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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Involution on monic cubic polynomials related to nesting/denesting of cubic radicals
Consider the involutive transformation
$$\mathbb{R}^3 \ni (a,b,c) \overset{\phi}{\mapsto} \left( \frac{a + 2 c}{\sqrt{3}}, \frac{a^2 + a c + c^2}{3} - b , \frac{a - c}{\sqrt{3}}\right)$$
Show that if $...
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Rational solutions for $x^3+y^3=1$ where both x and y are non-negative
How can I find rational solutions for $x^3+y^3=1$ where both x and y are non-negative?
Edit: One of the answer in this post for general form of solutions
$$(a,b) \mapsto \left( \frac{a(a^3 + 2b^3)}{a^...
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Involution on $2\times 2$ matrices
Show that the map on $2\times 2$ matrices
\begin{eqnarray}
\left( \begin{matrix} a & b\\ c & d \end{matrix} \right)\overset{\Phi}{\mapsto} \left( \begin{matrix} a & b\\ c & d \end{...
- 55k
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Vieta's Formula and Polynomials
The following below is a question i was asked. The question left me stunned as i was unable to solve it. The question is as follows:
If $A$,$B$ and $C$ are roots of the cubic equation $ax^3+bx^2+cx+d=...
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Homographic relation between two roots of a cubic
Consider a cubic equation $ x^3 + 3a x^2 + 3 b x + c=0$ with distinct roots. Show that any two roots $x$, $y$ are connected by a homographic relation
$$(a^2-b) x y + \frac{1}{2}\ (\ (a b-c+\delta) x +...
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3
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2
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Finding root of a cubic equation.
I was solving a physics statistical mechanics problem of an interacting system.
In that question, I have to find the eigenvalues of a matrix P whose elements are given by
$$P=
\begin{bmatrix}
e^{x} &...
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1
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Signs in the Cardano formula
When deriving the Cardano formula from
$x^{3}+px+q=0$ we let $x$ be a sum and compare coefficients. So
$x=u+v$, then we get a system for $u$ and $v$.
We get $(1) -q=u^{3}+v^{3}$ and $(2) u^{3}v^{3}=-(\...
3
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$\sqrt[3]{x_1} + \sqrt[3]{x_2} + \sqrt[3]{x_3} = \sqrt[3]{z}$ if $x_i$ are the real distinct roots of $(x+y)^3 - x^2 z + f x z( x + y + f^2/27 z)$
Show that
$$\sqrt[3]{x_1}+ \sqrt[3]{x_2} + \sqrt[3]{x_3} = \sqrt[3]{z}$$
where $x_1$, $x_2$, $x_3$ are the real distinct roots of a cubic polynomial in $x$ of the form
$$(x+y)^3 - x^2 z + f x z \left(...
- 55k
2
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2
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$\sqrt[3]{x_1} + \sqrt[3]{x_2} + \sqrt[3]{x_3} = \sqrt[3]{a},\,$ if $x_i$ are the real roots of $(x+b)^3 - a x^2$
Consider the equation,
$$(x+b)^3 = a x^2\tag1$$
with $a\ne 0$, and real roots $(x_1$, $x_2$, $x_3)$. Show that,
\begin{eqnarray}\sqrt[3]{x_1} &+& \ \sqrt[3]{x_2} &+ &\ \sqrt[3]{x_3} &...
- 55k
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Why do equilateral triangles relate to cubics
I found this question talking about the relation between an equilateral triangle and cubics with three distinct real roots.
Here's an image from the original post with an example:
What this post says ...
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Determine form of cubic polynomial given initial conditions [closed]
If I am given a function $P = ax^3 + bx^2 + cx + d$ and following:
$P(0) = P_1$,
$P(1) = P_2$,
$P^\prime(0) = v_0$,
$P^\prime(1) = v_1$.
How would I solve for $a$, $b$, $c$ and $d$?
Edit:
I have tried ...
2
votes
2
answers
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Given a cubic and a point S not on the cubic, how many tangent lines to F can we draw from S?
Let $F\in \mathbb{C}[x,y,z]$ be an irreducible homogeneous polynomial with total degree 3, defining a cubic in $\mathbb{CP}^2$. Given a point $S$ on the projective plane but not on $\mathbb{V}(F)$, ...
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If the roots of $x^3 − 6x^2 + 10x + 1$ are denoted as a, b, c, then find the value of $(a^2 + b^2 )(a^2 + c^2 )(b^2 + c^2 )$.
If the roots of $x^3 − 6x^2 + 10x + 1$ are denoted as a, b, c, then find the value of $(a^2 + b^2 )(a^2 + c^2 )(b^2 + c^2 )$.
I have tried factoring $x^3 − 6x^2 + 10x + 1$ but didn't get anything. ...
4
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Intersecting cubic equations and roots.
I know that it is possible to find all roots of a cubic using the cubic formula. Now when I have a cubic $P(x)$, I create the simultaneous equations:
$$P(x)=y \quad\text{and}\quad P(y)=x$$
I want to ...
user1162071
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What points in the plane of the graph $y=x^3$ have three tangents to the curve passing through them?
I’m studying high school math and encountered this question in the extension section for derivatives. The text says an algebraic solution to the problem is harder but possible. There is also a similar ...
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