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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.

-2 votes
0 answers
43 views

Add constraints to cubic or quintic polynomial [closed]

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
12 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
66 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
26 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
50 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
150 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
177 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
  • 115
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

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