“Cubic Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cubic Equation

释义

Cubic Equation

A cubic equation is a Polynomial equation of degree three. Given a general cubic equation

(1)

(the Coefficient

may be taken as 1 without loss of generality by dividing the entire equation through by

), first attempt to eliminate the

term by making a substitution of the form

(2)

Then

	(3)
	(4)
	(5)

The

is eliminated by letting

, so

(6)

Then

			(7)
			(8)
			(9)

so equation (1) becomes

(10)

(11)

(12)

Defining

			(13)
			(14)

then allows (12) to be written in the standard form

(15)

The simplest way to proceed is to make Vieta's Substitution

(16)

which reduces the cubic to the equation

(17)

which is easily turned into a Quadratic Equation in

by multiplying through by

to obtain

(18)

(Birkhoff and Mac Lane 1965, p. 106). The result from the Quadratic Equation is


			(19)

where

and

are sometimes more useful to deal with than are

and

. There are therefore six solutions for

(two corresponding to each sign for each Root of

). Plugging

back in to (17) gives three pairs ofsolutions, but each pair is equal, so there are three solutions to the cubic equation.

Equation (12) may also be explicitly factored by attempting to pull out a term of the form from the cubicequation, leaving behind a quadratic equation which can then be factored using the Quadratic Formula. This processis equivalent to making Vieta's substitution, but does a slightly better job of motivating Vieta's ``magic''substitution, and also at producing the explicit formulas for the solutions. First, define the intermediate variables

			(20)
			(21)

(which are identical to

and

up to a constant factor). The general cubic equation (12) then becomes

(22)

Let

and

be, for the moment, arbitrary constants. An identity satisfied by Perfect Cubicequations is that

(23)

The general cubic would therefore be directly factorable if it did not have an

term (i.e., if

). However, sincein general

, add a multiple of

--say

--to both sides of (23) to give the slightly messyidentity

(24)

which, after regrouping terms, is

(25)

We would now like to match the Coefficients

and

with those of equation (22), so we must have

(26)

(27)

Plugging the former into the latter then gives

(28)

Therefore, if we can find a value of

satisfying the above identity, we have factored a linear term from the cubic,thus reducing it to a Quadratic Equation. The trial solution accomplishing this miracle turns out to be the symmetrical expression

(29)

Taking the second and third Powers of

gives


			(30)







			(31)

Plugging

and

into the left side of (28) gives

(32)

so we have indeed found the factor

of (22), and we need now only factor the quadratic part. Plugging

into the quadratic part of (25) and solving the resulting

(33)

then gives the solutions



			(34)

These can be simplified by defining

			(35)


			(36)

so that the solutions to the quadratic part can be written

(37)

Defining

			(38)
			(39)
			(40)

where

is the Discriminant (which is defined slightly differently, includingthe opposite Sign, by Birkhoff and Mac Lane 1965) then gives very simple expressions for

and

, namely

			(41)
			(42)

Therefore, at last, the Roots of the original equation in

are then given by

			(43)
			(44)
			(45)

with

the Coefficient of

in the original equation, and

and

as defined above. These three equationsgiving the three Roots of the cubic equation are sometimes known as Cardano's Formula. Note that if the equationis in the standard form of Vieta

(46)

in the variable

, then

, and

, and the intermediate variables have the simple form (c.f. Beyer 1987)

			(47)
			(48)
			(49)

The equation for in Cardano's Formula does not have an appearing in it explicitly while and do,but this does not say anything about the number of Real and Complex Roots(since and are themselves, in general, Complex). However, determining whichRoots are Real and which are Complex can be accomplished bynoting that if the Discriminant , one Root is Realand two are Complex Conjugates; if , all Roots are Real and at least two are equal; and if , all Roots are Real and unequal. If, define

(50)

Then the Real solutions are of the form

			(51)
			(52)
			(53)

This procedure can be generalized to find the Real Roots for any equation in the standard form(46) by using the identity

(54)

(Dickson 1914) and setting

(55)

(Birkhoff and Mac Lane 1965, pp. 90-91), then

(56)

(57)

(58)

, then use

(59)

to obtain

(60)

and

, use

(61)

and if

and

, use

(62)

to obtain

(63)

The solutions to the original equation are then

(64)

An alternate approach to solving the cubic equation is to use Lagrange Resolvents. Let , define

			(65)
			(66)
			(67)

where

are the Roots of

(68)

and consider the equation

(69)

where

and

are Complex Numbers. TheRoots are then

(70)

for

, 1, 2. Multiplying through gives

(71)

(72)

where

			(73)
			(74)

The solutions satisfy Newton's Identities

			(75)
			(76)
			(77)

In standard form,

, and

, so we have the identities

			(78)
			(79)
			(80)

Some curious identities involving the roots of a cubic equation due to Ramanujan are given by Berndt (1994).

See also Quadratic Equation, Quartic Equation, Quintic Equation, Sextic Equation

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972.

Berger, M. §16.4.1-16.4.11.1 in Geometry I. New York: Springer-Verlag, 1994.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 22-23, 1994.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 9-11, 1987.

Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 3rd ed. New York: Macmillan, pp. 90-91, 106-107, and 414-417, 1965.

Dickson, L. E. ``A New Solution of the Cubic Equation.'' Amer. Math. Monthly 5, 38-39, 1898.

Dickson, L. E. Elementary Theory of Equations. New York: Wiley, pp. 36-37, 1914.

Dunham, W. ``Cardano and the Solution of the Cubic.'' Ch. 6 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 133-154, 1990.

Ehrlich, G. §4.16 in Fundamental Concepts of Abstract Algebra. Boston, MA: PWS-Kent, 1991.

Jones, J. ``Omar Khayyám and a Geometric Solution of the Cubic.'' http://jwilson.coe.uga.edu/emt669/Student.Folders/Jones.June/omar/omarpaper.html.

Kennedy, E. C. ``A Note on the Roots of a Cubic.'' Amer. Math. Monthly 40, 411-412, 1933.

King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Quadratic and Cubic Equations.'' §5.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 178-180, 1992.

Spanier, J. and Oldham, K. B. ``The Cubic Function and Higher Polynomials.'' Ch. 17 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 131-147, 1987.

van der Waerden, B. L. §64 in Algebra. New York: Frederick Ungar, 1970.

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