The Siebeck–Marden–Northshield Theorem and the Real Roots of the Symbolic Cubic Equation Emil M. Prodanov School of Mathematical Sciences, Technological University Dublin, Park House, Grangegorman, 191 North Circular Road, Dublin D07 EWV4, Ireland, e-mail: [email protected] Abstract The isolation intervals of the real roots of the symbolic monic cubic polynomial x3 ax2 bx c are determined, in terms of the coefficients of the polynomial, by` solving` the Siebeck–Marden–Northshield` triangle — the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial. Mathematics Subject Classification Codes (2020): 26C10, 12D10, 11D25. Keywords: Polynomials; Cubic equation; Siebeck–Marden–Northshield theorem; Roots; Isolation intervals; Root bounds. arXiv:2107.01847v1 [math.GM] 5 Jul 2021 1 1 Introduction The elegant theorem of Siebeck and Marden (often referred to as Marden’s theorem) [1]–[5] relates geometrically the complex non-collinear roots of a cubic polynomial with complex coefficients to a triangle whose vertices project onto them, on one hand, and, on the other, the critical points of the polynomial to the projections of the foci of the inellipse of this triangle. This ellipse is unique and is called Steiner inellipse [6]. It is inscribed in the triangle in such way that it is tangent to the sides of the triangle at their midpoints. The real version of the Siebeck–Marden Theorem, as given by Northshield [7], states that the three real roots (not all of which are equal) of a cubic polynomial are projections of the vertices of some equilateral triangle in the plane. However, it is the inscribed circle of the equilateral triangle that projects onto an interval the endpoints of which are the stationary points of the polynomial. The goal of this work is to consider a cubic equation with real coefficients and, using the Siebeck–Marden–Northshield theorem [7], solve the equilateral triangle and find the isolation intervals of the real roots of the symbolic monic cubic polynomial x3 ax2 ` ` bx c. ` 2 Analysis The vertices of the equilateral triangle that projects onto the three real roots of the cubic polynomial x3 ax2 bx c are points P , Q, and R with coordinates x , x x ?3 , ` ` ` p 1 p 2 ´ 3q{ q x , x x ?3 , and x , x x ?3 , respectively [7]. p 2 p 3 ´ 1q{ q p 3 p 1 ´ 2q{ q Lemma 1. The monic cubic polynomial p x x3 ax2 bx c with b a2 3 has only p q“ ` ` ` ą { one real root. Proof. The discriminant of the monic cubic polynomial x3 ax2 bx c is ` ` ` ∆ 27c2 18ab 4a3 c a2b2 4b3. (1) 3 “´ ` p ´ q ` ´ It is quadratic in c and the discriminant of this quadratic is ∆ 16 a2 3b 3 (2) 2 “ p ´ q As b a2 3, one has ∆ 0 for all a and thus ∆ 0 for all a and c. Hence, the cubic ą { 2 ă 3 ă polynomial p x x3 ax2 bx c with b a2 3 has only one real root (and a pair of p q“ ` ` ` ą { complex conjugate roots). Lemma 2. The monic cubic polynomial p x x3 ax2 bx c with b a2 3 has three p q“ ` ` ` ď { real roots, provided that c c , c , where c , are the roots of the quadratic equation P r 2 1s 1 2 4 2 1 4 x2 a3 ab x a2b2 b3 0, (3) ` ˆ27 ´ 3 ˙ ´ 27 ` 27 “ 2 namely: 2 c a, b c a2 3b 3, (4) 1,2 0 27 p q“ ˘ ap ´ q where 2 1 c a, b a3 ab. (5) 0p q“´27 ` 3 Proof. The discriminant ∆ 27c2 18ab 4a3 c a2b2 4b3 of the monic cubic 3 “ ´ ` p ´ q ` ´ polynomial x3 ax2 bx c is positive between the roots of the equation ∆ 0, which ` ` ` 3 “ is quadratic in c. This is exactly equation (3) and its roots are the ones given in (4) and (5). Lemma 3 (Solving the Siebeck–Marden–Northshield Triangle). The centre of the inscribed circle of the equilateral triangle that projects onto the three real roots of the monic cubic polynomial p x x3 ax2 bx c is point a 3, 0 , the projection p q “ ` ` ` p´ { q onto the abscissa of the inflection point of p x , and the radius of the inscribed circle is p q r 1 3 ?a2 3b. The radius of the circumscribed circle is 2r 2 3 ?a2 3b. “ p { q ´ “ p { q ´ Proof. The inflection point of the graph of the monic cubic polynomial p x x3 ax2 p q“ ` ` bx c occurs at the root of p x 6x 2a, namely at x φ a 3. ` 2p q“ ` “ “´ { Given that the vertices P , Q, and R of the triangle are points of coordinates x , x p 1 p 2 ´ x ?3 , x , x x ?3 , and x , x x ?3 , respectively, the centroid of the tri- 3q{ q p 2 p 3 ´ 1q{ q p 3 p 1 ´ 2q{ q angle is point of coordinates a 3, 0 — the first coordinate projection of the inflection p´ { q point. Each side of the triangle is equal to α ?12 3 ?a2 3b. The radius of a circle in- “ p { q ´ scribed in equilateral triangle with side α is r α ?12 1 3 ?a2 3b. The radius of “ { “ p { q ´ the circumscribed circle of an equilateral triangle with side α is 2r 2 3 ?a2 3b — “ p { q ´ see Figure 1. Lemma 4. The maximum distance between the three real roots of the monic cubic poly- nomial p x x3 ax2 bx c is ?12r ?12 3 ?a2 3b. In this case, one side p q “ ` ` ` “ p { q ´ of the equilateral triangle that projects onto the roots of the monic cubic polynomial p x x3 ax2 bx c is parallel to the abscissa. p q“ ` ` ` Proof. Given that the root x2 ν2 φ a 3 of the “balanced” cubic equation 3 2 “ 3“ “ ´ { x ax bx c0 0, where c0 2a 27 ab 3, is the midpoint between its other two ` ` ` “ 2 “´ { ` { roots x , ν , a 3 a 3 b, one has x x (the second coordinate of point R) 1 3 “ 1 3 “´ { ˘ { ´ 1 ´ 2 being equal to x x (thea second coordinate of point R). Hence P and R are both above 2 ´ 3 the abscissa and are equidistant from it. Thus PR is parallel to the abscissa. Hence, the distance between x and x is exactly equal to the length α ?12 3 ?a2 3b of the 3 1 “ p { q ´ side PR. In any other case of three real roots (c c , c and c c ), the side PR will P r 2 1s ‰ 0 not be parallel to the abscissa and hence the projection of PR onto the abscissa will be shorter than the length of PR, that is, the three real roots of the cubic polynomial will lie in an interval of length smaller than α ?12 3 ?a2 3b. “ p { q ´ 3 Figure 1 Siebeck–Marden–Northshield Theorem: Any three real numbers, not all equal, are the projections of the vertices of some equilateral triangle in the plane. For a cubic polynomial p x x3 ax2 bx c with three real roots (not all equal), the inscribed circle of the equilateral triangle thatp q“ projects` onto` those` roots itself projects to an interval with endpoints equal to critical points of p x [7]. The vertices of the triangle p q are points P , Q, and R with coordinates x1, x2 x3 ?3 , x2, x3 x1 ?3 , and x3, x1 x2 ?3 , respectively [7]. The centroid of the trianglep isp at the´ projq{ ectionq p ofp the´ inflectionq{ q pointp of p px .´ Theq{ radiusq of the inscribed circle is r 1 3 ?a2 3b. The radius of the circumscribed circle is 2r p 2q 3 ?a2 3b. The maximum distance between“ p { theq three´ real roots of the monic cubic polynomial p x “x3 p { axq 2 bx´ c is equal to the side of the triangle: α ?12r ?12 3 ?a2 3b. This occurs in thep caseq“ of the` “balanced”` ` 3 “ “ p { q ´ cubic (with c c0 2a 27 ab 3), when one of the sides of the triangle is parallel to the abscissa. For “ “´ { ` { any other c such that c2 c c1, the three real roots of the cubic lie within a shorter interval. ď ď Theorem 1. The monic cubic polynomial p x x3 ax2 bx c, for which b a2 3 p q“ ` ` ` ă { and c c , c , has three real roots x x x , at least two of which are different P r 2 1s 3 ď 2 ď 1 and any two of which are not farther apart than ?12 3 ?a2 3b, with the following p { q ´ isolation intervals: (I) For c c c : x ν ,µ , x µ , φ , and x ν ,ξ . 2 ď ď 0 3 P r 3 2s 2 P r 2 s 1 P r 1 2s (II) For c c c : x ξ ,ν , x φ, µ , and x µ ,ν , 0 ď ď 1 3 P r 1 3s 2 P r 1s 1 P r 1 1s where: 3 2 (i) µ1,2 is the double root and ξ1,2 is the simple root of p1,2 x x ax bx c1,2, 2 p q“ ` ` ` that is, µ1,2 are the roots of 3x 2ax b 0, namely: µ1,2 a 3 r 2 ` ` “ “ ´ { 2˘ “ a 3 1 3 ?a 3b and ξ , a 2µ , a 3 2r a 3 2 3 ?a 3b.