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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43
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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 ...
- 1
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 ...
- 2,452
2
votes
0
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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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0
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26
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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 +...
-1
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0
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46
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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 ...
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1
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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 '...
- 1,685
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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, ...
- 254
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 ...
- 254
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 ...
- 348
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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 ...
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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 ...
- 173
1
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0
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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 ...
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1
vote
2
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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, ...
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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 $\...
- 602
1
vote
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, ...
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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 ...
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3
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2
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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 ...
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2
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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
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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 ...
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2
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0
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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 ...
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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 +...
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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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1
answer
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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
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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
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1
answer
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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 ...
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1
vote
1
answer
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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
votes
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 ...
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0
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0
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39
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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 ...
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1
answer
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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 ...
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votes
1
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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 ...
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0
votes
0
answers
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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:
$$D=\{(a,b,c)\in \mathbb{R}^3\mid \text{the polynomial}\ p(x)\ \text{factors into linear factors over }\ \...
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2
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Can you tell me about the (probably) well known relationship between the coefficients of a cubic and some features of a rectangular solid?
If we look at the expansion of this
$$(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+...
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0
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76
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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 "...
0
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1
answer
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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 ...
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2
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109
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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 ...
1
vote
1
answer
45
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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,\ \...
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votes
2
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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 ...
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1
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1
answer
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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 ...
- 1,467
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0
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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 ...
- 119
4
votes
3
answers
177
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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 $\...
- 5,726
1
vote
2
answers
143
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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 ...
- 63
0
votes
0
answers
52
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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 ...
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3
votes
1
answer
109
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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}...
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votes
2
answers
120
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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 ...
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3
votes
4
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256
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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 $G$...
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0
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24
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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^...
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votes
0
answers
125
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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 ...
- 1
2
votes
2
answers
100
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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 ...
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1
vote
1
answer
108
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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 ...
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