An explosion is a rapid expansion in volume of a given amount of matter associated with an extreme outward release of energy, usually with the generation of high temperatures and release of high-pressure gases.
Explosions may also be generated by a slower expansion that would
normally not be forceful, but is not allowed to expand, so that when
whatever is containing the expansion is broken by the pressure that
builds as the matter inside tries to expand, the matter expands
forcefully. An example of this is a volcanic eruption created by the expansion of magma in a magma chamber as it rises to the surface. Supersonic explosions created by high explosives are known as detonations and travel through shock waves. Subsonic explosions are created by low explosives through a slower combustion process known as deflagration.
Causes
Explosions can occur in nature due to large influxes of energy. There are numerous ways explosions can occur naturally, such as volcanic or stellar processes of various sorts. Explosive volcanic eruptions occur when magma rises from below, it has dissolved gas in it. The reduction of pressure
as the magma rises causes the gas to bubble out of solution, resulting
in a rapid increase in volume. Explosions also occur as a result of impact events and in phenomena such as hydrothermal explosions (also due to volcanic processes). Explosions can also occur outside of Earth in the universe in events such as supernovae, or, more commonly, stellar flares. Explosions frequently occur during bushfires in eucalyptus forests where the volatile oils in the tree tops suddenly combust.
Astronomical
Among the largest known explosions in the universe are supernovae, which occur after the end of life of some types of stars. Solar flares
are an example of common, much less energetic, explosions on the Sun,
and presumably on most other stars as well. The energy source for solar
flare activity comes from the tangling of magnetic field lines resulting
from the rotation of the Sun's conductive plasma. Another type of large
astronomical explosion occurs when a very large meteoroid or an
asteroid impacts the surface of another object, such as a planet. For
example, the Tunguska event of 1908 is believed to have resulted from a meteor air burst.
Black hole mergers, likely involving binary black hole systems, are capable of radiating many solar masses of energy into the universe in a fraction of a second, in the form of a gravitational wave.
This is capable of transmitting ordinary energy and destructive forces
to nearby objects, but in the vastness of space, nearby objects are
rare. The gravitational wave observed on 21 May 2019, known as GW190521,
produced a merger signal of about 100 ms duration, during which time is
it estimated to have radiated away nine solar masses in the form of
gravitational energy.
Chemical
The most common artificial explosives are chemical explosives, usually involving a rapid and violent oxidation
reaction that produces large amounts of hot gas. Gunpowder was the
first explosive to be invented and put to use. Other notable early
developments in chemical explosive technology were Frederick Augustus Abel's development of nitrocellulose in 1865 and Alfred Nobel's invention of dynamite
in 1866. Chemical explosions (both intentional and accidental) are
often initiated by an electric spark or flame in the presence of oxygen.
Accidental explosions may occur in fuel tanks, rocket engines, etc.
Electrical and magnetic
A high current electrical fault can create an "electrical explosion" by forming a high-energy electrical arc which rapidly vaporizes metal and insulation material. This arc flash hazard is a danger to people working on energized switchgear. Excessive magnetic pressure within an ultra-strong electromagnet can cause a magnetic explosion.
Mechanical and vapor
Strictly
a physical process, as opposed to chemical or nuclear, e.g., the
bursting of a sealed or partially sealed container under internal
pressure is often referred to as an explosion. Examples include an
overheated boiler or a simple tin can of beans tossed into a fire.
Boiling liquid expanding vapor explosions
are one type of mechanical explosion that can occur when a vessel
containing a pressurized liquid is ruptured, causing a rapid increase in
volume as the liquid evaporates. Note that the contents of the
container may cause a subsequent chemical explosion, the effects of
which can be dramatically more serious, such as a propane
tank in the midst of a fire. In such a case, to the effects of the
mechanical explosion when the tank fails are added the effects from the
explosion resulting from the released (initially liquid and then almost
instantaneously gaseous) propane in the presence of an ignition source.
For this reason, emergency workers often differentiate between the two
events.
In addition to stellar nuclear explosions, a nuclear weapon is a type of explosive weapon that derives its destructive force from nuclear fission
or from a combination of fission and fusion. As a result, even a
nuclear weapon with a small yield is significantly more powerful than
the largest conventional explosives available, with a single weapon
capable of completely destroying an entire city.
Properties
Force
Explosive force is released in a direction perpendicular to the
surface of the explosive. If a grenade is in mid air during the
explosion, the direction of the blast will be 360°. In contrast, in a shaped charge
the explosive forces are focused to produce a greater local explosion;
shaped charges are often used by military to breach doors or walls.
Velocity
The
speed of the reaction is what distinguishes an explosive reaction from
an ordinary combustion reaction. Unless the reaction occurs very
rapidly, the thermally expanding gases will be moderately dissipated in
the medium, with no large differential in pressure and no explosion. As a
wood fire burns in a fireplace, for example, there certainly is the
evolution of heat and the formation of gases, but neither is liberated
rapidly enough to build up a sudden substantial pressure differential
and then cause an explosion. This can be likened to the difference
between the energy discharge of a battery, which is slow, and that of a flash capacitor like that in a camera flash, which releases its energy all at once.
Evolution of heat
The generation of heat in large quantities accompanies most explosive chemical reactions. The exceptions are called entropic explosives and include organic peroxides such as acetone peroxide. It is the rapid liberation of heat that causes the gaseous products of most explosive reactions to expand and generate high pressures.
This rapid generation of high pressures of the released gas constitutes
the explosion. The liberation of heat with insufficient rapidity will
not cause an explosion. For example, although a unit mass of coal yields
five times as much heat as a unit mass of nitroglycerin, the coal cannot be used as an explosive (except in the form of coal dust) because the rate at which it yields this heat is quite slow. In fact, a substance that burns less rapidly (i.e. slow combustion) may actually evolve more total heat than an explosive that detonates rapidly (i.e. fast combustion). In the former, slow combustion converts more of the internal energy (i.e.chemical potential) of the burning substance into heat released to the surroundings, while in the latter, fast combustion (i.e.detonation) instead converts more internal energy into work on the surroundings (i.e. less internal energy converted into heat); c.f.heat and work (thermodynamics) are equivalent forms of energy. See Heat of Combustion for a more thorough treatment of this topic.
When a chemical compound is formed from its constituents, heat
may either be absorbed or released. The quantity of heat absorbed or
given off during transformation is called the heat of formation.
Heats of formations for solids and gases found in explosive reactions
have been determined for a temperature of 25 °C and atmospheric
pressure, and are normally given in units of kilojoules per
gram-molecule. A positive value indicates that heat is absorbed during
the formation of the compound from its elements; such a reaction is
called an endothermic reaction. In explosive technology only materials
that are exothermic—that
have a net liberation of heat and have a negative heat of formation—are
of interest. Reaction heat is measured under conditions either of
constant pressure or constant volume. It is this heat of reaction that
may be properly expressed as the "heat of explosion."
Initiation of reaction
A
chemical explosive is a compound or mixture which, upon the application
of heat or shock, decomposes or rearranges with extreme rapidity,
yielding much gas and heat. Many substances not ordinarily classed as
explosives may do one, or even two, of these things.
A reaction must be capable of being initiated by the application of shock, heat, or a catalyst
(in the case of some explosive chemical reactions) to a small portion
of the mass of the explosive material. A material in which the first
three factors exist cannot be accepted as an explosive unless the
reaction can be made to occur when needed.
Fragmentation
Fragmentation
is the accumulation and projection of particles as the result of a high
explosives detonation. Fragments could originate from: parts of a
structure (such as glass, bits of structural material, or roofing material), revealed strata and/or various surface-level geologic features (such as loose rocks, soil, or sand),
the casing surrounding the explosive, and/or any other loose
miscellaneous items not vaporized by the shock wave from the explosion.
High velocity, low angle fragments can travel hundreds of metres with
enough energy to initiate other surrounding high explosive items, injure
or kill personnel, and/or damage vehicles or structures.
Classical Latin explōdō means "to hiss a bad actor off the stage", "to drive an actor off the stage by making noise", from ex- ("out") + plaudō ("to clap; to applaud"). The modern meaning developed later:
Classical Latin: "to drive an actor off the stage by making noise" hence meaning "to drive out" or "to reject"
In English:
Around 1538: "drive out or off by clapping" (originally theatrical)
Around 1660: "drive out with violence and sudden noise"
Around 1790: "go off with a loud noise"
Around 1882: first use as "bursting with destructive force"
In algebra, a cubic equation in one variable is an equation of the form
in which a 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 the coefficientsa, b, c, and d of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:
The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic
other than 2 and 3. The solutions of the cubic equation do not
necessarily belong to the same field as the coefficients. For example,
some cubic equations with rational coefficients have roots that are
irrational (and even non-real) complex numbers.
History
Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians. Babylonian (20th to 16th centuries BC) cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did. The problem of doubling the cube
involves the simplest and oldest studied cubic equation, and one for
which the ancient Egyptians did not believe a solution existed. In the 5th century BC, Hippocrates
reduced this problem to that of finding two mean proportionals between
one line and another of twice its length, but could not solve this with a
compass and straightedge construction, a task which is now known to be impossible. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui in the 3rd century. In the 3rd century AD, the Greek mathematicianDiophantus found integer or rational solutions for some bivariate cubic equations (Diophantine equations). Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections,
though historians such as Reviel Netz dispute whether the Greeks were
thinking about cubic equations or just problems that can lead to cubic
equations. Some others like T. L. Heath,
who translated all of Archimedes' works, disagree, putting forward
evidence that Archimedes really solved cubic equations using
intersections of two conics, but also discussed the conditions where the roots are 0, 1 or 2.
In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form x3 + px2 + qx = N, 23 of them with p, q ≠ 0, and two of them with q = 0.
In the 11th century, the Persian poet-mathematician, Omar Khayyam
(1048–1131), made significant progress in the theory of cubic
equations. In an early paper, he discovered that a cubic equation can
have more than one solution and stated that it cannot be solved using
compass and straightedge constructions. He also found a geometric
solution. In his later work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections. Khayyam made an attempt to come up with an algebraic formula for extracting cubic roots. He wrote:
“We have tried to express these roots by algebra but have failed. It may be, however, that men who come after us will succeed.”
In the 12th century, the Indian mathematician Bhaskara II attempted
the solution of cubic equations without general success. However, he
gave one example of a cubic equation: x3 + 12x = 6x2 + 35. In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), wrote the Al-Muʿādalāt (Treatise on Equations),
which dealt with eight types of cubic equations with positive solutions
and five types of cubic equations which may not have positive
solutions. He used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also used the concepts of maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations.
In his book Flos, Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation x3 + 2x2 + 10x = 20. Writing in Babylonian numerals he gave the result as 1,22,7,42,33,4,40 (equivalent to 1 + 22/60 + 7/602 + 42/603 + 33/604 + 4/605 + 40/606), which has a relative error of about 10−9.
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 one allows m and n to be negative, but negative numbers
were not known to him at that time. Del Ferro kept his achievement
secret until just before his death, when he told his student Antonio
Fior about it.
In 1535, Niccolò Tartaglia (1500–1557) received two problems in cubic equations from Zuanne da Coi
and announced that he could solve them. He was soon challenged by Fior,
which led to a famous contest between the two. Each contestant had to
put up a certain amount of money and to propose a number of problems for
his rival to solve. Whoever solved more problems within 30 days would
get all the money. Tartaglia received questions in the form x3 + mx = n, for which he had worked out a general method. Fior 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. In 1539,
Tartaglia did so only on the condition that Cardano would never reveal
it and that if he did write a book about cubics, he would give Tartaglia
time to publish. Some years later, Cardano learned about del Ferro's
prior work and published del Ferro's method in his book Ars Magna
in 1545, meaning Cardano gave Tartaglia six years to publish his
results (with credit given to Tartaglia for an independent solution).
Cardano's promise to Tartaglia said that he would not publish
Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as
to get around the promise. Nevertheless, this led to a challenge to
Cardano from Tartaglia, which Cardano denied. The challenge was
eventually accepted by Cardano's student Lodovico Ferrari (1522–1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and his income.
Cardano noticed that Tartaglia's method sometimes required him to
extract the square root of a negative number. He even included a
calculation with these complex 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.
François Viète (1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and René Descartes (1596–1650) extended the work of Viète.
Factorization
If the coefficients of a cubic equation are rational numbers, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a common multiple of their denominators. Such an equation
with integer coefficients, is said to be reducible if the polynomial on the left-hand side is the product of polynomials of lower degrees. By Gauss's lemma, if the equation is reducible, one can suppose that the factors have integer coefficients.
Finding the roots of a reducible cubic equation is easier than
solving the general case. In fact, if the equation is reducible, one of
the factors must have degree one, and thus have the form
with q and p being coprime integers. The rational root test allows finding q and p by examining a finite number of cases (because q must be a divisor of a, and p must be a divisor of d).
Thus, one root is and the other roots are the roots of the other factor, which can be found by polynomial long division. This other factor is
(The coefficients seem not to be integers, but must be integers if p / q is a root.)
are said to be depressed. They are much simpler than general cubics,
but are fundamental, because the study of any cubic may be reduced by a
simple change of variable to that of a depressed cubic.
Let
be a cubic equation. The change of variable
gives a cubic (in t) that has no term in t2.
After dividing by a one gets the depressed cubic equation
with
The roots of the original equation are related to the roots of the depressed equation by the relations
for .
Discriminant and nature of the roots
The nature (real or not, distinct or not) of the roots of a cubic can be determined without computing them explicitly, by using the discriminant.
Discriminant
The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root,
or, if it is divisible by the square of a non-constant polynomial. In
other words, the discriminant is nonzero if and only if the polynomial
is square-free.
If r1, r2, r3 are the three roots (not necessarily distinct nor real) of the cubic then the discriminant is
The discriminant of the depressed cubic is
The discriminant of the general cubic is
It is the product of
and the discriminant of the corresponding depressed cubic. Using the
formula relating the general cubic and the associated depressed cubic,
this implies that the discriminant of the general cubic can be written
as
It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are real,
the two discriminants have the same sign. In summary, the same
information can be deduced from either one of these two discriminants.
To prove the preceding formulas, one can use Vieta's formulas to express everything as polynomials in r1, r2, r3, and a. The proof then results in the verification of the equality of two polynomials.
Nature of the roots
If the coefficients of a polynomial are real numbers, and its discriminant is not zero, there are two cases:
If the cubic has one real root and two non-real complex conjugate roots.
This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate
is also a root. So the non-real roots, if any, occur as pairs of
complex conjugate roots. As a cubic polynomial has three roots (not
necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.
As stated above, if r1, r2, r3 are the three roots of the cubic , then the discriminant is
If the three roots are real and distinct, the discriminant is a product of positive reals, that is
If only one root, say r1, is real, then r2 and r3 are complex conjugates, which implies that r2 – r3 is a purely imaginary number, and thus that (r2 – r3)2 is real and negative. On the other hand, r1 – r2 and r1 – r3 are complex conjugates, and their product is real and positive. Thus the discriminant is the product of a single negative number and several positive ones. That is
Multiple root
If the discriminant of a cubic is zero, the cubic has a multiple root. If furthermore its coefficients are real, then all of its roots are real.
The discriminant of the depressed cubic is zero if If p is also zero, then p = q = 0 , and 0 is a triple root of the cubic. If and p ≠ 0 , then the cubic has a simple root
and a double root
In other words,
This result can be proved by expanding the latter product or retrieved by solving the rather simple system of equations resulting from Vieta's formulas.
By using the reduction of a depressed cubic, these results can be extended to the general cubic. This gives: If the discriminant of the cubic is zero, then
either, if the cubic has a triple root
and
or, if the cubic has a double root
and a simple root,
and thus
Characteristic 2 and 3
The above results are valid when the coefficients belong to a field of characteristic other than 2 or 3, but must be modified for characteristic 2 or 3, because of the involved divisions by 2 and 3.
The reduction to a depressed cubic works for characteristic 2,
but not for characteristic 3. However, in both cases, it is simpler to
establish and state the results for the general cubic. The main tool for
that is the fact that a multiple root is a common root of the
polynomial and its formal derivative.
In these characteristics, if the derivative is not a constant, it is a
linear polynomial in characteristic 3, and is the square of a linear
polynomial in characteristic 2. Therefore, for either characteristic 2
or 3, the derivative has only one root. This allows computing the
multiple root, and the third root can be deduced from the sum of the
roots, which is provided by Vieta's formulas.
A difference with other characteristics is that, in
characteristic 2, the formula for a double root involves a square root,
and, in characteristic 3, the formula for a triple root involves a cube
root.
As shown in § Nature of the roots, the two other roots are non-real complex conjugate numbers, in this case. It was later shown (Cardano did not know complex numbers) that the two other roots are obtained by multiplying one of the cube roots by the primitive cube root of unity and the other cube root by the other primitive cube root of the unity That is, the other roots of the equation are and
If there are three real roots, but Galois theory allows proving that, if there is no rational root, the roots cannot be expressed by an algebraic expression
involving only real numbers. Therefore, the equation cannot be solved
in this case with the knowledge of Cardano's time. This case has thus
been called casus irreducibilis, meaning irreducible case in Latin.
In casus irreducibilis, Cardano's formula can still be
used, but some care is needed in the use of cube roots. A first method
is to define the symbols and as representing the principal values
of the root function (that is the root that has the largest real part).
With this convention Cardano's formula for the three roots remains
valid, but is not purely algebraic, as the definition of a principal
part is not purely algebraic, since it involves inequalities for
comparing real parts. Also, the use of principal cube root may give a
wrong result if the coefficients are non-real complex numbers. Moreover,
if the coefficients belong to another field, the principal cube root is not defined in general.
The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be –p / 3. It results that a root of the equation is
In this formula, the symbols and
denote any square root and any cube root. The other roots of the
equation are obtained either by changing of cube root or, equivalently,
by multiplying the cube root by a primitive cube root of unity, that is
This formula for the roots is always correct except when p = q = 0, with the proviso that if p = 0, the square root is chosen so that C ≠ 0.
However, the formula is useless in these cases as the roots can be
expressed without any cube root. Similarly, the formula is also useless
in the other cases where no cube root is needed, that is when and when the cubic polynomial is not irreducible.
This formula is also correct when p and q belong to any field of characteristic other than 2 or 3.
General cubic formula
A cubic formula for the roots of the general cubic equation (with a ≠ 0)
can be deduced from every variant of Cardano's formula by reduction to a depressed cubic. The variant that is presented here is valid not only for real coefficients, but also for coefficients a, b, c, d belonging to any field of characteristic different of 2 and 3.
The formula being rather complicated, it is worth splitting it in smaller formulas.
Let
(Both and can be expressed as resultants of the cubic and its derivatives: is −1/8a times the resultant of the cubic and its second derivative, and is −1/12a times the resultant of the first and second derivatives of the cubic polynomial.)
Then let
where the symbols and are interpreted as any square root and any cube root, respectively (every nonzero complex number has two square roots and three cubic roots). The sign "±" before the square root is either "+" or "–"; the choice is almost arbitrary, and changing it amounts to choosing a different square root. However, if a choice yields C = 0 (this occurs if ), then the other sign must be selected instead. If both choices yield C = 0, that is, if a fraction 0/0 occurs in following formulas; this fraction must be interpreted as equal to zero (see the end of this section).
With these conventions, one of the roots is
The other two roots can be obtained by changing the choice of the cube root in the definition of C, or, equivalently by multiplying C by a primitive cube root of unity, that is –1 ± √–3/2. In other words, the three roots are
where ξ = –1 + √–3/2.
As for the special case of a depressed cubic, this formula
applies but is useless when the roots can be expressed without cube
roots. In particular, if the formula gives that the three roots equal which means that the cubic polynomial can be factored as A straightforward computation allows verifying that the existence of this factorization is equivalent with
Trigonometric and hyperbolic solutions
Trigonometric solution for three real roots
When
a cubic equation with real coefficients has three real roots, the
formulas expressing these roots in terms of radicals involve complex
numbers. Galois theory allows proving that when the three roots are real, and none is rational (casus irreducibilis),
one cannot express the roots in terms of real radicals. Nevertheless,
purely real expressions of the solutions may be obtained using trigonometric functions, specifically in terms of cosines and arccosines. More precisely, the roots of the depressed cubic
are
This formula is due to François Viète. It is purely real when the equation has three real roots (that is ).
Otherwise, it is still correct but involves complex cosines and
arccosines when there is only one real root, and it is nonsensical
(division by zero) when p = 0.
This formula can be straightforwardly transformed into a formula
for the roots of a general cubic equation, using the back-substitution
described in § Depressed cubic.
The formula can be proved as follows: Starting from the equation t3 + pt + q = 0, let us set t = u cos θ. The idea is to choose u to make the equation coincide with the identity
For this, choose and divide the equation by This gives
Combining with the above identity, one gets
and the roots are thus
Hyperbolic solution for one real root
When there is only one real root (and p ≠ 0), this root can be similarly represented using hyperbolic functions, as
If p ≠ 0 and the inequalities on
the right are not satisfied (the case of three real roots), the formulas
remain valid but involve complex quantities.
When p = ±3, the above values of t0 are sometimes called the Chebyshev cube root. More precisely, the values involving cosines and hyperbolic cosines define, when p = −3, the same analytic function denoted C1/3(q), which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted S1/3(q), when p = 3.
Geometric solutions
Omar Khayyám's solution
For solving the cubic equation x3 + m2x = n where n > 0, Omar Khayyám constructed the parabola y = x2/m, the circle that has as a diameter the line segment[0, n/m2] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis.
The solution is given by the length of the horizontal line segment from
the origin to the intersection of the vertical line and the x-axis (see the figure).
A simple modern proof is as follows. Multiplying the equation by x/m2 and regrouping the terms gives
The left-hand side is the value of y2 on the parabola. The equation of the circle being y2 + x(x − n/m2) = 0, the right hand side is the value of y2 on the circle.
Solution with angle trisector
A cubic equation with real coefficients can be solved geometrically using compass, straightedge, and an angle trisector if and only if it has three real roots.
A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root. This implies that the old problems of angle trisection and doubling the cube, set by ancient Greek mathematicians, cannot be solved by compass-and-straightedge construction.
Geometric interpretation of the roots
Three real roots
Viète's trigonometric expression of the roots in the three-real-roots
case lends itself to a geometric interpretation in terms of a circle. When the cubic is written in depressed form (2), t3 + pt + q = 0, as shown above, the solution can be expressed as
Here is an angle in the unit circle; taking 1/3 of that angle corresponds to taking a cube root of a complex number; adding −k2π/3 for k = 1, 2 finds the other cube roots; and multiplying the cosines of these resulting angles by corrects for scale.
For the non-depressed case (1) (shown in the accompanying graph), the depressed case as indicated previously is obtained by defining t such that x = t − b/3a so t = x + b/3a. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables t and x, without changing the angle relationships. This shift moves the point of inflection and the centre of the circle onto the y-axis. Consequently, the roots of the equation in t sum to zero.
One real root
In the Cartesian plane
When the graph of a cubic function is plotted in the Cartesian plane, if there is only one real root, it is the abscissa (x-coordinate) of the horizontal intercept of the curve (point R on the figure). Further, if the complex conjugate roots are written as g ± hi, then the real partg is the abscissa of the tangency point H of the tangent line to cubic that passes through x-intercept R of the cubic (that is the signed length OM, negative on the figure). The imaginary parts±h are the square roots of the tangent of the angle between this tangent line and the horizontal axis.
In the complex plane
With
one real and two complex roots, the three roots can be represented as
points in the complex plane, as can the two roots of the cubic's
derivative. There is an interesting geometrical relationship among all
these roots.
The points in the complex plane representing the three roots
serve as the vertices of an isosceles triangle. (The triangle is
isosceles because one root is on the horizontal (real) axis and the
other two roots, being complex conjugates, appear symmetrically above
and below the real axis.) Marden's theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse
of the triangle—the unique ellipse that is tangent to the triangle at
the midpoints of its sides. If the angle at the vertex on the real axis
is less than π/3
then the major axis of the ellipse lies on the real axis, as do its
foci and hence the roots of the derivative. If that angle is greater
than π/3, the major axis is vertical and its foci, the roots of the derivative, are complex conjugates. And if that angle is π/3,
the triangle is equilateral, the Steiner inellipse is simply the
triangle's incircle, its foci coincide with each other at the incenter,
which lies on the real axis, and hence the derivative has duplicate real
roots.
Galois group
Given a cubic irreducible polynomial over a field K of characteristic different from 2 and 3, the Galois group over K is the group of the field automorphisms that fix K of the smallest extension of K (splitting field). As these automorphisms must permute the roots of the polynomials, this group is either the group S3 of all six permutations of the three roots, or the group A3 of the three circular permutations.
The discriminant Δ of the cubic is the square of
where a is the leading coefficient of the cubic, and r1, r2 and r3 are the three roots of the cubic. As changes of sign if two roots are exchanged, is fixed by the Galois group only if the Galois group is
A3. In other words, the Galois group is A3 if and only if the discriminant is the square of an element of K.
As most integers are not squares, when working over the field Q of the rational numbers, the Galois group of most irreducible cubic polynomials is the group S3 with six elements. An example of a Galois group A3 with three elements is given by p(x) = x3 − 3x − 1, whose discriminant is 81 = 92.
Derivation of the roots
This section regroups several methods for deriving Cardano's formula.
This method applies to a depressed cubic t3 + pt + q = 0. The idea is to introduce two variables u and v such that u + v = t and to substitute this in the depressed cubic, giving
At this point Cardano imposed the condition 3uv + p = 0. This removes the third term in previous equality, leading to the system of equations
Knowing the sum and the product of u3 and v3, one deduces that they are the two solutions of the quadratic equation
so
The discriminant of this equation is , and assuming it is positive, real solutions to this equation are (after folding division by 4 under the square root):
So (without loss of generality in choosing u or v):
As u + v = t, the sum of the cube roots of these solutions is a root of the equation. That is
is a root of the equation; this is Cardano's formula.
This works well when but, if the square root appearing in the formula is not real. As a complex number
has three cube roots, using Cardano's formula without care would
provide nine roots, while a cubic equation cannot have more than three
roots. This was clarified first by Rafael Bombelli in his book L'Algebra (1572). The solution is to use the fact that uv = –p/3, that is v = –p/3u. This means that only one cube root needs to be computed, and leads to the second formula given in § Cardano's formula.
The other roots of the equation can be obtained by changing of
cube root, or, equivalently, by multiplying the cube root by each of the
two primitive cube roots of unity, which are
Vieta's substitution
Vieta's substitution is a method introduced by François Viète (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of § Cardano's method, and avoids the problem of computing two different cube roots.
Starting from the depressed cubic t3 + pt + q = 0, Vieta's substitution is t = w – p/3w.
The substitution t = w – p/3w transforms the depressed cubic into
Multiplying by w3, one gets a quadratic equation in w3:
Let
be any nonzero root of this quadratic equation. If w1, w2 and w3 are the three cube roots of W, then the roots of the original depressed cubic are w1 − p/3w1, w2 − p/3w2, and w3 − p/3w3. The other root of the quadratic equation is This implies that changing the sign of the square root exchanges wi and − p/3wi for i = 1, 2, 3,
and therefore does not change the roots. This method only fails when
both roots of the quadratic equation are zero, that is when p = q = 0, in which case the only root of the depressed cubic is 0.
Lagrange's method
In his paper Réflexions sur la résolution algébrique des équations ("Thoughts on the algebraic solving of equations"), Joseph Louis Lagrange
introduced a new method to solve equations of low degree in a uniform
way, with the hope that he could generalize it for higher degrees. This
method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it requires solving a resolvent polynomial of degree at least six.
Apart from the fact that nobody had previously succeeded, this was the
first indication of the non-existence of an algebraic formula for
degrees 5 and higher; as was later proved by the Abel–Ruffini theorem. Nevertheless, modern methods for solving solvable quintic equations are mainly based on Lagrange's method.
In the case of cubic equations, Lagrange's method gives the same
solution as Cardano's. Lagrange's method can be applied directly to the
general cubic equation ax3 + bx2 + cx + d = 0, but the computation is simpler with the depressed cubic equation, t3 + pt + q = 0.
Lagrange's main idea was to work with the discrete Fourier transform of the roots instead of with the roots themselves. More precisely, let ξ be a primitive third root of unity, that is a number such that ξ3 = 1 and ξ2 + ξ + 1 = 0 (when working in the space of complex numbers, one has but this complex interpretation is not used here). Denoting x0, x1 and x2 the three roots of the cubic equation to be solved, let
be the discrete Fourier transform of the roots. If s0, s1 and s2
are known, the roots may be recovered from them with the inverse
Fourier transform consisting of inverting this linear transformation;
that is,
By Vieta's formulas, s0 is known to be zero in the case of a depressed cubic, and −b/a for the general cubic. So, only s1 and s2 need to be computed. They are not symmetric functions of the roots (exchanging x1 and x2 exchanges also s1 and s2), but some simple symmetric functions of s1 and s2
are also symmetric in the roots of the cubic equation to be solved.
Thus these symmetric functions can be expressed in terms of the (known)
coefficients of the original cubic, and this allows eventually
expressing the si
as roots of a polynomial with known coefficients. This works well for
every degree, but, in degrees higher than four, the resulting polynomial
that has the si
as roots has a degree higher than that of the initial polynomial, and
is therefore unhelpful for solving. This is the reason for which
Lagrange's method fails in degrees five and higher.
In the case of a cubic equation, and are such symmetric polynomials (see below). It follows that and are the two roots of the quadratic equation Thus the resolution of the equation may be finished exactly as with Cardano's method, with and in place of u and v.
In the case of the depressed cubic, one has and while in Cardano's method we have set and Thus, up to the exchange of u and v, we have and
In other words, in this case, Cardano's method and Lagrange's method
compute exactly the same things, up to a factor of three in the
auxiliary variables, the main difference being that Lagrange's method
explains why these auxiliary variables appear in the problem.
Computation of S and P
A straightforward computation using the relations ξ3 = 1 and ξ2 + ξ + 1 = 0 gives
Marden's theorem states that the foci of the Steiner inellipse of any triangle can be found by using the cubic function whose roots are the coordinates in the complex plane of the triangle's three vertices. The roots of the first derivative of this cubic are the complex coordinates of those foci.
The area of a regular heptagon
can be expressed in terms of the roots of a cubic. Further, the ratios
of the long diagonal to the side, the side to the short diagonal, and
the negative of the short diagonal to the long diagonal all satisfy a
particular cubic equation. In addition, the ratio of the inradius to the circumradius of a heptagonal triangle is one of the solutions of a cubic equation. The values of trigonometric functions of angles related to satisfy cubic equations.
Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic.