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.
1,360
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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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Let a, b, c, d be complex numbers satisfying $a+b+c+d=a^3+b^3+c^3+d^3=0$. Prove that a pair of the a, b, c, d must add up to 0 [duplicate]
When doing this I tried using the identity
$x^3+y^3+z^3=3xyz$ if $x+y+z=0$
I take $x=a$, $y=b$, and $z=c+d$
So $a+b+(c+d)=0$
$a^3+b^3+(c+d)^3=3ab(c+d)$
$a^3+b^3+c^3+d^3+3cd(c+d)=3ab(c+d)$
$(a^3+b^3+c^...
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Can I use this algorithm for solving cubic equations?
I am trying to find the root solutions for a cubic equation including the eigenvalues of each root.
I tried to put the equation into my calcualtor but the calculator doesn't show solutions that has ...
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Convert an expression with radicals into simpler form
It was pointed out in a mathologer video on the cubic formula that $\sqrt[3]{20 + \sqrt{392}} + \sqrt[3]{20 - \sqrt{392}}$ is actually equal to $4$. Is there a series of transformations that can be ...
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Solving a cubic using triple angle for cos (i.e $\cos(3A)$)
a) Show that $x=2\sqrt{2}\cos(A)$ satisfies the cubic equation $x^3 - 6x = -2$ provided that $\cos(3A)$ = $\frac{-1}{2\sqrt{2}}$
I did not have a difficulty with this question, I have provided it for ...
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How do I find a cubic equation given only one root?
Given the root of a cubic equation $Z = \sqrt[3]{Y + \sqrt{Y^2 - \frac{X^6}{27}}} + \sqrt[3]{Y - \sqrt{Y^2 - \frac{X^6}{27}}} - X$ and the assumption that both $X$ and $Y$ are greater than zero, is ...
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Best way to solve $\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$
I was wondering what the best way to solve questions like these are?
$$\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$$
I can get the answer, which is $(-\infty,-1)\cup(1,3)$. But I'm not sure if I have ...
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Is there any faster way to factor $x^3-3x+2$?
$$x^3-3x+2$$
$$x^3-3x+x^2+2-x^2$$
$$x^2-3x+2+x^3-x^2$$
$$(x-2)(x-1)+x^2(x-1)$$
$$(x-1)[x^2+x-2]$$
$$(x-1)(x+2)(x-1)$$
Is there a better, faster way to factor this cubic trinomial?
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Classification of curves passing through 7 points. (Hartshorne III ex 10.7)
This is the exercise III 10.7 in Hartshorne's Algebraic Geometry
I am not sure if I misunderstood the question.
The seven points of the projective plane over $\mathbb{F}_2$, I think, means
$\{[x_0,...
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Prove $\lim_{n\to\infty}\int_0^{a} \left(\sqrt{2n/\pi-x^2}-\sqrt{2n/\pi-a^2}\right)dx=1/6$ where $a$ is the largest real root of $4x^6+x^2=2n/\pi$.
I've never seen anything like this before: an unsolvable cubic, within a definite integral, within a limit (which applies to the cubic and the integral), resulting in a simple closed form.
Prove $\...
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Condition for the existence of positive solution to cubic equation
In a physics textbook I have encountered a cubic equation of the form:
$$Ax^3-Bx+C=0$$
The book states that there exists a positive solution $x>0$ to this equation if and only if the following ...
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Find sum of all integral values of $r$ such that all roots of the equation $x^3-(r-1)x^2-11x+4r=0$ are also integers
Find sum of all integral values of $r$ such that all roots of the equation $$x^3-(r-1)x^2-11x+4r=0$$
are also integers.
What I could do was $$r=\frac{x^3+x^2-11x}{x^2-4}=x+1+\frac{4-7x}{x^2-4}$$
Since ...
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Find all real numbers $a$ for equation $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
Problem:
Find all real numbers $a$ for which the equation, $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
This problem is from an algebra round of a local high school math competition that has ...
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Why doesn't simultaneous equations work to find co-efficients of a cubic that passes through four points?
I'm trying to find the equation of a cubic that passes through three specific points (technically it's four but that point is y-intercept). The equation would look something like this:$f(x)=ax^3+bx^2+...
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Order $3$ linear transforms invariating a binary cubic form
Consider $P(x,y)$ a homogenous polynomial of degree $3$ in two variables (a binary cubic). To it we associate first the $2\times 2$ matrix
$$\frac{1}{2}\operatorname{Hess}(P) = \frac{1}{2}\cdot\left( ...
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