The Number of Real Roots of a Cubic Equation

Autor/inn/en	Kavinoky, Richard; Thoo, John B.
Titel	The Number of Real Roots of a Cubic Equation
Quelle	In: AMATYC Review, 29 (2008) 2, S.2-8 (7 Seiten) PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0740-8404
Schlagwörter	Calculus; Mathematics Instruction; Equations (Mathematics); Mathematical Concepts; Teaching Methods + Suchen Sie Ihr Suchwort? Analysis; Differenzialrechnung; Infinitesimalrechnung; Integralrechnung; Mathematics lessons; Mathematikunterricht; Equations; Mathematics; Gleichungslehre; Teaching method; Lehrmethode; Unterrichtsmethode
Abstract	To find the number of distinct real roots of the cubic equation (1) x[caret]3 + bx[caret]2 + cx + d = 0, we could attempt to solve the equation. Fortunately, it is easy to tell the number of distinct real roots of (1) without having to solve the equation. The key is the discriminant. The discriminant of (1) appears in Cardan's (or Cardano's) cubic formula. However, few students today are even aware of the cubic formula, let alone have seen it. We show how a student may come up with or be led to the discriminant of (1) without appealing to Cardan's cubic formula using ideas from a first calculus course--derivative, critical point, local extrema, and graphing--in an intuitive way. We also show how the discriminant defines a boundary in the plane across which the number of real roots of (1) changes, and apply the discriminant to determining the number of normals to the parabola y = x[caret]2 through a given point and the number of equilibrium solutions of dx/dt = (R-Rc)x-ax[caret]3, where Rc and a are positive constants and R is a parameter. (As Provided).
Anmerkungen	American Mathematical Association of Two-Year Colleges. 5983 Macon Cove, Memphis, TN 38134. Tel: 901-333-4643; Fax: 901-333-4651; e-mail: amatyc@amatyc.org; Web site: http://www.amatyc.org
Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2017/4/10