# 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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### Add constraints to cubic or quintic polynomial

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 ...
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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 ...
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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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### 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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### 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 ...
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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. ...
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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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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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