All Questions
Tagged with cubics roots-of-cubics
27
questions
12
votes
7
answers
5k
views
Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?
I am studying the history of complex numbers, and I don't understand the part on the screenshots. In particular, I don't understand why a cubic always has at least one real root.
I don't see why the ...
- 2,452
-1
votes
0
answers
46
views
Quicker and non-trivial methods for solving Cubic Equation
Motivation : There have been many elementary ways like Hit-and-trial method, Polynomial division and others used in teaching how to solve cubic equation. I wanted to find a method that is faster to ...
- 95
1
vote
2
answers
62
views
Prove that $a=0$ if and only if $b=0$ for the cubic $x^3 + ax^2 + bx + c=0$ whose roots all have the same absolute value.
Take three real numbers $a, b$ and $c$ such that the roots of equation $x^3+ax^2+bx+c=0$ have the same absolute value. We need to show that $a=0$ if and only if $b=0$.
I tried taking the roots as $p, ...
- 23
0
votes
0
answers
53
views
cubic equation edge cases
Working on general cubic equation solver in form ax^3+bx^2+cx+d=0 And have no clue for special cases:
In terms of cubic there should be one real root and two complex, or 3 real roots if coefficients ...
2
votes
3
answers
103
views
Prove that $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots
I'm trying to prove that the cubic equation
$a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$
has three real roots. The coefficients are
$a_3 = - 1 - \sigma - \tau - \chi$
$a_2 = -2 (\sigma +...
- 61
2
votes
1
answer
121
views
Three real roots of a cubic
Question: If the equation $z^3-mz^2+lz-k=0$ has three real roots, then necessary condition must be _______
$l=1$
$ l \neq 1$
$ m = 1$
$ m \neq 1$
I know there is a question here on stack about ...
- 497
0
votes
0
answers
42
views
Set of coefficients of degree three monic real polynomial with three real roots is connected.
Let $p(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients $a, b, c,$ and define:
$$D=\{(a,b,c)\in \mathbb{R}^3\mid \text{the polynomial}\ p(x)\ \text{factors into linear factors over }\ \...
- 483
1
vote
2
answers
143
views
How to solve $x^3−x+1=0$
I am interested in finding a solution for the equation:
$$ x^3 - x + 1 = 0 $$
I've noticed that there are numerous polynomial equations where one of the coefficients is zero. Could you provide ...
- 63
0
votes
0
answers
125
views
Can I use this algorithm for solving cubic equations?
I am trying to find the root solutions for a cubic equation including the eigenvalues of each root.
I tried to put the equation into my calcualtor but the calculator doesn't show solutions that has ...
- 1
1
vote
1
answer
108
views
Solving a cubic using triple angle for cos (i.e $\cos(3A)$)
a) Show that $x=2\sqrt{2}\cos(A)$ satisfies the cubic equation $x^3 - 6x = -2$ provided that $\cos(3A)$ = $\frac{-1}{2\sqrt{2}}$
I did not have a difficulty with this question, I have provided it for ...
- 15
1
vote
1
answer
203
views
How do I find a cubic equation given only one root?
Given the root of a cubic equation $Z = \sqrt[3]{Y + \sqrt{Y^2 - \frac{X^6}{27}}} + \sqrt[3]{Y - \sqrt{Y^2 - \frac{X^6}{27}}} - X$ and the assumption that both $X$ and $Y$ are greater than zero, is ...
- 1,861
3
votes
3
answers
165
views
Is there any faster way to factor $x^3-3x+2$?
$$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)$$
Is there a better, faster way to factor this cubic trinomial?
- 1,043
3
votes
6
answers
406
views
Find all real numbers $a$ for equation $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
Problem:
Find all real numbers $a$ for which the equation, $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
This problem is from an algebra round of a local high school math competition that has ...
- 167
1
vote
0
answers
50
views
Order $3$ linear transforms invariating a binary cubic form
Consider $P(x,y)$ a homogenous polynomial of degree $3$ in two variables (a binary cubic). To it we associate first the $2\times 2$ matrix
$$\frac{1}{2}\operatorname{Hess}(P) = \frac{1}{2}\cdot\left( ...
- 55k
3
votes
0
answers
64
views
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 $...
- 55k