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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Proof about intersection of three cubic curves
This is the proposition 3 page 124 from Algebraic Curves. An Introduction to Algebraic Geometry first edition by William Fulton. It appears in the paragraph of consequences of Max Noether's theorem.
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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 ...
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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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Interpolating with Hermite cubics in two dimensions
I want to estimate the value of the function $f(x,y)$ at a particular point. Suppose I am given two points, $(x_1,y_1)$ and $(x_2,y_2)$, along with the value of $f$ and its partial derivatives $f_x$ ...
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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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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 +...
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Do the second differences of the fifth powers count the sphere packing of a polyhedron?
If the third powers count the packing of a cube:
[1, 8, 27, 64, 125, 216, 343..]
And the first differences of the fourth powers form a cubic sequence that counts a packing of a rhombic dodecahedron (...
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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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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 '...
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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, ...
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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 ...
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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 ...
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How to derive the general formula to determine the equation of a given cubic function
My question is: When determining the equation of a cubic function, we can separate the general cubic equation into it's solutions and we end up with the equation
$y = a(x-r_1)(x-r_2)(x-r_3)$
We ...
CommunityBot
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When are the roots of a polynomial of degree 3 aligned?
Let $P \in \mathbb{C}[X]$ be a polynomial of degree 3. On what condition on the coefficients of $P$ are the three roots of $P$ aligned ?
To make things easier, we may assume that $P$ can be written as ...
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Inverse of $f(x) = x^{3}-x^{2}$
Can anybody find the inverse of $f:(-1,0) \to \mathbb{R}$ such that $f(x) = x^{3} - x^{2}$ ?
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