0
votes
1answer
46 views

Number of zeros of $ z^7+4z^4+z^3+1$

How many zeros does $z^7+4z^4+z^3+1$ have in each of the regions |z|<1 and |z|<2? I know I should use Rouche's Theorem but I can't find a $|f(z)| > |p(z)-f(z)|$ and $f(z)$ have equal number ...
0
votes
4answers
51 views

Proof that the coefficients of a polynomial are real

How does one prove that all the coefficients of this polynomial: $$(x+i)^{10}+(x-i)^{10}$$ are real numbers, without using Newton's Binomial Theorem?
0
votes
0answers
24 views

Divisibility by $z-z_0$ if $z_0\in \mathbb{C}$ [duplicate]

I have a problem I'm working on, and I'm just not getting it. Suppose that $z_0\in\mathbb{C}$ is fixed. Show that if $P(z)=c(z^k-z_0^k)$, then there exists a polynomial $Q(z)$ such that ...
1
vote
1answer
56 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
0
votes
2answers
44 views

Finding the Remainder of Complex Polynomials

Suppose $f(-1 + i) = 2 + 5i$ and $f(-2 - i) = -3$ determine the remainder of $f(x)$ divided by $(x + 1 - i)(x + 2 + i)$. I don't really know where to start any help would be great. Thanks :)
2
votes
0answers
51 views

Extension to complex numbers

Is there an extension to the complex numbers in which $zz^* = i$ has a solution? (The star denotes conjugation.) EDIT: I'm mathematically ignorant, but I'm guessing such an extension can't be a ...
4
votes
6answers
157 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
2
votes
1answer
33 views

Factorising a complex polynomial over C

If $f(z)=z^3+7z^2+16z+10$, find all factors of $f(z)$ over $C$. If I had at least one zero or factor I would be able to find the others, but I just don't know how to start.
1
vote
3answers
49 views

How to find number of real and complex roots?

Below is a question asked in JNU Entrance exam for M.Tech/PhD. I want to know if there is a fixed way to calculate it. I have failed to use the factor theorem. ...
0
votes
1answer
26 views

minimal polynomial and linear transformation

If $T:\Bbb{C} \to \Bbb{C}$ defined by $T(x)=x$ . T satisfity minimal poly is $x-1$. Is it correct. Any polynomial of degree $>1$ is a linear transformation on C .this type of transformation exist ...
12
votes
2answers
628 views

Prove that a polynomial has at least one nonreal complex root

Prove that the polynomial below has at least one nonreal complex root $$x^5+\frac{x^4}2+ \frac{x^3}3+\frac{x^2}4+\frac x{24}+\frac 1{120}$$ I have tried to prove that there exist $k\in \Bbb R$, such ...
0
votes
1answer
23 views

Proof that complex conjugate of polynomial result equals pynomial result with complex conjugated argument

This question feels uneasy to be expressed by words for me, however, I'm asked to prove this: $$P(\overline{a+bi}) = \overline{P(a+bi)}$$ Of course, $\overline{a+bi} = a-bi$.
5
votes
3answers
337 views

Polynomial of degree 4 with real coefficients, two complex roots given.m

Write in the form f(z) = 0, where f(z) is a polynomial of degree 4 with real coefficients, the equation having (3 + i) and (1 + 3i) as two of its roots. Can anyone help me? I'm guessing the two ...
1
vote
1answer
61 views

complex numbers, complex roots of equation.

$z_1=a+bi$ , $a,b\in\Bbb R$, $b\neq 0$ is a complex root of the equation $z^2-2z+25=0$. Without evaluating the roots, answer the following questions: i) show that $\overline{z_1}$, the conjugate of ...
3
votes
5answers
384 views

I'd like to get explain about complex roots

If $x^6+1=0$ so $x^6=-1$, then we have to find the roots at $\mathbb{C}$. I saw that the roots are $$\Large{e^{(\frac{\pi}{6}+\frac{2k\pi}{6})i}}\;\small{k=0,1,2,3,4,5}$$ this what I understand. ...
0
votes
1answer
82 views

Product of all complex roots of z^n=a+bi?

How can one prove that the product of all the roots of a complex equation is the same as one root to the power of equation? e.x. $z^n=a+bi$ has $n$ roots (from de Moivre's formula), prove that their ...
4
votes
4answers
76 views

$\omega^2+\omega+1$divides a polynomial

The question is Show that $f(n)=n^5+n^4+1$ is not prime for $n>4$. The solution is given as Let $\omega$ be the third root of unity. Then $\omega^2+\omega+1=0$. Since ...
1
vote
3answers
86 views

f(x) and g(x) are two polynomials, then choose the right option…

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^{3})+x^{2}g(x^{6})$ is divisible by $x^{2}+x+1$, then choose the correct option: $A. f(1)=g(1)$ $B. f(1) $ is not equal ...
1
vote
1answer
71 views

$\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
2
votes
2answers
31 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
4
votes
2answers
83 views

Use $\alpha, \beta, \gamma $ roots of a polynomial to construct another polynomial [duplicate]

Let $\alpha, \beta, \gamma $ be roots $\in \mathbb{C}$ of $x^3-3x+1$. Determinate a monic polynomial, degree $3$, witch roots are $1- \alpha^{-1},1-\beta^{-1},1-\gamma^{-1}$ The catch is that i can't ...
0
votes
2answers
114 views

Polynomial divisibility

Given $p(x) \in \mathbb Q[x] $ an irreducible polynomial, and $\alpha \in\mathbb C $ root of $p(x)$. Prove that if $q(x) \in \mathbb Q[x]$ it's a polynomial, such $q(\alpha) = 0$ then $p(x) \mid ...
1
vote
0answers
37 views

Show a polynomial is reducible to linear terms - check my answer

I have an exam tomorrow in linear algebra, and I want to make sure I answered this question correctly. Let $p \in \mathbb R[x], z \in \mathbb{C}$. We are given if $Im(z)>0$ then $p(z)\neq0$ Show ...
1
vote
1answer
86 views

Can a complex quadratic polynomial have real roots?

Where $a\in \textbf{Z}[i] $ and $a \not\in \textbf{Z}$, suppose for the quadratic formula, $ b^2-4ac = 0 \Rightarrow b^2 = 4 ac \Rightarrow c= \frac{b^2}{4a} $ and $ b=a $, so that $\displaystyle ...
0
votes
2answers
52 views

Finding the roots of 4096x^3-10496x^2+152576x - 961=0 (1 root and 2 complex)?

I don't know how to find the roots of 4096x^3-10496x^2+152576x - 961=0 I try using wolfram and http://en.wikipedia.org/wiki/Cubic_function. I don't really understand it can someone please explain how ...
0
votes
2answers
48 views

Polynomial with complex coefficients

I can't solve the following questions: Let $a,b$ be real numbers, $Z= a + ib$. How much polynomials with complex coefficients $q(x) = x^3 + b_2 x^2 + b_1 x + b_0$ there are so that $Z$ is a root of ...
4
votes
3answers
121 views

Show that the real part of the root of an equation is constant

I've been stuck for a while on the following question: Let $z$ be a root of the following equation: $$z^n + (z+1)^n = 0$$ where $n$ is any positive integer. Show that $$Re(z) = -\frac12$$ ...
0
votes
1answer
141 views

Are the Complex Numbers Isomorphic to the Polynomials Mod x^2+1?

My friend told me that the Complex Numbers are Isomorphic to the Polynomials Mod x^2+1, is this so? And how can this be proved?
10
votes
0answers
212 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
1
vote
1answer
73 views

Analog to bisection: Converging on complex roots of a polynomial

I am working on a Perl module that, among other features, will solve all the zeroes of a polynomial. Thus far, I am doing OK for $2$, $3$, $4$th degree using quadratic, Cardano's and Ferarri's ...
3
votes
1answer
31 views

Prove that $q(a_i)\in \{a_1,…, a_n\}$

Let $p(x)$ and $q(x)$ be polynomials with rational coefficients such that $p(x)$ is irreducible over $\mathbb{Q}$. Let $a_1,..., a_n\in \mathbb{C}$ be the complex roots of $p$, and suppose that ...
1
vote
2answers
88 views

Factoring any single-variable polynomial in $\mathbb C$

The fundamental theorem of algebra says $$ \forall p(x):\mathbb C \to \mathbb C,\ p(x) = a\prod_{n=0}^m\big(b_nx+c_n\big) $$ where $p(x)$ is a single-variable polynomial, and $\{a;m\}\cup\{\forall ...
0
votes
1answer
44 views

Why is this polynomial a function of $X^3$?

In studying that recent question, I noticed that curious (or perhaps not so curious) property : if $x,y$ are rational numbers and $a$ is the real part of a cubic root of $x+iy$, then $Q(a^3)=0$ where ...
13
votes
3answers
137 views

$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$

Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost. Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying: $$x_1 + x_2 \dots + x_k = 0$$ $$x_1^2 + x_2^2 ...
2
votes
1answer
84 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
112 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
63 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
40 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 ...
4
votes
1answer
91 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
76 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
133 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
186 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?
1
vote
2answers
64 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
200 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 ...
6
votes
2answers
197 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
92 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
125 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
375 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 ...
3
votes
1answer
286 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
100 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 ...