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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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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
33 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
  • 11
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
57 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
115 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
174 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
95 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
155 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
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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
50 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
101 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
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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
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-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
117 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
117 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
219 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
38 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
154 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
53 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
40 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
  • 471
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
73 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
42 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
108 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
44 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
117 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
  • 1,417
2 votes
0 answers
68 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
  • 5,444
1 vote
2 answers
140 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
  • 33
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
119 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
224 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
23 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
  • 121
0 votes
0 answers
119 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
97 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
106 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
1 vote
1 answer
197 views

How do I find a cubic equation given only one root?

Given the root of a cubic equation $Z = \sqrt[3]{Y + \sqrt{Y^2 - \frac{X^6}{27}}} + \sqrt[3]{Y - \sqrt{Y^2 - \frac{X^6}{27}}} - X$ and the assumption that both $X$ and $Y$ are greater than zero, is ...
Lawton's user avatar
  • 1,851
1 vote
1 answer
76 views

Best way to solve $\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$

I was wondering what the best way to solve questions like these are? $$\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$$ I can get the answer, which is $(-\infty,-1)\cup(1,3)$. But I'm not sure if I have ...
basket_case's user avatar
3 votes
3 answers
162 views

Is there any faster way to factor $x^3-3x+2$?

$$x^3-3x+2$$ $$x^3-3x+x^2+2-x^2$$ $$x^2-3x+2+x^3-x^2$$ $$(x-2)(x-1)+x^2(x-1)$$ $$(x-1)[x^2+x-2]$$ $$(x-1)(x+2)(x-1)$$ Is there a better, faster way to factor this cubic trinomial?
SirMrpirateroberts's user avatar
2 votes
1 answer
76 views

Classification of curves passing through 7 points. (Hartshorne III ex 10.7)

This is the exercise III 10.7 in Hartshorne's Algebraic Geometry I am not sure if I misunderstood the question. The seven points of the projective plane over $\mathbb{F}_2$, I think, means $\{[x_0,...
Xiong Jiangnan's user avatar
2 votes
1 answer
69 views

Prove $\lim_{n\to\infty}\int_0^{a} \left(\sqrt{2n/\pi-x^2}-\sqrt{2n/\pi-a^2}\right)dx=1/6$ where $a$ is the largest real root of $4x^6+x^2=2n/\pi$.

I've never seen anything like this before: an unsolvable cubic, within a definite integral, within a limit (which applies to the cubic and the integral), resulting in a simple closed form. Prove $\...
Dan's user avatar
  • 23.9k
2 votes
2 answers
130 views

Condition for the existence of positive solution to cubic equation

In a physics textbook I have encountered a cubic equation of the form: $$Ax^3-Bx+C=0$$ The book states that there exists a positive solution $x>0$ to this equation if and only if the following ...
Wild Feather's user avatar
1 vote
2 answers
534 views

Find sum of all integral values of $r$ such that all roots of the equation $x^3-(r-1)x^2-11x+4r=0$ are also integers

Find sum of all integral values of $r$ such that all roots of the equation $$x^3-(r-1)x^2-11x+4r=0$$ are also integers. What I could do was $$r=\frac{x^3+x^2-11x}{x^2-4}=x+1+\frac{4-7x}{x^2-4}$$ Since ...
Maverick's user avatar
  • 9,471
3 votes
6 answers
384 views

Find all real numbers $a$ for equation $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.

Problem: Find all real numbers $a$ for which the equation, $x^3 + ax^2 + 51x + 2023=0$, has two equal roots. This problem is from an algebra round of a local high school math competition that has ...
JHumpdos's user avatar
  • 167
4 votes
1 answer
259 views

Why doesn't simultaneous equations work to find co-efficients of a cubic that passes through four points?

I'm trying to find the equation of a cubic that passes through three specific points (technically it's four but that point is y-intercept). The equation would look something like this:$f(x)=ax^3+bx^2+...
sirOrange17's user avatar
1 vote
0 answers
47 views

Order $3$ linear transforms invariating a binary cubic form

Consider $P(x,y)$ a homogenous polynomial of degree $3$ in two variables (a binary cubic). To it we associate first the $2\times 2$ matrix $$\frac{1}{2}\operatorname{Hess}(P) = \frac{1}{2}\cdot\left( ...
orangeskid's user avatar
  • 54.4k

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