On an Improvement of the Numerical Application for Cardano’s Formula in Mathematica Software
Alicja Wr´obel∗, Edyta Hetmaniok∗, Mariusz Pleszczy´nski∗, and Roman Wituła∗,
∗ Institute of Mathematics, Silesian University of Technology, ul.Kaszubska 23, 44-100 Gliwice, Poland
Abstract—The aim of this paper is to develop a program for determining, by symbolic description, the roots of each real cubic polynomial on the basis of the well known Mathematica software. We have obtained a program completely satisfying our expectations and even more. For example, for many tested cases of the cubic polynomials, on the way of comparing the description of the roots of these polynomials received by using our program with their trigonometric form obtained as a result of geometric discussion or respective trigonometric transformations, we have got some new attractive relations of algebraic and trigonometric nature.
By applying our elaborated program we can also decide, symbolically, whether the given cubic polynomial is a Ramanujan cubic polynomial (one of two kinds). Moreover, in case of these polynomials we have got many new additional pieces of information essentially completing the facts discovered by now.
Additionally, we have solved some, posed ad hoc, theoretical problem.
Index Terms—cubic polynomials, Ramanujan cubic polynomi- als, Cardano’s formula.
I. INTRODUCTION
While preparing papers [12], [14] a part of computations has been done with the aid of Mathematica software. To our surprise, the program did not manage to derive, in the way as we expected, the symbolic transformations needed for determining the zeros of polynomials, especially the real cubic and quartic polynomials. We had to make by hand a part of the final transformations. For example, for the equation
x3−12x+ 13 = 0 (1)
Mathematica program gave us the following set of solutions 4
3
q1
2 −13 +i√ 87
+ 3 r1
2
−13 +i√ 87
,
− 2 1 +i√ 3
3
q1
2 −13 +i√ 87
−1 2
1−i√
3r3 1 2
−13 +i√ 87
,
− 2 1−i√ 3
3
q1
2 −13 +i√ 87
−1 2
1 +i√
3r3 1 2
−13 +i√ 87
.
We decided to change this situation which resulted in elaborat- ing the appropriate computer procedure calculating the roots of all real cubic polynomials starting from the Cardano’s formula.
Copyright c2016 held by the authors.
The program, developed by us, gave the following form of the solutions of equation (1):
−4 cos 1 3arctan
√87 13
!
, 4 sin π 6 ±1
3arctan
√87 13
! . The possibly excessive optimism about the effectiveness of symbolic computations realized by our program has been suppressed by other examples. For instance, for the equation
x3−12x+ 11 = 0, Mathematica answer are the following solutions
1, 1 2
−1±3√ 5
.
To the contrast our program produces the following trigono- metric form of the same solutions
−4 cos 1
3arctan3√ 15 11
!
, 4 sin π 6 ±1
3arctan3√ 15 11
! , but only numerically Mathematica verified the equality
1 = 4 sin π 6 −1
3arctan3√ 15 11
!
= 4 cos π 3 +1
3arctan3√ 15 11
! .
In the third section of this paper we present a number of such completely unexpected trigonometric relations obtained by comparing the ”model” decompositions of some cubic polynomials with the decompositions obtained by us by ap- plying the introduced Cardano’s type formulae (implemented for our computer procedure).
It should be also emphasized that the mentioned model polynomials and their decompositions were usually obtained in result of some trigonometric transformations of the known trigonometric relations (see [5], [17], [18], [19], [28], [33], [34]). In consequence, these two different ways of decom- posing the cubic polynomials resulted in many interesting equalities of trigonometric nature shedding a new light on many quantities, mysterious till now. For example, we dis- covered that the values of cos2k7π, k ∈ N, are strictly connected with the values ofarctan(3√
3), whereas values of
cos2k7π /
cos2k+27 π
, k ∈N, are strictly connected with the values of arctan√31
3 – see Section 3.
Finally, in the fourth section we describe the recently popu- lar subject matter concerning the Ramanujan cubic polynomi- als. Discussion executed for the goal of preparing this paper resulted in creating a new paper devoted to these polynomials [2] – the obtained there original result is announced at the end of Section 4.
As the Authors, we want to emphasize that our aim now is to initiate the investigations on the continuation of this work where we intend to concentrate on developing the algorithm for determining the roots of polynomials belonging to the selected families of polynomials of higher orders, for which such description of the roots is known (the examples are quartic polynomials, the modified Chebyshev polynomials [35]
and the selected quintics [8], [26]).
II. ZEROS OF CUBIC POLYNOMIALS–FINAL FORMULAE
From the Cardano procedure for finding the (complex) zeros of real cubic polynomial (see for example [11], [16], [20], [28], [35] – two last papers consider also because of their historical connotations):
p(z) :=z3+az+b
it follows that p(z) = 0 if and only if z = u+v where u∈ q3
−b2+√
∆,v ∈ q3
−2b−√
∆ and∆ = b22 + a33
, and where the respective complex roots (of the second and third order) are kept in mind. Let us discuss three cases with respect to the sign of discriminant ∆.
First we assume that∆<0 (which impliesa <0). Then
u∈ 3 r
−b 2 +i√
−∆, v∈ 3 r
−b 2 −i√
−∆
andp(z)possesses three real roots.
Now let us discuss two cases with respect to the sign of coefficientb.
First case, when b > 0: Additionally let us set ∆∗ :=
q
− 2b2
∆. We suppose that∆∗∈[0,∞), since−∆≥0⇔ (∆∗)2=−1−a3 2a3b2
≥0.
We have
−b 2 ±i√
−∆ = r
−a 3
3 exp
±i(π−arctan ∆∗) and (each sign from the circle corresponds with two signs± not included in the circle):
3
r
−b 2 ±i√
−∆ =
= r
−a 3exp
i
0
±23π
± 1
3(π−arctan ∆∗)
,
which implies (the formulae below are satisfied by all three real zeros ofp(z)):
z=u+v= 2 r
−a 3cos
0
±23π
+1
3(π−arctan ∆∗)
= 2 r
−a 3cos
1 3
±π 3π
−arctan ∆∗
(2)
=
−2p
−a3cos 13arctan ∆∗ , 2p
−a3cos 13(±π+ arctan ∆∗)
, (3)
=
−2p
−a3cos 13arctan ∆∗ , p−a3 cos 13arctan ∆∗
±√
3 sin 13arctan ∆∗ , (4) or in the equivalent form
z=u+v=
−2p
−a3cos 13arctan ∆∗ , 2p
−a3cos 13(π+ arctan ∆∗) , 2p
−a3sin 16(π+ 2 arctan ∆∗) ,
(5)
=
−2p
−a3cos 13arctan ∆∗ , 2p
−a3sin π6 ±13arctan ∆∗ .
The other form of zeros ofp(z)can be deduced if we introduce a new parameter ϕsetting
∆∗= tanϕ≥0.
Then −a33
=
b 2 cosϕ
2
. Hence, if we set a = −3α2, b= 2β3cosϕ, thenα=±β. Finally we obtain the following implication: iftanϕ≥0 and2αcosϕ >0, then
z3−3α2z+ (2 cosϕ)α3= 0⇔
⇔z=
−2αcosϕ3, α cosϕ3 ±√
3 sinϕ3 .
(6) For example, we get
z3−3α2z+α3= 0⇔z=
−2αcosπ9, α cosπ9 ±√
3 sinπ9 ,
= 2αcos 2kπ
9
, k= 1,2,3,
z3−3α2z+√
2α3= 0
⇔z=
−2αcos12π =−√
√3+1 2
α, α cos12π ±√
3 sin12π
= ( √
√2α,
√3−1 2 α,
= ( √
2α
±√
√3−1 2 α, z3−3α2z+√
3α3= 0⇔z=
−2αcos18π, α cos18π ±√
3 sin18π ,
= 2α(−1)k−1sin 2kπ
9
, k= 1,2,3.
Second case, when b <0: We get
−b 2 ±i√
−∆ = r
−a 3
3 exp
±iarctan ∆∗
and next
3
r
−b 2 ±i√
−∆ =
= r
−a 3exp
i
0
±23π
±1
3arctan ∆∗
, which implies
z=u+v= 2 r
−a 3cos
0
±23π
+1
3arctan ∆∗
(7)
= 2p
−a3cos 13arctan ∆∗ ,
−p
−a3 cos 13arctan ∆∗
∓√
3 sin 13arctan ∆∗ , (8)
=
2p
−a3cos 13arctan ∆∗ ,
−2p
−a3cos 13(π+ arctan ∆∗) ,
−2p
−a3sin 16(π+ 2 arctan ∆∗) ,
(9)
= 2p
−a3cos 13arctan ∆∗ ,
−2p
−a3sin π6 ±13arctan ∆∗ .
Similarly as in the previous case, if we set ∆∗ = tanϕ≥0, a=−3α2,b= 2α3cosϕ <0, then
z3−3α2z+(2 cosϕ)α3= 0⇔z=
−2αcosϕ3, α cosϕ3 ±√
3 sinϕ3 . For example, we obtain (below we assume that α > 0, moreover, the respective values of sine and cosine functions can be found in [20], [21], [28]):
z3−3α2z−√
2α3= 0
⇔z=
2αcos12π =
√√3+1 2 α,
−α cos12π ±√
3 sin12π
= ( −√
2α,
−√
√3+1 2 α,
= ( −√
2α,
1±√
√ 3 2 α, z3−3α2z−
√5 + 1 2 α3= 0
⇔z=
2αcos15π =α4√
5−1 +√ 3p
10 + 2√ 5
,
−α cos15π ±√
3 sin15π
=
= ( α
4
√5−1−√ 3p
10 + 2√ 5
,
α 2(1−√
5),
,
= ( α
2(1−√ 5),
α 4
√
5−1±√ 3p
10 + 2√ 5
.
We only need to discuss two more (rather trouble-free) cases.
If ∆ > 0, then we have precisely one real zero and two conjugate complex roots
z1=u−v,
z2=ei23πu−e−i23πv=−1
2(u−v) +i
√ 3
2 (u+v), z3=z2,
whereu:= 3 q
−b2+√
∆,v:= 3 qb
2+√
∆.
Hence, on the basis of the same formulae, if ∆ = 0, then we get
z1=−23 rb
2, z2=z3= 3 rb
2. III. UNEXPECTED IDENTITIES
By using the cubic polynomials (and their decompositions) from papers [17], [18], [28], [30], [33], [34], [35] and by applying the discussed here Cardano’s formulae for the roots of cubic polynomials, we obtained the completely unexpected trigonometric identities (of course we present here just few selected examples denoted by A – H). Let us explain only that all the presented equalities are the consequence of the fact that the decompositions of cubic polynomials, given in the cited above papers, result, in the first place, from the appropriate trigonometric transformations (also by geometric discussion) and not from the application of the Cardano’s formulae.
We obtain A)
z3+z2−2z−1 =
3
Y
k=1
z−2 cos2kπ 7
, which implies
6 cos2π
7 =−1 + 2√ 7 cos
1
3arctan(3√ 3)
,
6 cos4π
7 =−1−2√ 7 cos
1 3
π+ arctan(3√ 3)
,
6 cos8π
7 =−1−2√ 7 sin
1 6
π+ 2 arctan(3√ 3)
, B)
z3−3z2−4z−1 =
3
Y
k=1
z− cos2k7π cos2k+27 π
! , which follows
cos8π7
cos4π7 = 1 + 2 r7
3cos 1
3arctan 1 3√
3
, cos4π7
cos2π7 = 1−2 r7
3cos 1
3
π+ arctan 1 3√ 3
,
As an example we propose to examine the following polynomial
2
Y
k=0
x−2 cos
2kα
=x3+
1 + 2 cos 3α−sin(9α/2) sin(α/2)
x2
+
2 cos 3α−2 cos 4α−1 +sin(13α/2) sin(α/2)
x−sin 8α sinα, which gives forα= 2π/7,2π/9the polynomial of type A and polynomial
x3−3x+ 1 =
3
Y
k=1
x−2 cos 2kπ/9
which can be simply generated on the basis of discussion presented item E, respectively.
cos2π7
cos8π7 = 1−2 r7
3sin 1
6
π+ 2 arctan 1 3√
3
. We note that (see [34]):
3
X
k=1
3
v u u t
cos2k7π cos2k+27 π = 0.
C)
z3−7z2+ 7z+ 7 =
3
Y
k=1
z−√
7 cot2kπ 7
, which implies
3√
7 cot8π
7 = 7 + 4√ 7 cos
1
3arctan 3√ 3
,
3√
7 cot2π
7 = 7−4√ 7 cos
1 3
π+ arctan 3√ 3
,
3√
7 cot4π
7 = 7−4√ 7 sin
1 6
π+ 2 arctan 3√ 3
; D)
z3−z2−z−1 7 =
3
Y
k=1
z−1 + 4
√
7sin2kπ 7
, which follows
√6
7sin8π
7 = 1−2 cos 1
3arctan3√ 3 13
! ,
√6
7sin2π
7 = 1 + 2 cos 1
3 π+ arctan3√ 3 13
!!
,
√6
7sin4π
7 = 1 + 2 sin 1
6 π+ 2 arctan3√ 3 13
!!
. To the contrast, we note that
z3+z2−z+1 7 =
3
Y
k=1
z−
√7
7 tan2kπ 7
! . E)
z3+ 3√
3z2−3z−√ 3
=
z−tan2π
9 z+ tan4π
9 z−tan8π 9
, which implies the relation
tan2kπ
9 = 4 sin2kπ
9 + (−1)k√ 3, for k= 1,2,3, or the equivalent equalities
tan2π
9 = 4 sin2π 9 −√
3, tan4π
9 = 4 cos π 18+√
3, tan8π
9 = 4 sinπ 9 −√
3.
To the contrast, we note that z3+√
3z2−3z−
√ 3 3
=
z−cot2π
9 z+ cot4π
9 z−cot8π 9
, which implies the relation
(−1)k−1√
3 cot2kπ
9 = 4 cos2kπ 9 −1, for k= 1,2,3, or the equivalent equalities
cot2π
9 = 4 cos2π9 −1
√3 , cot4π
9 = 1−4 sin18π
√3 ,
−cot8π
9 =4 cosπ9 + 1
√
3 .
F)
z3+1−√ 13
2 z2−z+
√13 + 3 2
=
z−2 cos2π
13 z−2 cos6π
13 z−2 cos8π 13
, (10) which implies
12 cos2π
13 =−1 +√ 13 + 2
q
26−2√ 13×
×cos 1
3arctan5 + 2√ 13 3√
3
! ,
12 cos6π
13 =−1 +√ 13−2
q
26−2√ 13×
×cos 1
3 π+ arctan5 + 2√ 13 3√
3
!!
,
12 cos8π
13 =−1 +√ 13−2
q
26−2√ 13×
×sin 1
6 π+ 2 arctan5 + 2√ 13 3√
3
!!
.
G)
z3+1 +√ 13
2 z2−z−
√13 + 3 2
=
z−2 cos4π
13 z−2 cos10π
13 z−2 cos12π 13
, (11) from which we get
12 cos2π
13 =−1−√ 13 + 2
q
26 + 2√ 13×
×cos 1
3arctan5−2√ 13 3√
3
! ,
12 cos10π
13 =−1−√ 13−2
q
26 + 2√ 13×
×cos 1
3 π+ arctan5−2√ 13 3√
3
!!
,
12 cos12π
13 =−1−√ 13−2
q
26 + 2√ 13×
×sin 1
6 π+ 2 arctan5−2√ 13 3√
3
!!
. Moreover, surprisingly the following relation holds
√13−4
3 −2 cos8π
13 ≈0.577727
≈C=0.57721,
whereCis the Eulerian constant called also as the Euler- Mascheroni constant (see [13]).
H)
z3−z−1 = (z−τ0−1)(z−i√
τ0eiΨ)(z+i√
τ0e−iΨ).
The above polynomial is called the Perrin polynomial, also called as the Siegel’s polynomial (see [7], [34]).
Constant τ0−1 plays an important role in estimating the Mahler measureM(f)of polynomials f over C, which are not reciprocal. This estimation is of the formM(f)≥ τ0−1 and is optimal (see [23], [25]). To the contrast let us note that τ0 is the only positive root of polynomial τ3 +τ2 −1 (the same fact holds for the polynomial τ5+τ−1 = (τ3+τ2−1)(τ2−τ+ 1)) and
Ψ := arcsin 1 2p
τ03.
The number−τ0is the only real root of polynomialτ3− τ2+ 1. Furthermore, we get
6τ0=−2 +√3 4
3
q
25−3√ 69 + 3
q
25 + 3√ 69
,
√3
18τ0−1= 3 q
9 +√ 69 + 3
q 9−√
69,
√τ0sin Ψ = 1
2τ0−1, √
τ0cos Ψ = 1 2τ0−1
q 4τ03−1 and
i√
τ0eiΨ= 1 2τ0−1
−1 +i q
4τ03−1
,
i2√3 18√
τ0eiΨ=−
3
q 9 +√
69 + 3 q
9−√ 69
+i√ 3
q3
9 +√ 69− 3
q 9−√
69
. Hence we deduce the equality
q
4τ03−1 =√ 3
p3
9 +√ 69−p3
9−√ 69 p3
9 +√ 69 +p3
9−√ 69
.
Furthermore, from the equalityz3=z+1forz=τ0−1we obtain the following decomposition of τ0−1 in the nested third roots
3
r1 2 − 1
18
√ 69+3
r1 2+ 1
18
√ 69 = 3
r 1 + 3
q 1 +√3
1 +. . . We note that the convergence of the above nested radical from Theorem 3.2 in [15] holds. Moreover, we observe that the similar equality is fulfilled also for the only positive root zk of the polynomial zk−z−1, k ≥ 3, having the form zk= k
q 1 +pk
1 +√k 1 +. . ..
In order to emphasize the importance of this section let us introduce two more decompositions of the cubic polynomials, which we have not found in literature till now and which essentially complete the decompositions presented in items F) and G).
So from (10) and (11) we can obtain the decompositions
z−2 sin4π
13 z−2 sin10π
13 z−2 sin12π 13
=z3− s
13 + 3√ 13 2 z2+√
13z− s
13−3√ 13 2 and
z−2 sin2π
13 z−2 sin6π
13 z+ 2 sin8π 13
=z3− s
13−3√ 13 2 z2−√
13z+ s
13 + 3√ 13
2 .
Remark III.1. In this section we have dealt with expressions of the form cos 13arctan ∆∗
and sin 13arctan ∆∗ . Let us notice that one can find in literature some very interesting decompositions of such expressions on the nested square roots.
For example, in [3], [4] we can find the following Ramanujan formula
n→∞lim an= A−1 6 +2
3
√4a+Asin 1
3arctan2A+ 1 3√
3
,
where A:=√
4a−7,a≥2, and a1=√
a, a2= q
a−√
a, a3= r
a− q
a+√ a,
a4= s
a− r
a+ q
a+√ a, . . .
and where the sequence of signs −,+,+, . . ., appearing in this nested radicals, has period 3. In case a= 44 we obtain (see equalities in examples A, C, F for possible connections):
n→∞lim an= 2 + 2√ 21 sin
1
3arctan 3√ 3
.
IV. RAMANUJAN’S CUBIC POLYNOMIALS
V. Shevelev and R. Wituła in papers [1], [24], [29], [31]
have distinguished and discussed the so called Ramanujan’s cubic polynomials and Ramanujan’s cubic polynomials of the second kind, denoted for shortness by RCP and RCP2, respectively. And all this could happen thanks to the great Indian mathematician Srinivasa Ramanujan who proposed the proof of the following equalities (see [22]):
1 9
1/3
− 2
9 1/3
+ 4
9 1/3
= (√3
2−1)1/3,
cos2π 7
1/3
+
cos4π 7
1/3
+
cos8π 7
1/3
= 5−3√3 7 2
!1/3 ,
cos2π
9 1/3
+
cos4π 9
1/3 +
cos8π
9 1/3
= 3√3 9−6
2
!1/3
.
It is easy to connect the above equations with the following problem: for which cubic polynomials Q(x) (with all real roots) of the form
Q(x) = (x−ξ1)(x−ξ2)(x−ξ3) =x3+px2+qx+r there exists a functionf(p, q, r)that possesses possibly ”sim- ple” algebraic form and for which the following equality holds
p3
ξ1+p3 ξ2+p3
ξ3=p3
f(p, q, r).
Although some attempts to solve this interesting problem have been undertaken (see for example the respective Ramanujan Theorem in [4] or in the second Notebook of Ramanujan [22]), only the mentioned above V. Shevelev and R. Wituła succeeded in distinguishing the appropriate families of cubic polynomials and in describing properties of these polynomials (see also the paper [1]). Let us only recall that if the following conditions are satisifed
r6= 0, p√3
r+ 3√3
r2+q= 0,
b 2
2 + a33
<0,
(12) where
a:=q−p2
3 , b:= 2 27p3−1
3pq+r,
thenQ(x)is a RCP and the following identities hold (the last expressions in all three formulae below are prepared for the algorithmic applications):
√3
x1+√3 x2+√3
x3=
−p−6r1/3+ 3(9r−pq)1/31/3
= sgn
−p−6r1/3+ 3 sgn(9r−pq)|9r−pq|1/3
×
−p−6r1/3+ 3 sgn(9r−pq)|9r−pq|1/3
1/3
,
√3
x1x2+√3
x1x3+√3
x2x3= 3 q
q+ 6r2/3−3p3
9r2−pqr
= sgn
q+ 6r2/3−3 sgn(9r2−pqr)|9r2−pqr|1/3
×
q+ 6r2/3−3 sgn(9r2−pqr)|9r2−pqr|1/3
1/3
, as well as the so called Shevelev’s formula [24], [31]:
3
rx1
x2
+ 3 rx2
x1
+ 3 rx1
x3
+ 3 rx3
x1
+ 3 rx2
x3
+ 3 rx3
x2
= 1
√3
x1x2x3
3
q
x21x3+ 3 q
x1x23+ 3 q
x21x2+ 3 q
x1x22 +3
q
x22x3+ 3 q
x2x23
= sgnpq r −9
3
r
pq r −9
. To the contrast, if instead of conditions (12) the following conditions are fulfilled
r6= 0,
p3r+ 27r2+q3= 0,
b a
2 + a33
<0,
(13) thenQ(x)is a RCP2 and the following relations hold p3
ξ1+p3 ξ2+p3
ξ3=
−p−6√3 r
−33 q
3√3
r(q+p√3 r)−33
q
(p+ 3√3
r)(q+ 3√3 r2)
1/3 , as well as
3
s ξ1
ξ2
+ 3 s
ξ2
ξ1
+ 3 s
ξ1
ξ3
+ 3 s
ξ3
ξ1
+ 3 s
ξ2
ξ3
+ 3 s
ξ3
ξ2
= 3 r 3
r2/3(q+p√3 r) + 3
rpq r + 3
r2/3(q+p√3 r+ 33
√ r2).
Let us notice that a cubic polynomial p(x) =x3+px2+ qx+r, which is either RCP or RCP2, is simultaneously RCP and RCP2 if and only if pq= 0 which implies that either
p(x) =x3−3√3
r2x+r=
x−2√3
rcos2π 9
×
×
x−2√3
rcos4π
9 x−2√3
rcos8π 9
or
p(x) =x3−3√3
rx2+r=
x−1 2
√3
rsec2π 9
×
×
x−1 2
√3
rsec4π
9 x−1 2
√3
rsec8π 9
, where r ∈ R\ {0}. On the other hand we know that the polynomials belonging to families RCP and RCP2 share many common analytical-algebraic properties (see [14], [24]).
Let us present now the examples of cubic polynomials with indicating the Ramanujan classes of polynomials they belong:
1◦p(x) =x3+3x2−3√3
2x+1is the RCP2 polynomial which is not the RCP one. It has the following zeros (we apply here our formulae (5)):
x1=−1−2 q
1 +√3
2 cos 1
3arccot 3 p4√3
2−5
! ,
x2=−1 + 2 q
1 +√3
2 cos 1
3 π+ arccot 3 p4√3
2−5
!!
,
x3=−1 + 2 q
1 +√3
2 sin 1
6 π+ 2arccot 3 p4√3
2−5
!!
. Additionally we observe that
3 p4√3
2−5
= q
3(25 + 20√3
2 + 16√3 4) =
√ 3(√3
2 + 1)
√3
2−1 , so by identityarctanx= arccot1x for x >0we get
arccot 3 p4√3
2−5
= arctan
√3(√3 2 + 1)
√3
2−1 .
Using formulae (3)–(5) from [29] (or the respective ones from [31]) we obtain the following Ramanujan type equalities (for real roots of third order):
√3
x1+√3 x2+√3
x3= 0,
3
rx1 x2
+ 3 rx2
x1
+ 3 rx1
x3
+ 3 rx3
x1
+ 3 rx2
x3
+ 3 rx3
x2
=−3. (14) We also havex1+x2+x3=−3and
√3
x1x2+√3
x1x3+√3
x2x3=−3 q
3(√3 2 + 1).
Hence we deduce the relation P(x) = (x−√3
x1)(x−√3
x2)(x−√3 x3)
=x3− 3 q
3(√3
2 + 1)x+ 1
(15) and next, by applying the Cardano’s formulae (5) we get
√3
x1=− 2
√3
3
6
q 1 +√3
2 cos 1
3arctan1 3
q 4√3
2−5
,
√3
x2= 2
√3
3 q6
1 +√3 2 cos
1 3
π+ arctan1 3
q 4√3
2−5
,
√3
x3= 2
√3
3 q6
1 +√3 2 sin
1 6
π+ 2 arctan1 3
q 4√3
2−5
. By comparing these relations with the formulae for values of x1,x2,x3 we obtain the identity
xk+ 1
√3
xk
= 3 q
3(√3
2 + 1), k= 1,2,3,
It is well known that each polynomialq(x)from RCP2 class has the from (see [31]):
q(x) =x3+ 3√3
krx2−33 q
(k+ 1)r2x+r,
wherek, r∈R,r6= 0. Then for the rootsx1,x2,x3 ofq(x)the following Shevelev’s identity holds
3
rx1
x2
+ 3 rx2
x1
+ 3 rx1
x3
+ 3 rx3
x1
+ 3 rx2
x3
+ 3 rx3
x2
=√3 9
q3
√3
k−√3
k+ 1 + 3 q
(√3
k+ 1)(1−√3 k+ 1)
.
Hence and from (14) we deduce the equality
√3
3 =√3
2 + 1q3
√3
2−1.
(this identity results directly from formula (15)). It means that the following ”unexpected” polynomial identity holds
p
q3
3(√3
2 + 1)x−1
= 3(√3
2 + 1)P(x). (16) Moreover, by ”numerical experiment” we find that
√3
x1x2+√3
x1x3+√3
x2x3+√3 2 + 3
q
√3
2−1
=√3 2 + 3
q
√3
2−1− 3 q
3(√3
2 + 1)≈0.00545.
2◦q(x) =x3+x2−2x−1 = (x−2 cos2π7)(x−2 cos4π7)(x−
2 cos8π7 )– an example of the RCP polynomial which is not the RCP2 one.
3◦ The polynomials from examples B) – H) presented in Section 3 are neither RCP polynomials nor RCP2 ones.
Announcement
While preparing this section we have solved unexpectedly one more problem. We have proven that the polynomials
R(x;p) :=x3+ 9px2+ 23 6
1±√ 93
3
p2x
− 6
1±√ 93
3
p3= (x−x1)(x−x2)(x−x3), (17)
wherep∈R\ {0}and (for the upper signs):
x1
p =−3−6 q
6(√ 93−1)
23 cosarccot√ 31
3 ,
x2
p =−3 + 6√ 6(√
93−1)
23 cosπ+ arccot√ 31
3 , (18)
x3
p =−3 + 6√ 6(√
93−1)
23 sinπ+ 2arccot√ 31 6 are the only RCP such that polynomial
(x−√3
x1)(x−√3
x2)(x−√3 x3)
belongs to the RCP2 family (see [2]). By applying ”our”
Cardano’s formulae and a bit of Mathematica ”magic sim- plifications” we obtain the following relations
3
rx1
p =−1 +√
93 + 4p
106−9√
93 cos 13ϕ 2p3
182√
93−1497
,
3
rx2
p =−1−√
93 + 4p
106−9√
93 cos 13(π+ϕ) 2p3
182√
93−1497
,
3
rx3
p = −1−√
93 + 4p
106−9√
93 sin 16(π+2ϕ) 2p3
182√
93−1497
,
where ϕ= arctan 12√
3 9 +√ 93
x1, x2, x3 are defined by formulae (18).
V. CONCLUSION
In this paper we have presented the algorithms for determining the roots of cubic complex polynomials. The proposed algorithms generate the descriptions of these roots with the aid of radicals of trigonometric functions. The examples of testing polynomials, presented in this paper, have been originally generated by using the direct methods different than the given here algorithms. In consequence, by applying the algorithms presented here we received, the most often, the new and different symbolic descriptions of the sought roots of the given cubic polynomials. It gave us the possibility, by comparing the obtained and the testing descriptions, to reveal many new identities and relations. Additionally some conjecture arose about the possible existence of description of the values of functionscos 13arctanα
, sin 13arctanα in the form of radicals of variable α, whereα is also a radical defined on the set of rational numbers. We intend to discuss this problem in a separate paper. Moreover, let us notice that our algorithm verifies whether the given cubic polynomial belongs to the RCP or RCP2 class.
Also in a separate paper (see [36]) we intend, as we declared at the end of Introduction, to extend the discussion of the symbolic description of the roots of cubic polynomials, undertaken in this paper, for the roots of quartic polynomials and the selected polynomials of higher order. Next, we plan to adapt them numerically since we count on some better results then the ones obtained with the aid of Mathematica software.
REFERENCES
[1] S. Barbero, U. Cerruti, N. Murru, M. Abrate:Identities involving zeros of Ramanujan and Shanks cubic polynomials, J. Integer Seq.16(2013), article 13.8.1.
[2] B. Bajorska-Harapi´nska, M. Pleszczy´nski, R. Wituła,When RCP gener- ates RCP2, that is one more property of the Ramanujan polynomials, in preparation.
[3] B.C. Berndt:Ramanujan’s Notebooks, Part IV, Springer, New York 1994.
[4] B.C. Berndt, S. Bhargava: Ramanujan – for Lowbrows, Amer. Math.
Monthly100No 7 (1993), 644–656.
[5] W.A. Beyer, J.D. Louck, D. Zeilberger:Math Bite: A generalization of a curiosity that Feynmann remembered all his life, Math. Magazine69No 1 (1993), 43–44.
[6] B.N. Delone:Algebra (Theory of algebraic equations), in Mathematics, its subject, methods and significance, Part I, Academy of Science Press, Moscow 1956.
[7] A. Dubickas, K.G. Hare, J. Jankauskas:There are no two non-real conju- gates of a Pisot number with the same imaginary part, arXiv:1410.1600v1 [math.NT].
[8] D.S. Dummit:Solving solvable quintics, Math. Comp.57(1991), 387–
401.
[9] H.B. Dwight: Tables of Integrals and Other Mathematical Data, The MacMillan Company, New York 1961.
[10] S.K. Gupta, W. Szyma´nski:Cubic polynomials with rational roots and critical points, College Math. J.41No 5 (2010), 365–369.
[11] A. Heefer, T. Rothman:On remembering Cardano anew, Math. Intelli- gencer36No 4 (2014), 53–66.
[12] E. Hetmaniok, P. Lorenc, M. Pleszczy´nski, R. Wituła:Iterated integrals of polynomials, Appl. Math. Comput.249(2014), 389–398.
[13] J.C. Lagarias:Eulers constant: Eulers work and modern developments, Bull. Amer. Math. Soc.50(2013), 527-628; arXiv:1303.1856.
[14] P. Lorenc, M. Pleszczy´nski, R. Wituła:Representation of the sum of powers of the firstnpositive integers with multiple integral, [in review]
[15] C.D. Lynd:Using difference equations to generalize results for periodic nested radicals, Amer. Math. Monthly121No 1 (2014), 45–59.
[16] G.P. Matvievskaya:Ren´e Descartes 1596–1650, Nauka, Moscow 1976 (in Russian).
[17] P.S. Modenov:Problems in Geometry, Mir Publishers, Moscow 1981.
[18] S.G. Moreno, E.M. Garcia-Caballero: A Geometric proof of Morrie’s Law, Amer. Math. Monthly122(2015), 168.
[19] T.J. Osler: Cardan polynomials and the reduction of radicals, Math.
Mag.74(2001), 26–32.
[20] F.W.I. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark:NIST Handbook of Mathematical Functions, Cambridge Univ. Press, 2010.
[21] A.P. Prudnikov, A.J. Bryczkov, O.I. Mariczev: Integrals and series, Elementary Functions, Nauka, Moscov 1981 (in Russian).
[22] S. Ramanujan: Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay 1957.
[23] A. Schinzel:Solved and unsolved problems on polynomials, 149–159 in Panoramas of Mathematics Banach Center Publications, vol. 34, Warszawa 1995.
[24] V. Shevelev: On Ramanujan cubic polynomials, South East Asian J.
Math. & Math. Sci.8(2009), 113122.
[25] C.J. Smyth:On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc.3(1971), 169175.
[26] B.K. Spearman, K.S. Williams: Characterization of solvable quintics x5+ax+b, Amer. Math. Monthly101No. 10 (1994), 986992.
[27] J.-P. Tignol:Galois’ Theory of Algebraic Equations, World Scientific, New Jersey 2001.
[28] R. Wituła:Complex Numbers, Polynomials and Partial Fractions De- compositions, Parts 1, 2 and 3, Silesian University of Technology Press, Gliwice 2010 (in Polish).
[29] R. Wituła:Full description of Ramanujan cubic polynomials, J. Integer Seq.13(2010), article 10.5.7.
[30] R. Wituła:On Some Applications of Formulae for Sums of Unimodular Complex Numbers, PKJS Publishers, Gliwice 2011.
[31] R. Wituła:Ramanujan cubic polynomials of the second kind, J. Integer Seq.13(2010), article 10.7.5.
[32] R. Wituła:Ramanujan type trigonometric formulae, Demonstratio Math.
45No. 4 (2012), 779–796.
[33] R. Wituła, E. Hetmaniok, D. Słota, N. Gawro´nska:Sums of the rational powers of roots of the polynomials, International Journal of Pure and Applied Mathematics85No 1 (2013), 179–191.
[34] R. Wituła, P. Lorenc, M. R´o˙za´nski, M. Szweda: Sums of the rational powers of roots of cubic polynomials, Zesz. Naukowe Politechniki ´Sla¸skiej, Seria: Matematyka Stosowana z.4 (2010), 17–34, http://mat.polsl.pl/zn/zeszyty/z4/znps mat stos 04 2014 str 017- 034.pdf.
[35] R. Wituła, D. Słota:Cardano’s formula, square roots, Chebyshev poly- nomials and radicals, J. Math. Anal. Appl.363(2010), 639–647.
[36] A. Wr´obel, E. Hetmaniok, M. Pleszczy´nski, R. Wituła: Symbolic de- scription of roots of polynomials and their numerical implementation – better than in Mathematica software?, in preparation.
