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7votes

4answers

689views

Cubic polynomial with equal absolute values at $6$ points [duplicate]

Let $P \in {\Bbb R} [x]$ be a cubic polynomial with real coefficients such that $$ |P(1)| = |P(2)| = |P(3)| = |P(5)| = |P(6)| = |P(7)| = 12 $$ Find the value of $\frac19 P(0)$My approach so far:...

mukund

askedOct 7 at 10:59

4votes

8answers

347views

Factoring $(a + b + c)^3 - a^3 - b^3 - c^3$

I am doing I. M. Gelfand's "Algebra" problem 122 e), factoring$$(a + b + c)^3 - a^3 - b^3 - c^3$$So my solution is following:$$\begin{align}(a + b + c)^3 - a^3 - b^3 - c^3 &= (a + b)^...

Hugh Melee

askedSep 29 at 12:20

1vote

1answer

66views

Could hypercomplex systems analogous to generalized complex numbers be constructed with a higher order relation instead of a quadratic relation?

A hypercomplex number system is an algebra that expands the real numbers by adding a unit that is distinct from one and negative one.The most well known hypercomplex number system is the complex ...

Quinali Solaji

askedSep 20 at 18:57

6votes

0answers

219views

Roots of $x^3+ax^2+bx+1$, where $|a|=|b|=1$, satisfy $|z_1|\le3|z_2|$

Question. Let $a,b$ be complex numbers such that $|a|=|b|=1$. Let $z_1,z_2,z_3$ be roots of the polynomial $x^3+ax^2+bx+1$. Prove that $|z_1|\le 3|z_2|$.A high school student asked me this question, ...

Yizhen Chen

1,112

askedAug 24 at 20:14

2votes

1answer

127views

Why solving a cubic with this method yielding wrong result

let an equation be$x^3 - 15x^2 +75x - 125 = 0 $Step 1 : to solve this first check the condition $ b^2= 3ac$$(-15)^2=3(1)(75)$It holds true for this equation but the check had the terms a,b,c but ...

Prince

askedAug 16 at 17:11

0votes

2answers

71views

Third degree polynomial with small parameter

This is a follow-up to that question, so I will refer to it for the motivation. In continuation, I now have this polynomial obtained by inserting $x=\phi_2+\sqrt{\varepsilon}\cdot y$ in the polynomial ...

Gateau au fromage

1,385

askedJul 20 at 20:33

2votes

1answer

64views

Let $P = (x, x^2)$ and $Q$ be the intersection point of the line orthogonal to the tangent at $P$. Compute the minimum length of the segment $PQ$.

The following exercise comes from James Stewart's Calculus textbook, in the "Applications of Differentiation" chapter:Let $P$ be any point in $f(x) = x^2$, except for the origin, and $Q$ ...

musgo

askedJul 4 at 19:35

2votes

3answers

328views

Alternate derivation (without calculus) of a cubic with local maximum at $(x_1,y_1)$ and a local minimum at $(x_2,y_2)$

This problem came about as part of a pre-calculus seminar. The problem had specific coordinates instead of distinct $(x_1,y_1),(x_2,y_2)$ and the goal was to play with sliders to try to fit a cubic to ...

Integrand

7,704

askedJul 3 at 21:09

0votes

1answer

82views

Confusion regarding blowup calculation

The following is from Simon Peacock's lecture on blowing up:$2. 3.$ Cuspidal cubic. The cuspidal cubic is given by$$\mathbb{V}(Y^2-X^3)\subseteq\mathbb{C}^2.$$As before, using the blowup of $\...

Agaman

askedJun 26 at 21:33

3votes

0answers

52views

Simpler proof of a special case of Bezout‘s Theorem? [closed]

I am currently working on smooth complex projective cubics, i.e. the loci of complex polynomials $P(X,Y,Z)$ that are homogenous of degree 3. In particular, I would like to prove that a smooth cubic ...

John Cena

askedJun 20 at 20:19

4votes

1answer

88views

Balloons in 3-Regular Graphs: Tight Bound on Their Number?

Let $G$ be a connected 3-regular (cubic) graph on $n$ vertices. A balloon in $G$ is defined as a maximal subgraph of $G$ that has no cut-edges (i.e., is 2-edge-connected) and is connected to the rest ...

F. A. Mala

3,615

askedJun 18 at 14:01

2votes

2answers

252views

Diophantine $(a^3 - a - 1)b = c^3 - c - 1$

How to solve the diophantine equation$$(a^3 - a - 1)b = c^3 - c - 1$$for integers $a,b,c>1$ ?Is the expected number of solutions ,denoted $f(c)$, for $a_n$ such that $(a_n^3 - a_n - 1)b_n = c_n^...

mick

18.3k

askedJun 16 at 8:14

1vote

1answer

77views

Proof that: ($\forall a,b,c\in \mathbb{R}$)($ab=c^{2}\iff\exists k\in \mathbb{R}\;$s.t. $a=kc \wedge b=k^{-1}c$)

Was looking through a proof of a necessary and sufficient condition for the roots of a cubic to follow a geometric progression. It used the following argument for necessity.For the following cubic ...

iambadatnames

askedJun 14 at 7:19

1vote

2answers

197views

Solving the equation $x^3 - 3x = \sqrt{x + 2}$ over $\mathbb{C}$

I'm trying to solve the equation$$x^3 - 3x = \sqrt{x + 2}$$for $x \in \mathbb{C}$.The roots of the equation include $x=2$, $x=2\cos4\pi/5$, and $x=2\cos4\pi/7$.However, I want to see how exactly ...

Rutvaj Nehete

askedMay 19 at 0:14

-1votes

1answer

132views

How to find closed form for the sum of two complex cube roots

If you solve this equation $8x^3-6x-{1}=0$, by transforming it into the form of $x^3+px+q=0$, using Cardano's formula and, finding the trignometric solutions you get that one of the solutions is:\...

bruh Jardim

askedMay 9 at 0:11

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Questions tagged [cubics]

Cubic polynomial with equal absolute values at $6$ points [duplicate]

Factoring $(a + b + c)^3 - a^3 - b^3 - c^3$

Could hypercomplex systems analogous to generalized complex numbers be constructed with a higher order relation instead of a quadratic relation?

Roots of $x^3+ax^2+bx+1$, where $|a|=|b|=1$, satisfy $|z_1|\le3|z_2|$

Why solving a cubic with this method yielding wrong result

Third degree polynomial with small parameter

Let $P = (x, x^2)$ and $Q$ be the intersection point of the line orthogonal to the tangent at $P$. Compute the minimum length of the segment $PQ$.

Alternate derivation (without calculus) of a cubic with local maximum at $(x_1,y_1)$ and a local minimum at $(x_2,y_2)$

Confusion regarding blowup calculation

Simpler proof of a special case of Bezout‘s Theorem? [closed]

Balloons in 3-Regular Graphs: Tight Bound on Their Number?

Diophantine $(a^3 - a - 1)b = c^3 - c - 1$

Proof that: ($\forall a,b,c\in \mathbb{R}$)($ab=c^{2}\iff\exists k\in \mathbb{R}\;$s.t. $a=kc \wedge b=k^{-1}c$)

Solving the equation $x^3 - 3x = \sqrt{x + 2}$ over $\mathbb{C}$

How to find closed form for the sum of two complex cube roots

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