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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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
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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:
$$D=\{(a,b,c)\in \mathbb{R}^3\mid \text{the polynomial}\ p(x)\ \text{factors into linear factors over }\ \...
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Can you tell me about the (probably) well known relationship between the coefficients of a cubic and some features of a rectangular solid?
If we look at the expansion of this
$$(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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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 "...
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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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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 ...
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1
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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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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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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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3
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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 $\...
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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 ...
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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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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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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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