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Questions tagged [cubics]

This tag is for questions relating to cubic equations, these are polynomials with $~3^{rd}~$ power terms as the highest order terms.

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-1 votes
0 answers
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Add constraints to cubic or quintic polynomial

How can I add constraints to cubic or quintic polynomial such that the generated line is within a region. For example in the blue colored region below: Image_Graph For example if I generate a line ...
Pratham's user avatar
11 votes
7 answers
5k views

Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

I am studying the history of complex numbers, and I don't understand the part on the screenshots. In particular, I don't understand why a cubic always has at least one real root. I don't see why the ...
Tereza Tizkova's user avatar
2 votes
0 answers
62 views

Are the 28 bitangents on quartic curves bounded distance away from each other?

We know that the 28 bitangents on a smooth plane quartic curve over $\mathbb{C}$ are all distinct. Are the bitangents bounded distance away from each other? More precisely, is there a constant $d>0$...
Weiyan Chen's user avatar
0 votes
0 answers
25 views

Question about real inflection points of cubic curves in P^2(C)

An elementary question about real inflection points of cubics: Textbooks mention that non-singular cubics in $P^2(C)$ have 3 real and 6 complex inflections and show the Hesse normal form $ x^3 + y^3 +...
Uri Elias's user avatar
-1 votes
0 answers
46 views

Quicker and non-trivial methods for solving Cubic Equation

Motivation : There have been many elementary ways like Hit-and-trial method, Polynomial division and others used in teaching how to solve cubic equation. I wanted to find a method that is faster to ...
BeaconiteGuy's user avatar
0 votes
1 answer
82 views

What is a curve at $y=\infty$ mean?

On the wikipedia for the trident curve, $xy+ax^3+bx^2+cx=d$, two graphs are shown: Both are for the case where $a=b=c=d=1$, with the first matching what I find in desmos, but the latter being the '...
Eli Bartlett's user avatar
  • 1,685
1 vote
0 answers
49 views

Why is the locus of points given two segments AB and CD such that APB=CPD give a degree-3 curve (in complex proj plane)?

The isoptic cubic is defined as the locus of points given two segments AB and CD (similarily oriented) such that APB=CPD (directed angles). By elementary geometry this would go through AB $\cap$ CD, ...
user118161's user avatar
1 vote
0 answers
37 views

Proof of Thomson cubic pivotal property without coordinates

The Thomson cubic is defined as the cubic going through A,B,C, the three side midpoints, the three excenters. Is there a way to prove its pivotal property (any two isogonal conjugates on it have a ...
user118161's user avatar
0 votes
0 answers
31 views

Can axes $x$ and $y$ be rotated to eliminate the crossed product terms in a cubic form?

I've just learned that it is posible to rotate the axes $x$ and $y$ to obtain the axes $x'$ and $y'$ such that the quadratic form $$ax^2+bxy+cy^2$$ converts to $$\lambda _1x'^2+\lambda _2y'^2$$ So, is ...
Manuel Ocaña's user avatar
0 votes
0 answers
53 views

Solving a mixed system of 2 cubic and 2 quadratic equations with 4 unknowns

I tried plugging these cubic and quadratic equations into Wolfram Alpha and Symbolab but both said the same thing, too much computing time required. Now I am struggling to solve these equations and I ...
Kyle Liu's user avatar
0 votes
0 answers
49 views

For cubic surface, if $\dim(\operatorname{Sing}(X))\geq 1$ then a line is contained a in $\operatorname{Sing}(X)$

Let $X$ to be an irreducible cubic surface in $\mathbb{P}^3_{\mathbb{C}}$. Is it true that if $\dim(\operatorname{Sing}(X))\geq 1$, then a line is contained in $\operatorname{Sing}(X)$? I.e, does a ...
ben huni's user avatar
  • 173
1 vote
0 answers
40 views

Is there an isotomic analogous of circular points of infinity?

In isogonal pivotal (with pivot at the line of infinity) cubics with respect to a triangle $\triangle ABC$. By a suitable projective transformation, fixing $A$,$B$, and $C$, sending the incenter to ...
Curious's user avatar
  • 37
1 vote
2 answers
62 views

Prove that $a=0$ if and only if $b=0$ for the cubic $x^3 + ax^2 + bx + c=0$ whose roots all have the same absolute value.

Take three real numbers $a, b$ and $c$ such that the roots of equation $x^3+ax^2+bx+c=0$ have the same absolute value. We need to show that $a=0$ if and only if $b=0$. I tried taking the roots as $p, ...
user1299519's user avatar
2 votes
1 answer
56 views

Show that if $x=-1$ is a solution of $x^{3}-2bx^{2}-a^{2}x+b^{2}=0$, then $1-\sqrt{2}\le b\le1+\sqrt{2}$

$$x^{3}-2bx^{2}-a^{2}x+b^{2}=0$$ Show that if $x=-1$ is a solution, then $1-\sqrt{2}\le b\le1+\sqrt{2}$ I subbed in the solution $x=-1$, completed the square, and now I'm left with the equation $\...
LÜHECCHEgon's user avatar
1 vote
0 answers
59 views

Calculate on which side of a cuboid is a given point located?

Correct me if I'm using incorrect terms, I'm not well-versed with geometrical terminology I'm writing a code in which I have a point & I want to identify if the points lies on front, back, top, ...
Kuldeep J's user avatar
  • 111
2 votes
1 answer
149 views

Cubic equation coefficients from 4 points

For a cubic curve (Bezier) of the form: $ax^3 + bx^2 + cx + d = y$. I have a given set of four points $P_0, P_1, P_2, P_3$. Such that, $P_0$ is the origin and the other three are equidistant along the ...
Norma's user avatar
  • 23
3 votes
2 answers
176 views

How to draw graph of cubic function

I am taking a course in calculus and wanted to refresh my memory before the semester starts. And I have been working on drawing graphs from cubic functions. I am not that experienced with LaTex and ...
Mampenda's user avatar
  • 409
1 vote
2 answers
101 views

Minimizing a cubic polynomial over $\Bbb N$ [closed]

Let the polynomial function $f : \Bbb N \to \Bbb N$ be defined by $$f (M) := 2M^3N + 2M^3 - M^2N^2 - 3M^2N + 2M^2 - MN^3 + MN^2 - 2MN + \frac{N^4}{2} + \frac{N^2}{2}$$ where $N$ is given natural ...
Amirhossein Rezaei's user avatar
0 votes
1 answer
159 views

The Diophantine equation $P_1^3 + P_2^3 + P_3^3 = P_4^3$

Consider the Diophantine equation $$P_1^3 + P_2^3 + P_3^3 = P_4^3$$ Where $P_n$ are distinct odd primes. What are the smallest solutions ? Is there even a solution ? Or is there a reason no such ...
mick's user avatar
  • 16.4k
2 votes
0 answers
81 views

Failing to solve cubic equation

I'm trying to solve a more complex cubic equation but to simplify things as a start I picked this one: $$ 3\cdot 4^3+2\cdot 4-200=0 $$ Here $x$ is $4$. I'm looking at wikipedia and trying to solve ...
php_nub_qq's user avatar
0 votes
0 answers
53 views

cubic equation edge cases

Working on general cubic equation solver in form ax^3+bx^2+cx+d=0 And have no clue for special cases: In terms of cubic there should be one real root and two complex, or 3 real roots if coefficients ...
Vitaly Protasov's user avatar
2 votes
3 answers
103 views

Prove that $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots

I'm trying to prove that the cubic equation $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots. The coefficients are $a_3 = - 1 - \sigma - \tau - \chi$ $a_2 = -2 (\sigma +...
Rich T's user avatar
  • 61
5 votes
0 answers
115 views

There is a compass-like tool that can draw $y=x^2$ on paper. Is there one for $y=x^3$?

Is there a tool that can draw $y=x^3$ on paper? I'm referring to low-tech tools, e.g. not computers. I only know of tools that can draw $y=x^2$. The YouTube video "Conic Sections Compass" ...
Dan's user avatar
  • 25.7k
-2 votes
1 answer
42 views

Prove this potential cubic theorem/formula [closed]

Prove that $\dfrac{x³+y³+z³}{x+y+z}=x²+y+z$; if $x<y<z$; $y=x+1$; $z=y+1$; and $x$, $y$ and $z$ are positive whole numbers. If you prove this, I technically discovered a new formula since I ...
TheoRetical's user avatar
5 votes
3 answers
160 views

Prove $2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a)$ for $abc=1.$

Let $a,b,c>0: abc=1.$ Prove that: $$2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a). $$ I've tried to use a well-known lemma but the rest is quite complicated for me. ...
Anonymous's user avatar
2 votes
1 answer
121 views

Three real roots of a cubic

Question: If the equation $z^3-mz^2+lz-k=0$ has three real roots, then necessary condition must be _______ $l=1$ $ l \neq 1$ $ m = 1$ $ m \neq 1$ I know there is a question here on stack about ...
Darshit Sharma's user avatar
1 vote
1 answer
128 views

Why are all real inflection points on a cubic projective algebraic curve on 1 line?

Say we have $C\subset \mathbb{CP}^2$, a smooth curve of degree 3. I am aware of the group structure on cubics, what I don't get, is why are all inflection points with only real coordinates lie on a ...
Ilcu_elte_smurf's user avatar
7 votes
1 answer
230 views

More $a^3+b^3+c^3=(c+1)^3,$ and $\sqrt[3]{\cos\tfrac{2\pi}7}+\sqrt[3]{\cos\tfrac{4\pi}7}+\sqrt[3]{\cos\tfrac{8\pi}7}=\sqrt[3]{\tfrac{5-3\sqrt[3]7}2}$

I. Solutions In a previous post, On sums of three cubes of form $a^3+b^3+c^3=(c+1)^3$, an example of which is the well-known, $$3^3+4^3+5^3=6^3$$ we asked if there were polynomial parameterizations ...
Tito Piezas III's user avatar
0 votes
0 answers
39 views

Disjoint exceptional lines on non-minimal cubic surface

A diagonal cubic surface $\sum_{i=0}^3a_iT_i^3=0$ is not minimal if, for example, $a_1a_2a_3^{-1}a_4^{-1}\in(k^*)^3$. This should be because there is an exceptional line $D$ such that no element in ...
fp1's user avatar
  • 117
3 votes
1 answer
163 views

Prove $(1+a^3) (1+b^3)(1+c^3) \ge (\frac{ab+bc+ca+1}{2})^3$

This is a question from 4U maths (highest level of Y12 maths in Australia) from a generally difficult paper. The question itself does not define what a, b, and c are - based on the comments, we assume ...
socratic's user avatar
-1 votes
1 answer
55 views

Descartes folium

The geometry of Descartes' folium, $x^3+y^3=3axy$ has been well studied. Can someone tell me which geometric property characterizes the following cubic curve: $$bx^3+y^3=3axy$$ The previous curve is a ...
Felipe 's user avatar
0 votes
0 answers
42 views

Set of coefficients of degree three monic real polynomial with three real roots is connected.

Let $p(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients $a, b, c,$ and define: $$D=\{(a,b,c)\in \mathbb{R}^3\mid \text{the polynomial}\ p(x)\ \text{factors into linear factors over }\ \...
nkh99's user avatar
  • 483
9 votes
2 answers
571 views

Can you tell me about the (probably) well known relationship between the coefficients of a cubic and some features of a rectangular solid?

If we look at the expansion of this $$(x+a)(x+b)(x+c)=x^3+(a+b+c)x^2+(ab+bc+ca)x+abc$$ And consider a rectangular solid with length, width and height of $a, b, c$ respectively. Then $$l_{edges}=4(a+b+...
David Elm's user avatar
  • 566
0 votes
0 answers
76 views

For translation of axes, is there a definite equation for any of "the 27 lines" on the Clebsch Diagonal Cubic?

For the Clebsch Diagonal Cubic (related to a Quanta mag article on Hilbert's 13th Problem), I want to generate a point at will on any of these lines along the surface. Wolfram calls these "...
Oddly Specific Math's user avatar
0 votes
1 answer
43 views

Prove a relation between the coefficients of a depressed cubic.

The equation $x^{3}+px^{2}+q=0$ where p and q are non-zero constants, has three real roots $\alpha$, $\beta$ and $\gamma$. Given that the interval between $\alpha$ and $\beta$ is p and that the ...
druidmind's user avatar
0 votes
2 answers
109 views

find $a$ and $b$ where $x^3 - 4x^2 -3x + 18 = (x+a)(x-b)^2$

I have a problem solving this as when I match the coefficients to the expanded brackets I end up with $2$ unknowns $a \& b$. So cannot substitute any values to find the other. According to the ...
James Harding's user avatar
1 vote
1 answer
45 views

Proof of conditions for polynomials

Find the conditions for the roots $\alpha, \beta, \gamma$ of the equation $x^3-ax^2+bx-c=0$ to be in: $(i)$A.P.; $(ii)$G.P. If the roots are not in A.P. and if $\alpha+\lambda,\ \beta+\lambda,\ \...
J_dash's user avatar
  • 87
-2 votes
2 answers
118 views

Prime numbers $p$ such that $7p+1$ is a cube [closed]

I am stuck on this assignment. I have to find every prime number $p$ such that $7p+1$ is a cube number. After exploring enough I must say there is no prime $p$ satisfying this condition. I have tried ...
Enkt Enktson's user avatar
1 vote
1 answer
81 views

Why Can't Cubic Equation Have Fractional Solutions When Its Coefficients Are All Integers? [duplicate]

In Leonhard Euler's book, "The Elements of Algebra" he seems to say that if we convert any cubic equation into the form $x^3 + ax^2 + bx + c$, and make sure that $a$, $b$ and $c$ are integer ...
Camelot823's user avatar
  • 1,467
2 votes
0 answers
70 views

Finding rational coefficients of a cubic polynomial that fits 4 data points that have been floored to an integer

I have 4 data points: (204, 5422892) (205, 5722486) (207, 6343357) (213, 8386502) I have information that these data points were generated with a cubic polynomial $y = ax ^ 3 + bx ^ 2 + cx + d$ with ...
SeekingAnswers's user avatar
4 votes
3 answers
177 views

Prove that $\mathbb Q(\cos\tfrac\pi7)\neq\mathbb Q(\cos\tfrac\pi9)$

Let $\tau=2\pi$ be the full angle. (tau) For any integer $k$ and any angle $\theta$, $\cos(k\theta)$ is a polynomial in $\cos\theta$. In particular, $\cos(2\theta)=2\cos^2\theta-1$, which shows that $\...
mr_e_man's user avatar
  • 5,706
1 vote
2 answers
143 views

How to solve $x^3−x+1=0$

I am interested in finding a solution for the equation: $$ x^3 - x + 1 = 0 $$ I've noticed that there are numerous polynomial equations where one of the coefficients is zero. Could you provide ...
winter's user avatar
  • 63
0 votes
0 answers
52 views

Numerical Analysis - natural cubic spline and clamped cubin spline

a question from first exam period (A). True or false ( it is false, but I want to understand ). Given the following intersection points $x_0, x_1,...,x_n$ (interpolation nodes ) and the values of ...
LearningToCode's user avatar
3 votes
1 answer
109 views

Simplifying Coefficients of a Cubic Polynomial with Complex Roots

I am currently encountering difficulties while trying to solve the following question, and I would greatly appreciate any assistance you can provide. Let $a,b,c$ be complex numbers. The roots of $z^{3}...
MyLight's user avatar
  • 327
5 votes
2 answers
120 views

If the complex roots of $x^3-x-2=0$ are $r\pm si$, and $As^6 +Bs^4 + Cs^2 =26$ for integers $A$, $B$, $C$, find $A+B+C$

The question: In the cubic $x^3-x-2=0$, there is one real root and two complex roots of the form $r\pm si$, with $r$ and $s$ real. If there exists integers $A,B,$ and $C$ such that $As^6 +Bs^4 + Cs^2 ...
mathmule's user avatar
  • 159
3 votes
4 answers
256 views

Prove irreducible cubic polynomial over $\mathbb{Q}$ with a cyclic galois group has real roots

I want to prove the following: Let $f\in \mathbb{Q}[x]$ be an irreducible cubic polynomial, whose Galois group is cyclic. Prove that all of the roots of $f$ are real. I know that the Galois group $G$...
Ariel Yael's user avatar
0 votes
0 answers
24 views

Let a, b, c, d be complex numbers satisfying $a+b+c+d=a^3+b^3+c^3+d^3=0$. Prove that a pair of the a, b, c, d must add up to 0 [duplicate]

When doing this I tried using the identity $x^3+y^3+z^3=3xyz$ if $x+y+z=0$ I take $x=a$, $y=b$, and $z=c+d$ So $a+b+(c+d)=0$ $a^3+b^3+(c+d)^3=3ab(c+d)$ $a^3+b^3+c^3+d^3+3cd(c+d)=3ab(c+d)$ $(a^3+b^3+c^...
Namura's user avatar
  • 123
0 votes
0 answers
125 views

Can I use this algorithm for solving cubic equations?

I am trying to find the root solutions for a cubic equation including the eigenvalues of each root. I tried to put the equation into my calcualtor but the calculator doesn't show solutions that has ...
Ryan's user avatar
  • 1
2 votes
2 answers
100 views

Convert an expression with radicals into simpler form

It was pointed out in a mathologer video on the cubic formula that $\sqrt[3]{20 + \sqrt{392}} + \sqrt[3]{20 - \sqrt{392}}$ is actually equal to $4$. Is there a series of transformations that can be ...
Neeraj's user avatar
  • 23
1 vote
1 answer
108 views

Solving a cubic using triple angle for cos (i.e $\cos(3A)$)

a) Show that $x=2\sqrt{2}\cos(A)$ satisfies the cubic equation $x^3 - 6x = -2$ provided that $\cos(3A)$ = $\frac{-1}{2\sqrt{2}}$ I did not have a difficulty with this question, I have provided it for ...
Mikhael's user avatar
  • 15

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