Tagged Questions
2
votes
1answer
51 views
Is there a geometric relationship between plane geometry and polynomials?
It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a ...
2
votes
2answers
36 views
Roots of cubic polynomial lying inside the circle
Show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle
$|z|=max{\{1,|a|+|b|+|c| \}}$
Now this problem is given in Beardon's Algebra and Geometry third chapter on complex numbers.
What might ...
1
vote
1answer
36 views
Lower bound for polynomial with complex coefficient
Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$.
...
1
vote
2answers
35 views
Help please on complex polynomials
I wanted to know if there's any good approaches to these questions
a)By considering $z^9-1$ as a difference of two cubes, write $1+z+z^2+z^3+z^4+z^5+z^6+z^7+z^8$
as a product of two real factors one ...
5
votes
1answer
47 views
Roots of a polynomial and its derivative
All roots of a complex polynomial have positive imaginary part. Prove that all roots of its derivative also have positive imaginary part.
It's not a homework. This issue has been proposed in the ...
2
votes
3answers
56 views
Determine all $z \in\Bbb C$ such that $z^8 + 3iz^4 + 4 = 0$
Trying to study for my final, and this question came up.
Any hints as how to how to begin would be greatly appreciated.
-edit-
thank you all for your help. I would have never thought of that in a ...
2
votes
3answers
91 views
How can complex polynomials be represented?
I know that real polynomials (polynomials with real coefficients) are sometimes graphed on a 3D complex space ($x=a, y=b, z=f(a+bi)$), but how are polynomials like $(1+2i)x^2+(3+4i)x+7$ represented?
3
votes
2answers
73 views
Write in polynomial in factored form in complex number
Write the following polynomial in factored form(in complex number):
$$1+z+z^2+z^3+z^4+z^5+z^6$$
Also, is there general solution of factoring for $1+z+z^2...z^n$ types of polynomial?
0
votes
2answers
41 views
number of roots of polynomial of order n
from theorem of algebra,it is well know that polynomial of order n has exactly n roots,for exmaple quadratic equation like $ax^2+bx+c$ has three cases
let $D=b^2-4ac$ ,so we have
...
2
votes
1answer
61 views
Solutions to $(z+1)^n = z^n$ using conformal maps.
I'm doing a homework problem where I have to find all roots of $(z+1)^7 - (z)^7 = 0$ using the roots of unity for $z^7$
I noticed that if $a$ is a root of unity for $z^7$, then $1/(a-1)$ maps the ...
4
votes
2answers
102 views
A ‘strong’ form of the Fundamental Theorem of Algebra
Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial
$$
p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in ...
3
votes
0answers
68 views
Prove (*) by induction on k.
Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form
$$\sum_{i=1}^m ...
1
vote
1answer
35 views
Is the Fujiwara bound the most precise bound on maximum absolute value of complex roots of real polynomials?
Is the Fujiwara bound the most precise bound on maximum absolute value of complex roots of real polynomials ? Or does it exist some improved version for this special case of real polynomials ?
3
votes
2answers
150 views
geometric interpretation of quadratic equation with complex coefficients
When an equation has real coefficients and non-negative discriminant, the geometric meaning of it's roots is intersection of the function with the x-axis.
I know how to get roots of quadratic ...
2
votes
1answer
124 views
Recursive FFT java implementation
Given below is my java program for FFT. For the input {0,2,3,-1} its returns a false output in complex point representation.
...
-3
votes
4answers
93 views
Complete instead of Complex, Irregular instead of Imaginary
Will the terms complex and imaginary ever be replaced? At least within beginning classes?
I imagine it is more of a kind of hazing into the "mathemitician's club" to allow the terms to confuse ...
1
vote
2answers
45 views
Multiplying imagionary roots of a polynomial
I am trying to answer the following question:
The roots of the quadratic equation $ax^2-16x+25$ are $2+mi$ and $2-mi$, where $m>0$. Compute the sum of $a+m$.
Should the zeros of the equation ...
2
votes
1answer
73 views
Evaluate a certain derivative
Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let
$$ h_j(x)=\prod_{i\neq j,i=1,\dots, n} ...
1
vote
1answer
150 views
How to factor a polynomial in real numbers?
Which of the following polynomials can't be factored in real numbers?
Multiple choice question from an old test.
Got the following polynomials:
$x^8$
$(x-3)(x^2+x+1)$
$(x-2)^3(x^2-1)$
$x^8(x^2-1)$
...
0
votes
3answers
897 views
How to find the imaginary roots of polynomials
I'm looking for a simple way to calculate roots in the complex numbers.
I'm having the polynomial $2x^2-x+2$
I know that the result is $1/4-$($\sqrt{15}/4)i$.
However I'm not aware of an easy way to ...
3
votes
3answers
125 views
How to find the roots of $x³-2$?
I'm trying to find the roots of $x^3 -2$, I know that one of the roots are $\sqrt[3] 2$ and $\sqrt[3] {2}e^{\frac{2\pi}{3}i}$ but I don't why.
The first one is easy to find, but the another two roots?
...
1
vote
1answer
44 views
Polynomial $P\in\mathbb{R}[x]$ with $\overline{P(\overline{x})}=P(x),\forall x\in\mathbb{C}$
I have already shown that any polynomial $P\in\mathbb{R}[x]$ satisfies $\overline{P(\overline{x})}=P(x),\forall x\in\mathbb{C}$
My question is, given a polynnomial $P\in\mathbb{C}[x]$, how I can ...
1
vote
1answer
62 views
what are the solutions of $x^n+a^n=0$?
For what values $a$, the equation $x^n+a^n=0$ has $n$ different solutions? what are the solutions?
(the question refers to complex solutions).
1
vote
2answers
109 views
Gre Question Complex Number (plug and chug)
This seems like it should be easy, but I can't seem to simplify it: If $z=e^{i\frac{2\pi}{5}}$, then what is $1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9$. The choices are $0, 4e^{i\frac{3\pi}{5}}, ...
1
vote
1answer
48 views
Understanding a theorem of Marden's on the moduli of zeros of polynomials
My question is concerning Theorem 3.2 in this paper of Marden's. The gist of the theorem is stated below.
Theorem 3.2.
Every polynomial of the form
$$ f(z) = \sum_{j=0}^{n} (b_j - ...
3
votes
2answers
240 views
Cubic with complex roots
I have a problem figuring out how exactly I find the cube roots of a cubic with complex numbers.
I need solve the cubic equation $z^3 − 3z − 1 = 0$.
I've come so far as to calculate the two complex ...
2
votes
2answers
87 views
Solutions to $z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2 = 0, b\in \mathbb R, z \in \mathbb C$
$z_1 = 1+i$ is a given solution.
I guess what I have to find is $z_2$ and $z_3$ in
$(z - (1 + i))(z - z_2)(z-z_3) = z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2$.
I tried to divide the polynomial by $(z - (1 ...
0
votes
1answer
130 views
How to solve this by galois theory?
please focus on the concept to solve this problem, because i can't handle to research on diffcult terminology.Thanks in advance.
Find all real roots by galois theory and find all other root to this ...
1
vote
2answers
90 views
Induction (concerning $1+z+\dots+z^n$) and follow up question
I am doing a review of stuff from earlier in the semester and I can't prove this by induction:
Use induction on $n$ to verify that $1+x+\cdots+z^n= \frac{1-z^{n+1}}{1-z}$ (for $z\not=1)$. Use this ...
4
votes
2answers
429 views
If z is one of the fifth roots of unity, not 1…
If z is one of the fifth roots of unity, not 1, show that:
$1+z+z^2+z^3+z^4=0$
Which wasn't too bad, but the next part is killing me: show that:
$z-z^2+z^3-z^4=2i(sin(2\pi/5)-sin(\pi/5))$
Can ...
0
votes
1answer
6k views
Where to find information on shadow functions?
I happen to give some private lessons to an IB (International Baccalaureate) student. He asked me for help with writing some kind of a project on a set topic, given some materials (containing the ...
3
votes
3answers
274 views
Is $\mathbb{R}[X]/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$
Trying to solve a problem, I got stuck on the following question.
If $P\in\mathbb R[X]$ is an irreducible polynomial and $\operatorname {deg} P=2$, then is it true that ...
1
vote
2answers
167 views
Complex Polynomial transformation
I'm studying for an exam and professor gave us to create a little program that automatically does a transformation for a polynomial with complex coefficients, I don't have many problems doing the ...
5
votes
3answers
283 views
An application of Vandermonde determinant
Let $\lambda_1,\ldots,\lambda_n$ be complex numbers such that for each positive integer $k\geq 0$,
$$\sum_{i=1}^n \lambda_i^k=0.$$
Here I am supposed to show that $\lambda_i=0$ for each $i\in ...
1
vote
2answers
88 views
simplify complex polynomial $p(t) \in \mathbb{C}[t]$
How to simplify the following polynomial?
$$
\begin{align}
(t - \sqrt{3} \; e^{ \frac{\pi}{3} i }) (t - \sqrt{3} \; e^{ -\frac{\pi}{3} i }) &= t^2 - \sqrt{3} \; e^{ \frac{\pi}{3} i } \; t - ...
2
votes
1answer
51 views
A problem with polynomials.
This is a problem from a test in my course in analytic functions. I didn't manage to solve it. Could you please give me a hint? The problem is:
Calculate the third root of the sum of the coefficients ...
2
votes
1answer
220 views
Bound the complex roots of a polynomial above
We consider $P(z)=a_{0}+a_{1}z+\cdot+a_{n-1}z^{n-1}+a_{n}z^n$, with $a_{0},\ldots,a_{n-1},a_{n} \in \mathbb{C}$ and $a_{n}\neq0$.
Let $R=\max_{0\leq k\leq n-1}\left | \frac{a_k}{a_n} \right |$ and ...
4
votes
1answer
100 views
Existence problem for a polynomial with complex coefficients
Let $n$ be a nonnegative integer and $a_{0}, a_{1}, ..., a_{n}$ real numbers. For any real number $t$ let $f(t)= \sum_{k=0}^{n}a_{k}\cos(kt)$.
Could you help me with the following two questions ?
a) ...
5
votes
1answer
515 views
Using the fifth roots of unity to find the roots of $(z+1)^5=(z-1)^5$
The question I am working on starts of with:
Find the five fifth roots of unity and hence solve the following problems
I have done that and solved several questions using this, however ...
-1
votes
3answers
71 views
Complex Numbers Question
1) let $Z_0$ be a solution of $Z^{13}-13Z^{7}+7Z^{3}-3Z+1=0$, Is it true that $Z_0$'s conjugate is also a solution?
2) let $Z_0$ be a solution of $Z^{2}+iZ+2=0$, Is it true that $Z_0$'s conjugate is ...
5
votes
1answer
110 views
Preimage of discs under a complex polynomial
Let $a_0, \ldots, a_n \in \mathbb{C}$, with $a_n \neq 0$. Consider set
$$U_R = \{~z \in \mathbb{C} ~:~ |a_nz^n + \dots + a_1z + a_0| < R~\}$$
for each $R > 0$. How do I prove that $U_R$ is ...
1
vote
1answer
81 views
Polynomial equations in 2 variables with symmetry
Suppose $P(x,y)$ is a polynomial with real coefficients.
Is it true that any solution $(x_0,y_0)$ of the system $P(x,y)=P(y,x)=0$
has the property that $y_0 = \overline{x_0}$ (i.e. they are ...
11
votes
2answers
278 views
A property of roots of the truncated series for $\sin(x)$
Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$
In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$.
...
0
votes
2answers
178 views
Rational function, sequences, polynomials and roots of unity
Let $$f(x) = \sum_{n\geq 0} a_n x^n = \frac{P(x)}{(1-x)^d}$$ be a rational function.
(a) Prove: There is a polynomial $P_2(x)$ so $$\sum\limits_{n\geq 0} a_{2_n} x^n = \frac{P_2(x)}{(1-x)^d}$$
(b) ...
10
votes
5answers
389 views
imaginary numbers - how can I understand them - especially as they occur in 'roots' of polynomials?
In another question here, about roots of equations being imaginary, the accepted answer
said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I ...
15
votes
5answers
1k views
How fundamental is the fundamental theorem of algebra?
Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial ...
