# 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 ...
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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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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 ...
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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 ...
1 vote
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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 ...
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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 ...
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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 ...
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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
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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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1 vote
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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