On the Cubic Equation with its Siebeck–Marden–Northshield Triangle and the Quartic Equation with its Tetrahedron

The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck–Marden–Northshield equilateral triangle and regular tetrahedron. The Vi`ete trigonometric formulæ for the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classiﬁcation for the quartic x 4 ` ax 3 ` bx 2 ` cx ` d is proposed for which the conditions are imposed on the individual coeﬃcients a , b , c , and d . The maximum and minimum lengths of the interval containing the four real roots of the quartic are determined in terms of a and b . The upper and lower root bounds for a quartic with four real roots are also found: no root can lie farther than p? 3 { 4 q ? 3 a 2 ´ 8 b from ´ a { 4. The real roots of the quartic are localized by ﬁnding intervals containing at most two roots. The end-points of these intervals depend on a and b and are roots of quadratic equations — which makes this localization helpful for quartic equations with complicated parametric coeﬃcients.


Introduction
Presented in this work is a geometrical study of the roots of the cubic and the quartic which reveals new features.Viète [1] and Descartes [2] first related the case of three real roots of a cubic polynomial (the casus irreducibilis) to circle geometry [3].The cubic polynomial x 3 `ax 2 `bx`c with three real roots (not all equal) is associated with an equilateral triangle, the Siebeck-Marden-Northshield triangle [4]- [11], the vertices of which have x-coordinate projections equal to the roots of the cubic.The centroid of the triangle is at the x-coordinate projection of the inflection point ´a{3 of the cubic and the inscribed circle crosses the abscissa at the x-coordinate projections of the stationary points of the cubic.The radius of the inscribed circle is p2{3q ? a 2 ´3b, hence a necessary condition for the existence of the Siebeck-Marden-Northshield triangle (or three real roots of the cubic) is b ă a 2 {3.If, additionally, the free term c of the polynomial is between the roots of a specific quadratic equation, then one has a necessary and sufficient condition for three real roots of the cubic.This makes the study of the real roots easier than when considering if the cubic discriminant is positive.The variation of the free term results in rotation of the Siebeck-Marden-Northshield triangle and from that a relationship with Viète trigonometric formulae for the roots of the cubic is established.On the other hand, a quartic polynomial with four real roots (not all equal) is associated with a regular tetrahedron and its symmetries [4], [12]- [14].The four real roots of the quartic x 4 `ax 3 `bx 2 `cx `d are the x-coordinate projections of a regular tetrahedron in I R 3 .The inscribed sphere of this tetrahedron projects onto an interval with endpoints given by the x coordinates of the two inflection points of the quartic.The equilateral triangle in the xy-plane around that sphere is exactly the Siebeck-Marden-Northshield triangle for the cubic polynomial 4x 3 `3ax 2 `2bx `c and its vertices project onto the x coordinates of the stationary points of the quartic.The continuous variation of the free term of the quartic allows to trace how the discriminant of the quartic changes sign and, from there, get the nature of the roots.This discriminant is cubic in the free term and the discriminant of this cubic is very simple.This allows the presentation of a very detailed complete root classification of the quartic with all possible cases listed (e.g. two pairs of equal real roots) and the conditions for these cases are conditions on the individual coefficients of the polynomial.Some further new results are also reported here: (i) the maximum and minimum lengths of the interval containing the four real roots of the quartic are determined -the roots lie in an interval of length between p ? 2{2q ?3a 2 ´8b and p ? 3{3q ?3a 2 ´8b; (ii) it is shown that a quartic with four real roots cannot have a root greater than ´a{4 `p? 3{4q ?3a 2 ´8b and cannot have a root smaller than ´a{4 ´p? 3{4q ?3a 2 ´8b; (iii) the real roots of the quartic are localized by finding intervals containing at most two roots and the end-points of these intervals depend on a and b and are roots of quadratic equations -which makes this localization helpful for quartic equations with complicated parametric coefficients.First steps towards the complete root classification for the quintic equation are also provided.Due to the Siebeck-Marden-Northshield theorem [5]- [11], if the cubic polynomial p 3 pxq " x 3 `ax 2 `bx `c has three real roots x 1 , x 2 , and x 3 , not all of which equal, these can be found geometrically as the x-coordinate projections of the vertices px 1 , px 2 ´x3 q{ ?3q, px 2 , px 3 ´x1 q{ ?3q, and px 3 , px 1 ´x2 q{ ?3q of an equilateral triangle -the Siebeck-Marden-Northshield triangle.The inscribed circle of this triangle is centered at x " ´a{3, the x-coordinate projection of the inflection point of p 3 pxq, while the intersection points of the inscribed circle with the x-axis are the x-coordinate projections of the critical points µ 1,2 " ´a{3 ˘p1{3q ?
a 2 ´3b of the cubic polynomial.Thus, the radius of the inscribed circle is r " p1{3q ? a 2 ´3b and the radius of the circumscribed circle is 2r " p2{3q ? a 2 ´3b.Each side l of the Siebeck-Marden-Northshield triangle is equal to p ? 12{3q ? a 2 ´3bsee Figure 1.Instead of requesting a non-negative discriminant δ 3 of the cubic polynomial to ensure three real roots, one can find another necessary and sufficient condition for their existence.
Proof.Clearly, a 2 ´3b ą 0 is a necessary condition for the existence of the Siebeck-Marden-Northshield triangle -as this ensures that p 3 pxq will have two distinct critical points.But p 3 pxq will not necessarily have three real roots in this case.Consider the discriminant of p 3 pxq " x 3 `ax 2 `bx `c.It is quadratic in c and the discriminant of this quadratic is As a 2 ´3b ą 0, one has δ 2 ą 0 and thus δ 3 pcq ě 0 for c 2 ď c ď c 1 , where c 1,2 are the two distinct real roots (distinct, since δ 2 ą 0) of δ 3 pcq " 0: where c 0 pa, bq " ´2 27 a 3 `1 3 ab. (4) The Siebeck-Marden-Northshield triangle and its rotation upon variation of c Hence, the cubic polynomial ppxq " x 3 `ax 2 `bx `c with a 2 ´3b ą 0 and c 2 ď c ď c 1 has three real roots.Note that if a 2 ´3b ą 0 and c " c 1,2 , then the cubic will have a double real root and single real root.If a 2 ´3b ą 0 and c 2 ă c ă c 1 , the three real roots will be distinct.Finally, if a 2 " 3b, then the cubic will have a triple real root ´a{3 if c " a 3 {27, for any other value of c, cubic with a 2 " 3b will have a single real root.
Theorem 2. The maximum length of the interval containing the three real roots of the cubic x 3 `ax 2 `bx `c is ?12r " p ? 12{3q ? a 2 ´3b " l and this is achieved when a 2 ´3b ą 0 and c " c 0 P rc 2 , c 1 s.The minimum length of this interval is 3r " ? a 2 ´3b, occurring when a 2 ´3b ą 0 and c " c 1,2 [11,15].
Proof.The roots of the cubic x 3 `ax 2 `bx `c , with a 2 ´3b ą 0 and c " c 0 " ´2a 3 {27 `ab{3 are ν 1,3 " ´a{3 ˘l{2 " ´a{3 ˘p? 3{3q ? a 2 ´3b and ν 2 " ´a{3.Note that ν 1,3 are symmetric with respect to the centroid ´a{3 (point H on Figure 1) of the Siebeck-Marden-Northshield triangle onto which the middle root projects.The ycoordinates of the vertices of the Siebeck-Marden-Northshield triangle, which project onto the biggest and smallest roots, are both equal to p ? 3{3q ? a 2 ´3b, that is, the triangle has a side parallel to the abscissa.On Figure 1, this is the triangle M 0 N 0 P 0 with M 0 P 0 parallel to the x-axis.The centroid of the Siebeck-Marden-Northshield triangle is at point H and the angle which P 0 H forms with the abscissa is π{6.The distance between the roots ν 3 and ν 1 is exactly equal to l " ?12r " p ? 12{3q ? a 2 ´3b.The roots of the cubic x 3 `ax 2 `bx `c , with a 2 ´3b ą 0 and c " c 1,2 are the double root µ 1,2 " ´a{3 ˘r " ´a{3 ˘p1{3q ? a 2 ´3b (points K{J on Figure 1), which are roots to p 1 3 pxq " 3x 2 `2ax `b " 0, and the single root ξ 1,2 " ´a ´2µ 1,2 " ´a{3 ¯2r " ´a{3 ¯p2{3q ?
a 2 ´3b (points M 1 {P 2 ).In these cases, the respective equilateral triangles have a side perpendicular to the abscissa: these are M 1 N 1 P 1 with P 1 N 1 K Ox, when c " c 1 , and M 2 N 2 P 2 with M 2 N 2 K Ox, when c " c 2 -see Figure 1.This yields the shortest possible interval, containing the three real roots of the cubic (one of which is repeated).It is equal to the height of Siebeck-Marden-Northshield triangle or 3r " ? a 2 ´3b.The angle which P 1 H forms with the abscissa is π{3 and the angle which P 2 H forms with the abscissa is 0 (as P 2 is on the x-axis).For any other c such that c 2 ď c ď c 1 , the three real roots of the cubic lie within an interval of length between 3r and ?12r « 3.4641r.On Figure 1, these roots are represented by M N P -the vertices of the general Siebeck-Marden-Northshield triangle.The angle θ, which P H forms with the x-axis, satisfies 0 ď θ ď π{3.Note that c " c 2 corresponds to θ " 0; c " c 0 corresponds to θ " π{6; and c " c 1 corresponds to θ " π{3.

Roles Played by the Coefficients of the Cubic
The coefficient a of the quadratic term of x 3 `ax 2 `bx`c selects the centroid of Siebeck-Marden-Northshield triangle.This is where the inflection point of x 3 `ax 2 `bx `c projects onto the abscissa.For any given a, the coefficients b ă a 2 {3 of the linear term of x 3 `ax 2 `bx `c determines the radius r " p1{3q ? a 2 ´3b of the inscribed circle (the triangle exists only for a 2 ´3b ą 0) and, hence, the size of Siebeck-Marden-Northshield triangle.The inscribed circle projects to an interval on the abscissa with endpoints equal to the projections of the critical points of the cubic, the distance between which is always 2r " p2{3q ? a 2 ´3b -the radius of the circumscribed circle.The inflection point of the cubic is always in the middle between the two critical points.For a 2 ´3b ą 0, the variation of the free term c of x 3 `ax 2 `bx `c rotates the Siebeck-Marden-Northshield triangle counterclockwise from the position of M 2 N 2 P 2 , when c " c 2 (θ " 0), through that of M 0 N 0 P 0 , when c " c 0 (θ " π{6), up to that of M 1 N 1 P 1 , when c " c 1 (θ " π{3).

Relationship to the Viète Trigonometric Formulae for the Roots of the Cubic
Under the coordinate translation x Ñ x `a{3, the centroid of the Siebeck-Marden-Northshield triangle gets at the origin of the new coordinate system (this also depresses the cubic).It is straightforward to determine the x-coordinate projections of the vertices of the triangle in this new coordinate system and hence find that the real roots of the cubic are given by: x and from this, one easily gets that the angle θ of rotation of the Siebeck-Marden-Northshield triangle around its centroid is Note that when θ " 0, formula (8) yields c " c 2 ; when θ " π{6, one gets c " c 0 ; and when θ " π{3, from (8) one gets c " c 1 .That is, the expression in the square brackets above takes continuous values from ´1 (when c " c 1 ), through 0 (when c " c 0 ), to 1 (when c " c 2 ).When a 2 ´3b ą 0, but c R rc 2 , c 1 s, the value of the expression in the square brackets of (9) has absolute value greater than 1.In this case, θ is a complex number (purely imaginary, if c ă c 2 ).However, ( 5)-( 7) still produce the roots of the cubic -with only one of them real and the other two -complex-conjugates of each other.When a 2 ´3b ă 0, there will again be only one real root and a pair of complex-conjugate roots.However, formulae (5)- (7) no longer work.θ is again a complex number.One needs to get the roots, symmetric to those in ( 5)- (7) with respect to the y-axis in the x Ñ x `a{3 coordinate system: In this case, one has p x 1 `p x 2 `p x 3 " ´a and p

Isolation Intervals of the Roots of the Cubic
From the Siebeck-Marden-Northshield triangle, one can immediately read geometrically the isolation intervals of the three real roots of the cubic.These are as follows [11,15]: 3 Nature of the Roots of the Quartic Equation It has been long since the complete root classification for the quartic equation was established -in 1861, Cayley [16] used the Sturmian constants (the leading coefficients in the Sturm functions modulo positive multiplicative factors) for the quartic (in binomial form) to achieve this.In this work, Cayley also proposed the complete roots classification for the cubic equation.
For the monic quartic polynomial the sequence of Sturm functions for p 4 pxq are [17]: π 0 pxq " p 4 pxq, π 1 pxq " p 1 4 pxq, π i pxq " remrπ i´2 pxq{π i´1 pxqs (i " 2, 3, 4) -the remainder of the division of the polynomial π i´2 pxq by the polynomial π i´1 pxq.The Sturmian constants for p 4 pxq are: S 0 " 1, S 1 " 1, S 3 " 3a 2 ´8b, S 4 " ´3a 3 c pb 2 ´6dqa 2 `14abc ´4b 3 `16bd ´18c 2 , and S 5 " ∆ -the discriminant of the quartic: The number of real roots of p 4 pxq is equal to the excess of the number of variations of sign in the sequence of π j pxq at ´8 over the number of variations of sign of these at `8 (with vanishing terms ignored).Denoting in round brackets the sequence of the signs of the Sturm functions π j pxq at `8 and in square brackets -those at ´8, one has [16]: (i) p`````q, r`´`´`s -four real roots.
(ii) p``´``q, r`´´´`s -no real roots.
(iii) p```´`q, r`´```s -no real roots.
(iv) p``´´`q, r`´´``s -no real roots.
(v) p````´q, r`´`´´s -two real roots.
(vii) p```´´q, r`´``´s -two real roots.
(viii) p``´´´q, r`´´`´s -two real roots.
It is clear from this complete root classification, that positive discriminant of the quartic is associated with either four real roots or no real roots, while a negative one -with two real roots.
In 1908, Petrovich [18] proposed a very interesting approach for finding a necessary and sufficient condition for reality of all roots of a polynomial with real coefficients, a 0 `a1 z `¨¨¨`a p z p , together with the roots of any polynomial composed of the set of any number of its first terms, i.e. a 0 `a1 z `¨¨¨`a n z n , n ď p.The Sturm method allows the immediate determination of the nature of the roots of a polynomial as, from the given coefficients, one can immediately determine the Sturm functions and hence establish their sequences of signs at `8 and ´8.However, one would pose the following, in some sense, inverse, problem: would it be possible to "reverse-engineer" a quartic, with a priori specified nature of its roots, by working at the level of the individual coefficients of the polynomial, that is, by knowing how to choose the coefficients of the quartic suitably [and not by merely expanding px ´x1 qpx ´x2 qpx ´x3 qpx ´x4 q in which the roots x i are chosen]?In this work, an alternative method is proposed with which not only complete root classification of the quartic is made (in the following Section), but which also allows one to "reverse-engineer" quartics.For example, could a quartic with a " 5 have four real roots?(The answer is yes.)And, if a " 5, what should b, c, and d sequentially be so that the quartic will have four real roots?Indeed, a cubic (with guaranteed tree real roots in this case) has to be solved in the process in order to achieve this, but the Viète trigonometric formulae for the roots of the cubic are quite easy to apply (as in the previous Section).
The discriminant of a monic polynomial of degree n is ś iăj px i ´xj q 2 , where x i are all n roots (real or complex) of the polynomial.If the discriminant is positive, then, obviously, the number of complex roots of the polynomial is a multiple of 4. If the discriminant is negative, then there are 2m `1 pairs of complex-conjugate roots, where m ď pn ´2q{4.If the quartic p 4 pxq has a positive discriminant ∆, there are either four real roots or there are two pairs of complex-conjugate roots.On the other hand, if ∆ is negative, then the quartic has two real roots and a pair of complex-conjugate roots.Vanishing discriminant means a repeated real root (of multiplicity 2, 3, or 4).One can find criteria to determine which of these three cases (zero, two, or four real roots) applies by deriving conditions on the coefficients of the quartic, rather than studying the discriminant ∆ alone.The polynomial p 4 pxq can be viewed as an element of a congruence of quartic polynomials tx 4 `ax 3 `bx 2 `cx `D | D P IRu, the graphs of which foliate the xy-plane by the continuous variation of the foliation parameter D -the free term.All polynomials in the congruence have the same set of stationary points M i -the roots of the polynomial p 1 4 pxq " 4x 3 `3ax 4 pxq has a double real root and a single real root] -as these are the determined by finding the roots of a cubic.Quartic polynomials with one stationary point can have either no real roots or two real roots, while the rest can have either zero, or two, or four real roots.Within this congruence of quartic polynomials, there are either one or three (not necessarily all different in the latter case) polynomials P piq 4 pxq " p 4 pxq ´p4 pM i q, (16) for which the abscissa is tangent to their graph at the stationary point M i .Each polynomial P piq 4 pxq has a zero discriminant as each one of them has (at least one) at least double real root.These polynomials are shown on Figure 2 (for the case of polynomials with three distinct stationary points) and on Figure 3 (for those with a single stationary point).When the free term d of p 4 pxq is large enough, the quartic will have no real roots and, for as long as the free term d satisfies d ą p 4 pM i q for all i, the graph of this polynomial will lie entirely in the upper half-plane and there will be no real roots -these are the polynomials in the band above the uppermost solid curve on Figure 2 and those above the single solid curve on Figure 3. Hence the band of polynomials whose graphs lie above the (uppermost) solid curve on Figures 2 and 3 all have positive discriminants.
If the free term d of p 4 pxq is less than the smallest of p 4 pM i q, but larger than the other two values of p 4 pM i q, if these exist, then the graph of this quartic will cross the abscissa twice.Hence, all polynomials is this band (bounded by the two uppermost solid curves on Figure 2) will have two real roots and a negative discriminant.One should note however that in the case of a quartic polynomial with two double real roots (not shown on Figure 2), when d is less than the two equal smallest values of p 4 pM i q (at the two minima), but greater than the value of p 4 pxq at the maximum, there will be four real roots: the discriminant will not become negative, but will "bounce back" from zero.
If the free term d is less than p 4 pM i q at the two minima, but greater than the value of p 4 pxq at the maximum, there will be four real roots and all polynomials in this band (bounded by the two lowermost solid curves on Figure 2) will be with positive discriminants.
Finally, if d is smaller than all p 4 pM i q, then the polynomial will have two real roots and a negative discriminant -the band below the lowermost curve on Figure 2.
Clearly, with the variation of the free term, the discriminant of the quartic changes sign three times (in case of three distinct stationary points) or once (in case of either a single global minimum or a saddle and a global minimum).This is indicative of the presence of a polynomial cubic in d and, indeed, the discriminant ∆ of the quartic [given in (15)] is such polynomial [but it is quartic with respect to any other coefficient of p 4 pxq].
Consider the discriminant ∆ 3 of this cubic in d.It has a very simple form: The sign of the discriminant ∆ 3 and hence, the number of roots of the cubic in d equation ∆pdq " 0, depend on the sign of the term in the square brackets.The discriminant ∆ 3 could also be zero [leading to a multiple root of ∆pdq " 0] -either by virtue of c " ´a3 {8 `ab{2 or due to the term in the square brackets.
The term in the square brackets is quadratic in c.The discriminant of this quadratic is also very simple: and ∆ 2 is positive for 3a 2 ´8b ą 0. In such case, the term in the square brackets of ∆ 3 is negative when c is between the roots C 1,2 of the quadratic 4c 2 `apa 2 ´4bqc ṕb 2 {3q pa 2 ´32b{9q, where and The discriminant ∆ 3 , given by (17), can be written conveniently as Should ∆ 3 happen to be zero, either by virtue of c " C 0 or due to c " C 1,2 (the latter being real only when b ď 3a 2 {8), then the cubic equation ∆pdq " 0 will have a repeated root, that is, either a double real root d : and a single real root r d, or a triple real root r d.In either case, the discriminant ∆pdq of the quartic will change sign only once -at r d.In the case of a triple root r d, the quartic will have the form px ´x0 q 4 with x 0 being a quadruple root.If c ‰ C 0 and c ‰ C 1,2 , the discriminant ∆pdq of the quartic will not be zero.It will have either 3 distinct roots d 1 ą d 2 ą d 3 (in the case of three stationary points of the quartic) or a single real root d 0 (in the case of a single stationary point (minimum) of the quartic).In the case of three distinct real roots d 1 , d 2 , and d 3 or a single real root d 0 of the discriminant ∆pdq, these can be easily found using the Viète trigonometric formilae ( 5)-( 7): where: B " C " and ˆ´19683a 12 ´314928a 10 b `419904a 9 c `1959552a 8 b 2 ´5038848a 7 bc provided that A 2 ´3B " p1{65536q p3a 2 ´8bq p243a 6 ´1944a 4 b `3456a 3 c `4032a 2 b 2 13824abc ´512b 3 `13824c 2 q is positive.The expression in the last pair of parenthesis is quadratic in c and its discriminant is ´55296p3a 2 ´8bq 3 .Hence, if 3a 2 ´8b ą 0, then this quadratic in c is positive.This, in turn, leads to A 2 ´3B ą 0. That is, 3a 2 ´8b ą 0 is a sufficient condition for A 2 ´3B ą 0. Depending on C -whether it lies in the closed interval between the roots ´2A 3 {27 `AB{3 ˘p2{27q a pA 2 ´3Bq 3 or not, see (3), there will be either three distinct real roots d 1 ą d 2 ą d 3 or a single real root (denoted by d 0 ), respectively.However, if A 2 ´3B ă 0, then, in order to determine the roots of the cubic equation ∆pdq " 0, one should use the Viète formulae ( 22)-( 24) with the signs in front of the radicals inverted -as in ( 10)- (12).Only one of these roots, denote it by d 0 again, will be real.

Complete Root Classification for the Quartic
With the results obtained so far, one can now summarize that the nature of the roots of the quartic x 4 `ax 3 `bx 2 `cx `d can be completely classified, for any a, as follows: In this case, ∆ 3 ă 0. Thus, the discriminant ∆pdq of the quartic has only one real root d 0 .The quartic has one stationary point and positive discriminant ∆.There are no real roots.Figure 3 applies -dotted curve.
In this case, one again has ∆ 3 ă 0 and the discriminant ∆pdq of the quartic having only one real root d 0 .The quartic has one stationary point and zero discriminant ∆.There are two equal real roots, but not a quadruple root as b ‰ 8a 2 {3.When c " C 0 , one has ∆ 3 " 0. Thus, the discriminant ∆pdq of the quartic either has a repeated real root d : and a single real root r d (it may also have a treble real root r d).The discriminant ∆pdq of the quartic changes sign only once (at r d) and, hence, the quartic has one stationary point and positive discriminant ∆, except for d " d : , where the discriminant ∆ is zero.There are no real roots (if d " d : , there are two double complex roots).Figure 3 applies -dotted curve.Again, as c " C 0 , one has ∆ 3 " 0. The quartic now has negative discriminant ∆.There are two distinct real roots.Figure 3 applies -dashed curve.
The situation is the same as the one in case (vii) with the only difference that this time the quartic has zero discriminant ∆.There are two equal real roots.Figure 3 applies -solid curve.
Same situation as in cases (vii) and (viii), with the difference that the quartic now has negative discriminant ∆.There are two distinct real roots.Figure 3 applies -dashed curve.
The quartic in this case is px `a{4q 4 .It has one stationary point, zero discriminant ∆ and quadruple root ´a{4.Figure 3 applies -solid curve.
Same situation as in cases (x) and (xi), with the difference that the quartic now has negative discriminant ∆.There are two distinct real roots.Figure 3 applies -dashed curve.
In this case, one has ∆ 3 ą 0 and the discriminant ∆pdq of the quartic having three distinct real roots d 1 ą d 2 ą d 3 .With d ą d 1 , the quartic has three stationary point and positive discriminant ∆.There are no real roots.Figure 2 applies -curves in the band above the uppermost curve.
This time the discriminant ∆pdq of the quartic is zero.There are two equal real roots (the abscissa is tangent to the quartic at one of its minima).Figure 2 applies -the uppermost curve.
This time the discriminant ∆pdq of the quartic is negative.There are two distinct real roots.Figure 2 applies -curves between the two uppermost curves.
d " d 2 -four real roots, two of which (the two largest or the two smallest) are equal The quartic has zero discriminant ∆ again.There are four real roots, two of which are equal (the abscissa is tangent to the quartic at the other of its minima, see case xiii).Figure 2 applies -the second curve from top.
The discriminant of the quartic is now positive.There are four distinct real roots.Figure 3 applies -the dashed curve.

the middle two of which are equal
The quartic has zero discriminant ∆ again.There are four real roots, with the middle two being equal (the abscissa is tangent to the quartic at its local maximum).Figure 2 applies -the lowermost curve.
The quartic has negatuve discriminant ∆ again.There are two distinct real roots.Figure 2 applies -curves below the lowermost curve.The discriminant of the quartic is negative and there are two distinct real roots.The minimum is below the x-axis, while the saddle is above it.The discriminant of the quartic is zero.As the quartic"sinks" with d becoming smaller, the two local minima "land" simultaneously on the abscissa, that is, when d " d : , the x-axis is tangent to the graph of the quartic at both its minima.There are two pairs of equal real roots.The discriminant of the quartic "bounces back" from zero and is again positive.The two local minima are now below the abscissa, while the local maximum is above it.There are four distinct real roots.
where r d is the single root of ∆pdqs -four real roots, the middle two of which -equal The discriminant of the quartic is again zero.The abscissa is tangent to the quartic at its local maximum.There are four real roots and the two in the middle are equal -at the local maximum.
The discriminant of the quartic is negative (it changes sign at r d).The local maximum now lies below the abscissa.There are two distinct real roots.
When c R rC 2 , C 1 s, one has ∆ 3 ă 0. The discriminant ∆pdq of the quartic has one real root d 0 .Thus it changes sign only once.The quartic has one stationary point.When d ą d 0 , the quartic has a positive discriminant ∆pdq and no real roots.Figure 3 applies -the dotted curve.
As c R rC 2 , C 1 s, one has ∆ 3 ă 0 again.This time the quartic has a zero discriminant ∆pdq and there are two equal real roots.Figure 3 applies -the solid curve.
The discriminant ∆pdq of the quartic is negative and there are two distinct real roots.Figure 3 applies -the dashed curve.

The Quartic Equation and its Tetrahedron
From the Complete Root Classification in the previous Section, one can immediately identify the necessary and sufficient conditions for the quartic to have four real roots.These are given, according to the root multiplicities, by the following four Theorems: Theorem 3. The quartic polynomial x 4 `ax 3 `bx 2 `cx `d has four distinct real roots if, and only if, its coefficients satisfy all of the following three conditions: piiiq pc ‰ C 0 and d 3 ă d ă d 2 q or pc " C 0 and r rThese are cases (xvii) and (xxvii), respectively, of the Complete Root Classification.sTheorem 4. The quartic polynomial x 4 `ax 3 `bx 2 `cx `d has four real roots, exactly two of which equal, if, and only if, its coefficients satisfy any one of the following three conditions:  The quartic polynomial and its regular tetrahedron regular tetrahedron and its symmetries [12], [13], [14], [4].
The following Theorem is stated by Northshiled in [4]: Given a quartic p 4 pxq with four real roots (at least two distinct), those roots are the first coordinate projections of a regular tetrahedron in IR 3 .That tetrahedron has a unique inscribed sphere, which projects onto an interval whose endpoints are the two roots of p 2 pxq.The vertices of an equilateral triangle around that sphere project onto the roots of p 1 pxq.This equilateral triangle is exactly the Siebeck-Marden-Northshield triangle for the cubic polynomial p 1 4 pxq " 4x 3 `3ax 2 `2bx `c.It lies in the xy-plane -this is the triangle MNP on Figure 4. Clearly, the Siebeck-Marden-Northshield triangle exists when b ă 3a 2 {8 and C 2 ď c ď C 1 .In particular, when b ă 3a 2 {8 and C 2 ă c ă C 1 , then ∆ 3 will be positive and p 1 4 pxq will have three distinct real roots.Hence, the quartic will have three distinct stationary points E 1 , E 2 , and E 3 (see Figure 4) and it will have either four distinct real roots, or four real roots with two of them equal, or two pairs of equal real roots -Theorem 3, Theorem 4, and Theorem 6, respectively.When c " C 0 , the side MP of the Siebeck-Marden-Northshield triangle will be parallel to the abscissa.In the case of b ă 3a 2 {8 and c " C 1,2 , the discriminant ∆ 3 will be zero and p 1 4 pxq will have three real roots, two of which -equal (Theorem 6).The Siebeck-Marden-Northshield triangle will have a side, perpendicular to the x-axis: NP , if c " C 1 , or MN , if c " C 2 .tetrahedron -as one cannot have two roots of the quartic between a minimum and a maximum, that is, for a quartic polynomial with four real roots, there could be only one extremum between two neighbouring roots.Of course, in the case of a double middle root, it coincides with the local maximum, while in the case of a triple root, it coincides with the saddle point of the quartic.And these are exactly the two extreme cases in the premise of this theorem.
(1) The four real roots of the quartic will lie in an interval of maximum length when they are at length L " p ? 2{2q ?3a 2 ´8b apart, as this is the maximum possible length of the x-coordinate projection of the tetrahedron upon rotations about the y-axis and the z-axis.Namely, one edge of the tetrahedron must have x-coordinate projection with the same length L, that is, the tetrahedron must have an edge parallel to the abscissa.There are two situations in which this can be realized.One can either have: (a) A double root of the quartic exactly at the x-coordinate projection ´a{4 of the centroid of the tetrahedron (which coincides with the centroid of Siebeck-Marden-Northshield triangle) and two simple roots on either side of the double root -at distance L{2.Hence, the four real roots of the quartic in this case are x 1,4 " ´a{4 p? 2{4q ?3a 2 ´8b and x 2,3 " ´a{4.This is achieved when 3a 2 ´8b ą 0, c " C 0 , and d " r d, that is, the third case of Theorem 4.
(b) Two pairs of double roots of the quartic (in this case, the projection of the tetrahedron onto the xy-plane is a square of side L, with its diagonals drawn).Both pairs of equal roots are equidistant from the centroid: x 1 " x 2 " ´a{4 `p1{4q ?3a 2 ´8b " ´a{4`p ?2{4qL " ´a{4`l{2 and x 3 " x 4 " ´a{4´p1{4q ?3a 2 ´8b " ´a{4´p ?2{4qL " ´a{4 ´l{2.[The Viète formulae for a quartic with two pairs of equal real roots are: 2px 1 `x2 q " ´a, px 1 `x2 q 2 `2x 1 x 2 " b, 2x 1 x 2 px 1 `x2 q " ´c, and x 2 1 x 2 2 " d.From the first two of these, one gets x 1 " r1{p2x 2 qspb ´a2 {4q to eliminate x 1 from 2px 1 `x2 q " ´a and get the quadratic equation 2x 2  2 `ax 2 `b ´a2 {4 " 0 from which all roots are determined.]This is achieved when 3a 2 ´8b ą 0, c " C 0 , and d " d : -the case of Theorem 5.The Siebeck-Marden-Northshield triangle in both cases (a) and (b) has a side parallel to the abscissa (as c " C 0 ) and the three real roots of p 1 4 pxq are σ 2 " ´a{4 and -the latter equidistant form σ 2 (which lies on the centroid).Note that in case (b), the double roots of the quartic coincide with the smallest and the largest root of p 1 4 pxq, that is, two of the vertices of the tetrahedron have x-coordinate projections equal to that of one the vertices of the equilateral triangle, while the other two vertices of the tetrahedron have x-coordinate projections equal to that of another vertex of the triangle.The remaining vertex of the triangle projects onto the x axis in the middle of the projections of the other two -exactly where the centroid of the tetrahedron projects.
(2) The four real roots of the quartic will lie in an interval of minimum length when the x-coordinate projection of the tetrahedron has a minimum length.Clearly, this (ii) For C 0 ď c ď C 1 : φ 1 ď p x 3 ď σ 3 , ´a{4 ď p x 2 ď ρ 1 , and ρ 1 ď p x 1 ď σ 1 .
Consider first C 2 ď c ď C 0 .Under Theorem 8, the smallest possible root of any quartic with four real roots is λ min " ´a{4 ´p? 3{4q ?3a 2 ´8b and the biggest possible root of any quartic with four real roots is λ max " ´a{4 `p? 3{4q ?3a 2 ´8b (when c " C 2 ).The smallest possible root of the cubic p 1 4 pxq (the left minimum of the quartic) is smaller than or equal to ρ 2 , hence the smallest root x 4 of the quartic is greater than or equal to λ min and smaller than or equal to ρ 2 .The next root x 3 of the quartic is between the left minimum and the maximum, that is, it is greater than or equal to σ 3 and smaller than or equal to ´a{4.The root x 2 is between the maximum and the right minimum, namely, it is greater than or equal to ρ 2 and smaller than or equal to φ 2 .Finally, the biggest root x 1 of the quartic is greater than or equal to σ 1 and smaller than or equal to λ max .Namely, for C 2 ď c ď C 0 one has: In a similar manner, for C 0 ď c ď C 1 one gets: As there is an overlap in the above intervals, these cannot be referred as isolation intervals of the roots of the quartic -as two roots can occur in any one of them.Consider, as first example, the quartic x 4 `3x 3 `2x 2 ´x ´19{20.For it, one has: a " 3, b " 2 ă 3a 2 {8, c " 2 which is between C 2 " ´1.2526 and C 0 " ´0.3750 (also, C 1 " 0.5026) and d " ´19{20 which is between d 3 " ´1.000 and d 2 " ´0.9288 (also, d 1 " 0.0967).Hence, under Theorem 3, there are four distinct real roots.Indeed, these are: x 1 " 0.6094, x 2 " ´0.7928, x 3 " ´1.2787, and x 4 " ´1.5379.One further has: L " 2.3452, R " 0.4787, λ min " ´5.0584, ρ 2 " ´1.2287, σ 3 " ´1.5792, ´a{4 " ´0.7500, φ 2 " 0.2074, σ 1 " 0.0792, and λ max " 3.5584.Clearly, the roots are within their prescribed intervals.
As second example, the quartic x 4 ´4x 3 `5x 2 ´p7{4qx ´1{5 has a " ´4, b " 5 ă 3a 2 {8, c " ´7{4 which is between C 0 " ´2.0000 and C 1 " ´1.4557 (also, C 2 " ´2.5443) and d " ´1{5 which is between d 3 " ´0.2659 and d 2 " ´0.1681 (also, d 1 " 0.1840).Again, under Theorem 3, there are four distinct real roots and they are: x 1 " 1.7679, x 2 " 1.4535, x 3 " 0.8682, and x 4 " ´0.0896.Additionally: L " 2.0000, R " 0.4082, λ min " ´2.6742, σ 3 " 0.2929, φ 1 " 0.1835, ρ 1 " 1.4082, ´a{4 " 1.0000, σ 1 " 1.7071, and λ max " 4.6742 -again, the roots are within their prescribed intervals.Similar considerations can be extended to the cases of a quartic with only two real roots, when the Siebeck-Marden-Northshield triangle still exists, but the tetrahedron does not -one can bound the two real roots with the help of the bounds of the stationary points of the quartic (the vertices of the Siebeck-Marden-Northshield triangle), which are invariant under the variation of the free term d, and the nearest roots of the quartic when it has four real roots, i.e. simply within different ranges of d.

2
The Siebeck-Marden-Northshield Triangle and Viète's Trigonometric Formulae for the Roots of the Cubic 2.1 The Siebeck-Marden-Northshield Triangle

Figure 2 Figure 3
Figure 2Figure3Quartic polynomials with three stationary points

Figure 3
applies -solid curve.(iii) b ą 8a 2 {3, c ‰ C 0 , d ă d 0 -two distinct real roots Again, one has ∆ 3 ă 0 and the discriminant ∆pdq of the quartic having only one real root d 0 .The quartic has one stationary point and negative discriminant ∆.There are two distinct real roots.Figure 3 applies -dashed curve.(iv) b ą 8a 2 {3, c " C 0 , d ą r d [where r d is the single root of ∆pdqs -no real roots

(v) b ą 8a 2
{3, c " C 0 , d " r d [where r d is the single root of ∆pdqs -two equal real roots One again has ∆ 3 " 0 as c " C 0 .The quartic has zero discriminant ∆.There are two equal real roots.Figure 3 applies -solid curve.(vi) b ą 8a 2 {3, c " C 0 , d ă r d [where r d is the single root of ∆pdqs -two distinct real roots

(xx) b ă 8a 2
{3, c " C 1,2 , d ą r d [where r d is the single root of ∆pdq] -no real roots When c " C 1 or c " C 2 , one has ∆ 3 " 0. The discriminant ∆pdq of the quartic has a double real root d : and a single real root r d (it may also have a treble real root r d).Thus it changes sign only once -at r d.The stationary points of the quartic are a global minimum and a saddle (the latter to the right of the minimum if c " C 1 and to the left of it if c " C 2 ).When d ą r d, the quartic has a positive discriminant ∆pdq and no real roots.(xxi) b ă 8a 2 {3, c " C 1,2 , d " r d [where r d is the single root of ∆pdq] -two equal real roots This time the quartic has a zero discriminant ∆pdq and there are two equal real roots.The abscissa is tangent to the graph at the double root -the minimum.The saddle point is above the abscissa.(xxii) b ă 8a 2 {3, c " C 1,2 , d : ă d ă r d [where r d is the single root of ∆pdq and d : is the double root of ∆pdqs -two distinct real roots

(
xxiii) b ă 8a 2 {3, c " C 1,2 , d " d : [where d : is the double root of ∆pdq]four real roots, three of which equal The discriminant of the quartic is zero.There are four real roots and three of them are equal.The quartic has a saddle point and it lies on the abscissa -at the triple root.The minimum is below the x-axis.(xxiv) b ă 8a 2 {3, c " C 1,2 , d ă d : [where d : is the double root of ∆pdq]two distinct real roots The discriminant of the quartic "bounces back" from zero and is again negative.There are two real roots.The saddle and the minimum of the quartic are both below the abscissa.(xxv) b ă 8a 2 {3, c " C 0 , d ą d : [where d : is the double root of ∆pdq] -no real roots When c " C 0 , one has ∆ 3 " 0. The discriminant ∆pdq of the quartic has a double real root d : and a single real root r d (it may also have a treble real root r d).The quartic has three stationary points, but its discriminant changes sign only once -at r d ă d : .The quartic is a symmetric curve with respect to its local maximum.When d ą d : , the quartic has a positive discriminant ∆pdq and no real roots.(xxvi) b ă 8a 2 {3, c " C 0 , d " d : [where d : is the double root of ∆pdq] -two pairs of equal real roots

(xxvii) b ă 8a 2
{3, c " C 0 , r d ă d ă d : [where d : is the double root of ∆pdq and r d is the single root of ∆pdqs -four distinct real roots

Figure 4
Figure 4 2 `2bx `c.The stationary points of p 4 pxq are either a single global minimum [when p 1 4 pxq has a single real root or a treble real root], or a local maximum, sandwiched by two local minima [when p 1 4 pxq has three distinct real roots], or a saddle and a global minimum [when p 1 Theorem 5.The quartic polynomial x 4 `ax 3 `bx 2 `cx `d has two pairs of equal real roots, if, and only if, its coefficients satisfy: Theorem 6.The quartic polynomial x 4 `ax 3 `bx 2 `cx `d has four real roots, exactly three of which equal, if, and only if, its coefficients satisfy:The quartic polynomial p 4 pxq " x 4 `ax 3 `bx 2 `cx `d with four real roots (at least two distinct) -subject to any of Theorem 3 to Theorem 6 -is associated with a