Questions tagged [roots-of-cubics]
For questions related to roots of a cubic equation. All of the roots of the cubic equation can be found by the following means: algebraically, trigonometrically or numerical approximations of the roots.
81 questions
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How is $\alpha_1 + \omega \alpha_2 + \omega^2 \alpha_3$ a Ruffini radical?
This comes from exercise 8.10 in Ian Stewart’s Galois Theory, where it is desired to show that the general cubic (with roots $\alpha_1, \alpha_2, \alpha_3$) is solvable in Ruffini radicals. A hint ...
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Simplification of Cardano's Method
Context:
I'm having a lot of trouble handling Cardano's last equation, symbolically. Given the depressed cubic $x^3+px+q$, one can define $u_{1, 2}=\sqrt[3]{ -\cfrac{q}{2} \pm \sqrt{\cfrac{q^2}{4} + \...
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Descartes' Rule of Signs with a Parametric Cubic
Consider a cubic polynomial (in $\mu$), where the coefficients depending parametrically on real variables $\chi,\tau > 0$:
\begin{equation}
P(\mu;\chi,\tau) = a(\chi,\tau)\mu^3 + b(\chi,\tau)\mu^2 +...
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Discriminant of a Real Polynomial vs Complex Polynomial
Suppose $f(x) = ax^3 + bx^2 + cx + d$, with $a,b,c,d \in \mathbb{R}$.
Let $\Delta$ denote the discriminant. If $\Delta > 0$, then $f$ has 3 distinct, real roots. If $\Delta = 0$, then $f$ has a ...
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Prove that $f(x)=ax^3+bx^2-2bx+4b$ has at most one real root.
To Prove: The function $f(x)=ax^3+bx^2-2bx+4b$ has at most one real root.
My approach:
I know I could use the extrema $x=\frac{-b\pm\sqrt{b^{2}+6ab}}{3a}$ but substituting back into $f(x)$ yields the ...
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General solution for $x^3+px^2+q=0$
There's a fairly well-known formula for cubic equations of the form $x^3+px+q=0$: The transformation $x = z-\frac{p}{3z}$ gives $z^3+q-\frac{p^3}{27}z^{-3}$, and after multiplying by $z^3$, the ...
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Cubic equation for closest point to a function
We have $y = x^2 + 1$ and are trying to find the point closest to $(4, 0)$. For context, this is for a high school calculus class.
Clearly, $l = \sqrt{(x - 4)^2 + ((x^2 + 1) - 0)^2}$, from which we ...
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A neusis construction with tilt angle 27° that approximates cube root of 2: coincidence or hidden reason?
In a previous MSE question I described a neusis-style technique with tilted circles and squares. While experimenting further I found a construction that produces a tilted segment of length $(\approx 1....
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Show $\int_0^\sqrt[3]2(u^3f(u)+u^7/f(u))du=\ln(1+x)-(\sqrt x-\sqrt2)^2+\frac43$ where $f(u)=\sqrt[3]{27+u^6+\sqrt{27(27+2u^6)}}$ and $x^3-2x^2-4x-2=0$
Inspired by this question, I've come up with an exercise that I hope you will find amusing:
Show that
$$\begin{eqnarray*} &&\int_{0}^{\sqrt[3]{2}}\left[u^3\sqrt[3]{27+u^6+\sqrt{27}\sqrt{27+2u^...
- 374k
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1
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How to find different sets of coefficients of the cubic function?
Suppose that $f(x) = x^3 + mx^2 + nx + 8$ is a real-valued function.
I am determining some integers m and n such that $f(x) = 0$ has only one real root (the other two are complex conjugates). ...
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Third degree polynomial with small parameter
This is a follow-up to that question, so I will refer to it for the motivation. In continuation, I now have this polynomial obtained by inserting $x=\phi_2+\sqrt{\varepsilon}\cdot y$ in the polynomial ...
- 1,425
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Singular expansion of a root of a polynomial
I find the following issue when dealing with a problem concerning a PDE.
Consider the polynomial equation for $x$ given by
$$
(1-x)(2b+x^2)=2\phi_2(1-\phi_2)^2,\;\;\frac{1}{3}<\phi_2<{1}.
$$
At ...
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Proof that: ($\forall a,b,c\in \mathbb{R}$)($ab=c^{2}\iff\exists k\in \mathbb{R}\;$s.t. $a=kc \wedge b=k^{-1}c$)
Was looking through a proof of a necessary and sufficient condition for the roots of a cubic to follow a geometric progression. It used the following argument for necessity.
For the following cubic ...
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Seeking a Simple Proof of Two Conjugate Complex Roots for a Cubic Equation
I'm currently working on a complex analysis problem and I’m trying to show that the polynomial
$$
f(z) = z^3 + (-8 - 3i)z^2 + (17 + 24i)z - 51i
$$
has two complex conjugate roots.
While I’ve seen one ...
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Finding value of $f(4)$ for a cubic polynomial under given derivative and sign conditions
Let $f(x)$ be a monic cubic polynomial (i.e., leading coefficient is $1$) such that it satisfies the following conditions:
There exists no integer $k$ such that $f(k+1) \cdot f(k-1) < 0$
$f^\...
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