In algebra, a quartic function is a function of the form
where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial.
A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form
where . The derivative of a quartic function is a cubic function.
Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form
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. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. 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.
History Lodovico Ferrari 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this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately.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 in the book Ars Magna.
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 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois prior to dying in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result.Stewart, Ian, Galois Theory, Third Edition (Chapman & Hall/CRC Mathematics, 2004)
Applications Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus. It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics. Here are examples of other geometric problems whose solution involves solving a quartic equation.
In computer-aided manufacturing, the torus is a shape that is commonly associated with the endmill 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, 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 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 eigenvalues of a 4×4 matrix are the roots of a quartic polynomial which is the characteristic polynomial of the matrix.
The characteristic equation of a fourth-order linear difference equation or differential equation is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending.
Intersections between spheres, cylinders, or other quadrics can be found using quartic equations.
Inflection points and golden ratio Letting and be the distinct inflection points of the graph of a quartic function, and letting be the intersection of the inflection secant line and the quartic, nearer to than to , then divides into the golden section:
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
with real coefficients and the nature of its roots is mainly determined by the sign of its discriminant
This may be refined by considering the signs of four other polynomials:
such that is the second degree coefficient of the associated depressed quartic (see below);
such that is the first degree coefficient of the associated depressed quartic;
which is 0 if the quartic has a triple root; and
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 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, but in fact they cannot occur. For example, , = 0 and ≤ 0 is not a possible case. In fact, if and = 0 then > 0, since so this combination is not possible.
General formula for roots
The four roots , , , and for the general quartic equation
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 equations.
where and are the coefficients of the second and of the first degree respectively in the associated depressed quartic
and where
(if or , see , below)
with
and where is the aforementioned discriminant. 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 the value of 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 is also real, despite being expressed in terms of 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, as follows:
where
If and the sign of has to be chosen to have that is one should define as maintaining the sign of If then one must change the choice of the cube root in in order to have This is always possible except if the quartic may be factored into 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 of is zero, in which case the associated depressed quartic is biquadratic; it may thus be solved by the method described below. If and and thus also at least three roots are equal to each other, and the roots are rational functions of the coefficients. The triple root is a common root of the quartic and its second derivative it is thus also the unique root of the remainder of the Euclidean division of the quartic by its second derivative, which is a linear polynomial. The simple root can be deduced from If and 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
It is reducible if , where and are non-constant polynomials with rational coefficients (or more generally with coefficients in the same field as the coefficients of ). Such a factorization will take one of two forms:
or
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 or cubic function.
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) 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) are based upon finding such factorizations.
Biquadratic equation If then the function
is called a biquadratic function; equating it to zero defines a biquadratic equation, which is easy to solve as follows
Let the auxiliary variable . Then becomes a quadratic in : . Let and be the roots of . Then the roots of the quartic are
Quasi-palindromic equation The polynomial
is almost palindromic, as (it is palindromic if ). The change of variables in produces the quadratic equation . Since , the quartic equation may be solved by applying the quadratic formula 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
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
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
This depressed quartic can be solved by means of a method discovered by Lodovico Ferrari. The depressed equation may be rewritten (this is easily verified by expanding the square and regrouping all terms in the left-hand side) as
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 of this quadratic equation is zero, that is is a root of the equation
which may be rewritten as
This is the resolvent cubic 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