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.
1,360
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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 ...
12
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7
answers
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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 ...
2
votes
0
answers
66
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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 $d>0$...
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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 +...
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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 ...
0
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1
answer
82
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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 '...
1
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0
answers
50
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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, ...
1
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0
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37
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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 ...
0
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0
answers
31
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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 ...
0
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53
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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 ...
0
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49
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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 ...
1
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0
answers
40
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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 ...
1
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2
answers
62
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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, ...
2
votes
1
answer
56
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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 $\...
1
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0
answers
59
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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, ...
2
votes
1
answer
150
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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 ...
3
votes
2
answers
177
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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 ...
1
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2
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101
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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 ...
0
votes
1
answer
159
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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 ...
2
votes
0
answers
81
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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 ...
0
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0
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53
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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 ...
2
votes
3
answers
103
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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 +...
5
votes
0
answers
115
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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" ...
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votes
1
answer
42
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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 ...
5
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3
answers
160
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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.
...
2
votes
1
answer
121
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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 ...
1
vote
1
answer
128
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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 ...
7
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1
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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 ...
0
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0
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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 ...
3
votes
1
answer
163
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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 ...