All Questions
Tagged with cubics complex-numbers
95
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
3
votes
1
answer
109
views
Simplifying Coefficients of a Cubic Polynomial with Complex Roots
I am currently encountering difficulties while trying to solve the following question, and I would greatly appreciate any assistance you can provide.
Let $a,b,c$ be complex numbers.
The roots of $z^{3}...
- 327
2
votes
2
answers
220
views
Determine whether the roots of a cubic equation have positive real part
Consider the cubic equation
$a x^3 + b x^2 + cx + d =0 $,
where all coefficients depend on three parameters
$$a=a(i, j, k), b=b(i, j, k),\cdots$$
and $a, b, c \in \mathbb{R}$ for all $(i, j, k)$. The ...
2
votes
1
answer
76
views
Spivak, Ch, 25, 2(v): Is there some specific technique to factorize $x^3-x^2-x-2$ or must one guess that 2 is root?
The following problem appears in Ch. 25, "Complex Numbers" of Spivak's Calculus
2 (v) Solve the equation $x^3-x^2-x-2=0$.
Is there some specific technique to factorize this? Must one ...
- 5,021
0
votes
0
answers
105
views
When we say the odd degree polynomial has odd number of real roots, is there any condition on the coefficients?
I read that if the degree of a polynomial equation is odd then the number of real roots will also be odd.
I took the example of a cubic equation. If it has imaginary roots then that will occur in pair....
- 8,338
3
votes
3
answers
214
views
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
My Attempt:
On rearranging, I get, $(x^2+x+1)(3x+7)+2=0$
Or, $3x^3+10x^2+10x+9=0$
Derivative of the cubic ...
- 8,338
2
votes
1
answer
233
views
Collinearity of the three roots of a cubic equation in the complex plane
The three roots of the equation $x^3+bx+c=0\;(b,c\in\Bbb C,c\ne0)$ in the complex plane are collinear iff$$k\in\Bbb R\land k\le-\frac{27}4$$where $k=\frac{b^3}{c^2}$.
Original post on Math ...
- 3,047
1
vote
2
answers
101
views
Missing solutions from complex cubic root
From Cardano's work in the 1600s, we have this famous example of a cubic polynomial equation:
$$x^3-15x-4=0.$$
Plugging the coefficients into the Cardano-Tartaglia formula, we get an expression for ...
- 769
2
votes
3
answers
473
views
How does one show this complex expression equals a natural number?
We have:
$$\left(\frac{10 }{3^{3/2}}i-3\right)^{1/3}+ \frac{7}{3 \left(\frac{10}{3^{3/2}}i-3\right)^{1/3}}=2$$
This comes from solving the cubic equation of $x^3-7x+6=0$ which factors as $(x-2)(x-1)(x+...
- 4,425
0
votes
1
answer
78
views
Polynomial and Complex Roots Problem
Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$ find all possible ...
user1068511
4
votes
0
answers
104
views
A cute way to solve the quadratic. How to extend it to the cubic?
Playing around with complex numbers, I found a cute way of solving the quadratic equation.
Let's start with the (monic) equation
\begin{equation}
z^2+pz+q = 0
\end{equation}
where $z$ and the ...
- 3,386
0
votes
3
answers
395
views
Find the roots of $z^3 +3z^2 +3z +3=0$
Hello I have this problem:
Find all $z \in C$
$z^3 +3z^2 +3z +3=0.$
With Mathematica I get 3 roots
$z_1 = -1 -\sqrt[3]{2}$
$z_2 = -1 + (1 + i \sqrt{3})/2^{(2/3)}$
$z_3 = -1 + (1 - i \sqrt{3})/2^{(2/3)}...
- 157
0
votes
2
answers
48
views
How to factor $z^3-3\sqrt{3}iz^2-9z+3\sqrt{3}i$ strictly via the method of grouping
I really don't have much of my own attempt to show for this question. All I managed to get to was:
$$z^2(z-3\sqrt{3}i)-3(3z-3\sqrt{3}i)$$
By simply factorising. It is asked in the question to be ...
- 189
2
votes
3
answers
94
views
Stuck on simplifying expressions involving trig and inverse trig functions
TL; DR
Using Mathcad and Wolfram I can see that
$$\sqrt{7}\cos\frac{\tan^{-1}\left(\frac{9\sqrt{3}}{10}\right)}{3}=2.5$$
The decimal value seems to be exact because Mathcad displays it like that with ...
- 173
-1
votes
1
answer
150
views
Solving Non-Homogenous Recurrence Relation
I was interested in Solution of this Non-Homogenous Recurrence Relation
$f(n)=f(n-1) + f(n-3) + 1$
The Base conditions are:
$f(0)=1$
$f(1)=2$
$f(2)=3$
Kindly help me in solving this Recurrence ...
- 1,143