0
votes
1answer
14 views

Euclidean division - For what values of a, does the polynomial g(t) get divided by f(t) in the complex ring

They want to find the values of a where g(t) can be divided by f(t). $f(t) = t^2 + it − ai$ $g(t) = t^4 + (1 − i)t^3 + (1 − 2i)t^2 − 3at − (4 + 2i)a$ Euclidean algorithm: $g(t) = f(t)q(t) + ...
0
votes
1answer
19 views

weighted inner product of polynomials, can weight function be complex?

I am just learning about inner-products on polynomial space, where the coefficients of the polynomials may be complex: $P_m(\mathbf{F})$ The inner-product given by: $\langle p,q \rangle = \int_0^1 ...
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
2
votes
2answers
62 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
0
votes
3answers
60 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
1
vote
2answers
86 views

Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
0
votes
1answer
34 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
1
vote
0answers
35 views

Maximum of $P$ in the disk $|z|=1$ depending on co-efficients

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that ...
2
votes
1answer
41 views

Prove that $\bar{P}_{\bar{z}}=P_z,\ (P_z,P_z)=(P_{\bar{z}},P_{\bar{z}})$ with $P_z=\dfrac{\partial{P}}{\partial{z}}$

I have a problem: For $P$ is a nonzero real valued homogeneous polynomial of degree $k$: $$P(z,\bar{z})=\sum_{j=1}^{k-1}a_jz^j\bar{z}^{k-j}$$ where $a_j \in \Bbb C,\ a_j=\bar{a}_{k-j}$. ...
0
votes
1answer
26 views

What is the special thing about |z|=2, will this point lie in the mandelbrot set?

For the quadratic iteration $z \to z^2+4$, if you perform a few iterations letting $z_0 =0.5+1.936491673i$, the modulus of the points will be 2 ( or closer 2 two because of the inaccuracy of the ...
1
vote
1answer
30 views

Convert complex logarithms to inverse tangents

I'm doing an exercise from the book "Algorithms for Computer Algebra" by Keith O. Geddes. I'm asked to show that if $u$ and $v$ are two relatively prime polynomials in $\mathbb{Q}[x]$ and $s$ and $t$ ...
0
votes
2answers
50 views

Show that $(x-\alpha)(x-\overline{\alpha})$ is a also a factor of $p(t)$ over the complex numbers

Here is the full question. Lots of struggles: Let $p(t)$ belong to $P(R)$. a) If $(x − \alpha)$ is a factor of $p(t)$ over the complex numbers (i.e. $p(t) = (x − \alpha)\cdot q(t)$, for ...
1
vote
2answers
88 views

Evaluation of complex real numbers

The much anticipated math.SE community blog will $\tiny\mathrm{hopefully}$ contain a contribution from Alex Becker with the topic The Complex Real Roots of $x^3-3x+1$, which I'm really looking forward ...
0
votes
1answer
54 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
59 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?
1
vote
0answers
25 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 ...
5
votes
1answer
194 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
47 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
56 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
187 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
36 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
91 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. ...
12
votes
2answers
678 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
916 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
66 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
415 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
93 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
89 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
92 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
81 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
33 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
140 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
125 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
100 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
58 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
51 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
127 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
181 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?
15
votes
0answers
319 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
96 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
93 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
141 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
86 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
129 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
66 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 ...