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.
1,404 questions
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$f(x)$ is a cubic polynomial $f(k-1)f(k+1)<0$ is NOT true for any integer $k$. Find $f(8)$
$f(x)$ is a monic cubic polynomial and $f(k-1)f(k+1)<0$ is NOT true for any integer $k$.
Also $f'\left(-\frac{1}{4}\right)=-\frac{1}{4}$ and $f'\left(\frac{1}{4}\right)<0$.
Find value of $f(8)$
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Not all quadratic extensions over $\mathbb{Q}$ are contained in the compositum of all the splitting fields of irreducible cubics in $\mathbb{Q}[X]$
Question: Let $F$ be the composite of all the splitting fields of irreducible cubics over $\mathbb{Q}$. Prove that
$F$ does not contain all quadratic extensions of $\mathbb{Q}$.
(This is exercise 16 ...
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A doubt in an IIT-JEE-Adv-2025-P1 question
If $a_i,b_i \in \mathbb R$ for $i\in\{1,2,3\}$, define $f:\mathbb R \to \mathbb R, g: \mathbb R\to \mathbb R, h: \mathbb R \to \mathbb R$
$$f(x)=a_1+10x+a_2x^2+a_3x^3+x^4, \quad g(x)=b_1+3x+b_2x^2+...
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Solving system of equations with one affine and two cubic equations
From Hall & Knight's Higher Algebra:
Solve the system of equations $$ \begin{aligned} x^3 + y^3 + z^3 &= 495 \\ x + y + z &= 15 \\ x y z &= 105 \end{aligned} $$
What I tried
We know ...
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Differences in implementation of Cardano's method [closed]
Given a cubic
$$ax^3+bx^2+cx+d=0$$
You can divide by $a$, and replace $x$ with $w - \frac{b}{3a}$ to center the cubic and remove the quadratic term
$$
\begin{aligned}
ax^3+bx^2+cx+d &= 0 \\
x^3+\...
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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:
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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)^...
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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 ...
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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, ...
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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 ...
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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 ...
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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$ ...
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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 ...
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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 $\...
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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 ...
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