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is a "surface" that can be flattened onto a plane without distortion . Conversely, it is a surface which can be made by transforming a plane...
s.
Another frequent generator of quartics is the intersection of two ellipses.
In
computer-aided manufacturingComputer-aided manufacturing is the use of computer software to control machine tools and related machinery in the manufacturing of workpieces. This is not the only definition for CAM, but it is the most common; CAM may also refer to the use of a computer to assist in all operations of a...
, the torus is a common shape associated with the
endmillAn endmill 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...
cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the Z-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. Over 10% of the computational time in a CAM system can be consumed simply calculating the solution to millions of quartic equations.
A program demonstrating various analytic solutions to the quartic was provided in Graphics Gems Book V.
However, none of the three algorithms implemented are unconditionally stable.
In an updated version of the paper, which compares the 3 algorithms from the original paper and 2 others, it is demonstrated that computationally stable solutions exist only for 4 of the possible 16 sign combinations of the quartic coefficients.
Degenerate case
If then , and so is a solution. It follows that
Q(
x) may be factorised as The remaining three roots – see
Fundamental Theorem of AlgebraThe fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
– can be found by solving the
cubic equationIn mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
.
Evident roots: 1 and −1 and −k
If
then
,
so
is a root.
Similarly, if
that is,
then
is a root.
When
is a root, we can divide
by
and get
where
is a
cubic polynomial,
which may be solved to find
's other roots.
Similarly, if
is a root,
where
is some cubic polynomial.
If
then −
k is a root
and we can factor out
,
And if
then both
and
are roots
Now we can factor out
and get
To get
Q 's other roots, we simply solve the quadratic factor.
Biquadratic equations
If
then
We call such a polynomial a
biquadratic, which is easy to solve.
Let
Then
Q becomes a
quadraticA quadratic function, in mathematics, is a polynomial function of the formf=ax^2+bx+c,\quad a \ne 0.The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis....
q in
Let
and
be the roots of
q.
Then the roots of our quartic
Q are
Quasi-symmetric equations
-
Steps:
1) Divide by
x 2.
2) Use variable change
z =
x +
m/
x.
The general case, along Ferrari's lines
To begin, the quartic must first be converted to a
depressed quartic.
Converting to a depressed quartic
Let
be the general quartic equation we want to solve. Divide both sides by
A to produce a monic polynomial,
The first step should be to eliminate the
x3 term. To do this, change variables from
x to
u, such that
.
Then
Expanding the powers of the binomials produces
Collecting the same powers of
u yields
Now rename the coefficients of
u. Let
The resulting equation is
which is a
depressed quartic equation.
If
then we have a biquadratic equation, which (as explained above) is easily solved; using reverse substitution we can find our values for
.
If
then one of the roots is
and the other roots can be found by dividing by
, and solving the resulting depressed cubic equation,
Using reverse substitution we can find our values for
.
Ferrari's solution
Otherwise, the depressed quartic can be solved by means of a method discovered by
Lodovico FerrariLodovico Ferrari was an Italian mathematician.Born in Milan, Italy, grandfather, Bartholomew Ferrari was forced out of Milan to Bologna. He settled in Bologna, Italy and he began his career as the servant of Gerolamo Cardano. He was extremely bright, so Cardano started teaching him mathematics...
. Once the depressed quartic has been obtained, the next step is to add the valid identity
to equation (1), yielding
The effect has been to fold up the
u4 term into a
perfect squareIn mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
: (
u2 + α)
2. The second term, α
u2 did not disappear, but its sign has changed and it has been moved to the right side.
The next step is to insert a variable
y into the perfect square on the left side of equation (2), and a corresponding 2
y into the coefficient of
u2 in the right side. To accomplish these insertions, the following valid formulas will be added to equation (2),
and
These two formulas, added together, produce
which added to equation (2) produces
This is equivalent to
The objective now is to choose a value for
y such that the right side of equation (3) becomes a perfect square. This can be done by letting the discriminant of the quadratic function become zero. To explain this, first expand a perfect square so that it equals a quadratic function:
The quadratic function on the right side has three coefficients. It can be verified that squaring the second coefficient and then subtracting four times the product of the first and third coefficients yields zero:
Therefore to make the right side of equation (3) into a perfect square, the following equation must be solved:
Multiply the binomial with the polynomial,
Divide both sides by −4, and move the −
β2/4 to the right,
This is a cubic equation for
y. Divide both sides by 2,
Conversion of the nested cubic into a depressed cubic
Equation (4) is a cubic equation nested within the quartic equation. It must be solved to solve the quartic. To solve the cubic, first transform it into a depressed cubic by means of the substitution
Equation (4) becomes
Expand the powers of the binomials,
Distribute, collect like powers of
v, and cancel out the pair of
v2 terms,
This is a depressed cubic equation.
Relabel its coefficients,
The depressed cubic now is
Solving the nested depressed cubic
The solutions (any solution will do, so pick any of the three complex roots) of equation (5) are computed as (see Cubic equation)
where
-
-
and
V is computed according to the two defining equations
and
, so
-
-
Folding the second perfect square
With the value for y given by equation (6), it is now known that the right side of equation (3) is a perfect square of the form
-
- (This is correct for both signs of square root, as long as the same sign is taken for both square roots. A ± is redundant, as it would be absorbed by another ± a few equations further down this page.)
so that it can be folded:
.
-
- Note: If β ≠ 0 then α + 2y ≠ 0. If β = 0 then this would be a biquadratic equation, which we solved earlier.
Therefore equation (3) becomes
.
Equation (7) has a pair of folded perfect squares, one on each side of the equation. The two perfect squares balance each other.
If two squares are equal, then the sides of the two squares are also equal, as shown by:
.
Collecting like powers of u produces
.
-
- Note: The subscript s of
and 
is to note that they are dependent.
Equation (8) is a quadratic equationIn mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
for u. Its solution is
Simplifying, one gets
This is the solution of the depressed quartic, therefore the solutions of the original quartic equation are
-
- Remember: The two
come from the same place in equation (7'), and should both have the same sign, while the sign of 
is independent.
Summary of Ferrari's method
Given the quartic equation
its solution can be found by means of the following calculations:
If 
then
-
Otherwise, continue with
(either sign of the square root will do)
(there are 3 complex roots, any one of them will do)
Ferrari was the first to discover one of these labyrinthIn Greek mythology, the Labyrinth was an elaborate structure designed and built by the legendary artificer Daedalus for King Minos of Crete at Knossos...
ine solutions. The equation he solved was:
which was already in depressed form. It has a pair of solutions that can be found with the set of formulas shown above.
Ferrari's solution in the special case of real coefficients
If the coefficients of the quartic equation are real then the nested depressed cubic equation (5) also has real coefficients, thus it has at least one real root.
Furthermore the cubic functionIn mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
where P and Q are given by (5) has the properties that
and
where α and β are given by (1).
This means that (5) has a real root greater than 
,
and therefore that (4) has a real root greater than 
.
Using this root the term 
in (8) is always real, which ensures that the two quadratic equations (8) have real coefficients.
Obtaining alternative solutions by factoring out complex conjugate solutions
It could happen that one only obtained one solution through the seven formulae above, because not all four sign patterns are tried for four solutions, and the solution obtained is complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
. It may also be the case that one is only looking for a real solution. Let x1 denote the complex solution. If all the original coefficients A, B, C, D and E are real — which should be the case when one desires only real solutions — then there is another complex solution x2, which is the complex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of x1. If the other two roots are denoted as x3 and x4 then the quartic equation can be expressed as
but this quartic equation is equivalent to the product of two quadratic equations:
and
Since
then
Let
so that equation (9) becomes
Also let there be (unknown) variables w and v such that equation (10) becomes
Multiplying equations (11) and (12) produces
Comparing equation (13) to the original quartic equation, it can be seen that
and
Therefore
Equation (12) can be solved for x yielding
These two solutions are the desired real solutions if real solutions exist.
Factorization into quadratics
One can solve a quartic by factoring it into a product of two quadraticsIn mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
. Let
By equating coefficients, this results in the following set of simultaneous equations:
This can be simplified by starting again with a depressed quartic where 
, which can be obtained by substituting 
for 
, then 
, and:
It's now easy to eliminate both 
and 
by doing the following:
If we set 
, then this equation turns into the resolvent cubic equation
which is solved elsewhere. Then:
The symmetries in this solution are easy to see. 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 the quartic polynomial with a zero coefficient on the cubic term is factorable into quadratics with rational coefficients if and only if the resolvent cubic 
has a root which is the square of a rational; this can readily be checked using the rational root test.
Galois theory and factorization
The symmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
S4 on four elements has the Klein four-groupIn mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
as a normal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
. This suggests using a whose roots may be variously described as a discrete Fourier transform or a Hadamard matrixIn mathematics, an 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...
transform of the roots; see Lagrange resolvents for the general method. Suppose ri for i from 0 to 3 are roots of
If we now set
then since the transformation is an involution we may express the roots in terms of the four si in exactly the same way. Since we know the value s0 = -b/2, we really only need the values for s1, s2 and s3. These we may find by expanding the polynomial
which if we make the simplifying assumption that b=0, is equal to
This polynomial is of degree six, but only of degree three in z2, and so the corresponding equation is solvable. By trial we can determine which three roots are the correct ones, and hence find the solutions of the quartic.
We can remove any requirement for trial by using a root of the same resolvent polynomial for factoring; if w is any root of (3), and if
then
We therefore can solve the quartic by solving for w and then solving for the roots of the two factors using the quadratic formula.
Algebraic geometry
An alternative solution using algebraic geometry is given in , and proceeds as follows (more detailed discussion in reference). In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three reducible quadratic curvesIn mathematics, a degenerate conic is a conic that fails to be an irreducible curve...
(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 x coordinates of the intersections of the two quadratic equations 
i.e., using the substitution 
that two quadratics intersect in four points is an instance of Bézout's theoremBézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...
. 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) passing through these points. Writing the projectivization of the two quadratics as quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s in three variables:
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 
different ways. Denote these 
Given any two of these, their intersection is exactly the four points.
The reducible quadratics, in turn, may be determined by expressing the quadratic form 
as a 3×3 matrix: reducible quadratics correspond to this matrix being singular, which is a 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
- Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
- Quadratic function
A quadratic function, in mathematics, is a polynomial function of the formf=ax^2+bx+c,\quad a \ne 0.The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis....
- Cubic function
In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
- Quintic function
- Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
- Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
External links