This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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4
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1answer
117 views

Number of complex roots of a degree 6 polynomial

Given some degree 6 polynomial $f(x) \in \mathbb{Q}[x]$, is there any invariant of the polynomial (depending on the coefficents) that will tell you if this polynomial has 6 complex roots or just 2 ...
0
votes
1answer
18 views

Question about irreducible polynomials?

Is this polynomial: $irr(\sqrt{3 -\sqrt{6}}, \mathbb{Q})$ irreducible? Here is what I did $ a = \sqrt{3 -\sqrt{6}}$ $a^2 = 3 - \sqrt{6}$ $a^2 - 3 = -\sqrt{6}$ $(a^2 - 3)^2 = 6$ Our polynomial ...
0
votes
2answers
48 views

Suppose $f(x)$ is a polynomial of degree 5, and with leading coefficient 1. [closed]

Suppose $f(x)$ is a polynomial of degree $5$, and with leading coefficient 1. If further that $f(1)=1, \ f(2)=2, \ f(3)=3, \ f(4)=4, \ f(5)=5$. What is the value of $f(6)$?
0
votes
1answer
48 views

Residue class ring $\mathbb{Z}[x]$/I and $\mathbb{Z}[x]$/J

$I = \left\lbrace \sum_{i=1}^{n} a_ix^i : n \in \mathbb{N}, a_1, ..., a_n \in \mathbb{Z} \right\rbrace$ beeing an ideal of $\mathbb{Z}[x]$ with polynomials without a constant term and $J = \left\...
4
votes
6answers
71 views

Prove this polynomial falls within $\mathbb R[x]$

[ The problem below is from Yao Musheng (姚慕生), Wu Quanshui (吴泉水), Advanced Algebra (高等代数学) Ed $2$, Fudan University Press, page $207$. ] Suppose $f(x)\in \mathbb C[x]$. If $\forall c\in \mathbb R$,...
0
votes
2answers
55 views

What exactly is the purpose of the evaluation homomorphism?

I just don't understand the point of terming the evaluation of a polynomial by a map like this? And what's more, the map is going into a larger field than the field the polynomial is in anyway. What ...
4
votes
2answers
110 views

How to show $f(x)$ has no root within $\Bbb Q$

A polynomial problem from my old algebra textbook: $f(x)\in\Bbb Z[x]$ with leading coefficient $1$, $\deg f(x)\ge 1$, and both $f(0)$ and $f(1)$ are odd numbers, prove: $f(x)$ has no root within ...
1
vote
1answer
37 views

Help finishing proof with polynomial discriminant?

Prove that the discriminant of $$f(x) = x^n + nx^{n-1} + n(n-1)x^{n-2} + \cdots + n(n-1)\ldots (3)(2)x + n!$$ is $(-1)^{n(n-1)/2}(n!)^n$. So far, I let $\alpha_1,\ldots, \alpha_n$ be the roots of $f(...
1
vote
0answers
51 views

Two-term asymptotic approximation for roots of a polynomial (dominant balance)

I'm trying to find the roots to the following equation: $t^5 - \epsilon t^3 + \epsilon^3 = 0$ as $\epsilon \rightarrow 0$. From expansion $t= \epsilon^{\alpha}t_1 + \epsilon^{2\alpha}t_2 + \...
2
votes
2answers
66 views

Solve the equation -

Solve $$ 3-\frac{4}{9^x}-\frac{4}{81^x}=0 $$ I had this question for an exam today and I want to find out if my answer was correct.
0
votes
1answer
33 views

On a question about polynomial ring

Let the ring $ R$ define as the following $R=\{a_1+a_2x^2+a_3x^3+...+a_x^n;a_i\in \mathbb R,\,n\gt 2\}$ and Let the ideal $I$ generated by $<x^2+1,x^3+1>$. Is $I=R$ or not?
1
vote
1answer
16 views

quadratic form polynomial divisibility vs. matrix pointwise multiplication.

Given matrix $V',W',Y'$ is of $d\times m (d\le m)$ ; column vector $c$ is of size $m$; $r_i, i=1,...,d$ are distinct; and each row of the matrix A is $A_i=(r_i^0 ... r_i^{d-1})$. So, A is of $d\times ...
0
votes
1answer
33 views

Do Bezier control points aproximate their curve?

I was just reading here about degree elevation in Bezier curves and I noticed that in the diagrams of the progressively higher degree curve, that the control points began to approximate the curve ...
4
votes
3answers
63 views

Let $f(x) =7x^{32}+5x^{22}+3x^{12}+x^2$. Then find its remainder in the following cases.

Let $f(x) =7x^{32}+5x^{22}+3x^{12}+x^2$. (i) Then find the remainder when $f(x)$ is divided by $[x^2+1]$. (ii) Also find the remainder when $xf(x)$ is divided by $[x^2+1]$. Given both the ...
1
vote
0answers
53 views

Question about principle ideals and polynomials and quotient ring construction?

Say I have a ring of polynomials in $R[x]$. I wish to define the quotient group $R[x]/<x^2+1>$. My question lies in the ideal generated by $<x^2 + 1>$. This is the set of all numbers such ...
3
votes
3answers
64 views

Finding real coefficients of equation given that $a+ib$ is a root

Below is the question present in a past examination paper. I'll be giving my attempts and how I thought it through. Do feel free to point out any mistakes I make throughout my working even if ...
29
votes
5answers
1k views

Conjecture: Every analytic function on the closed disk is conformally a polynomial.

Here is my conjecture, any proof, counter-example, or intuitions? If $f$ is analytic on $\text{cl}(\mathbb{D})$ (that is, analytic on some open set containing $\text{cl}(\mathbb{D})$), then there is ...
1
vote
1answer
69 views

If a degree $n$ polynomial has $f(k)=k/(k+1)$ for $k=0,1,\ldots,n$, what is $f(n+1)$?

Suppose $f(x)\in K[x]$, $\deg f(x)=n$ and $f(k)=k/(k+1)$ for any $k=0,1,2,\ldots ,n$, what is $f(n+1)$? I have a vague memory that there is a very clever trick that can solve this problem easily, ...
1
vote
3answers
93 views

Distinct roots of $z^n-z$

How would we prove that for any positive integer $n$ the complex roots of $z^n-z$ are all distinct? In the case that $n=1,2,3$ I have factored it directly but how can we do it in general?
1
vote
1answer
51 views

How can one solve an equation over over a specific finite field?

How can one solve an equation of the following form where the coefficients are in $GF(2^{128})$? $Az^3 + Bz^2 + Cz + D = 0$ The operations are defined over the same field.
1
vote
1answer
55 views

Minimal polynomials

Can someone explain to me how the minimal polynomials in page 4 of this document are obtained? Please help me. http://web.ntpu.edu.tw/~yshan/BCH_code.pdf It should be something standard about ...
1
vote
1answer
75 views

finding the remainder of $(x+1)^7+x^5+(x-1)^3$ divided by $ x+2$

How can i find the remainder of $(x+1)^7+x^5+(x-1)^3$ divided by $x+2$? I tried long division but it's really messy. Also i saw that $x=0$ is a root but it still difficult.
2
votes
1answer
51 views

Factorising polynomials over $\mathbb{Z}_2$

Is there some fast way to determine whether a polynomial divides another in $\mathbb{Z}_2$? Is there some fast way to factor polynomials in $\mathbb{Z}_2$ into irreducible polynomials? Is there a ...
2
votes
1answer
47 views

Total derivative for a polynomial

I refer to Rudin's (Principles of Mathematical analysis, 3rd ed.) definition of differentiability: Suppose E is an open set in $R^n$ and f maps E into $R^m$ and $x \in E$. If there exists a linear ...
1
vote
2answers
164 views

Please find the exact equation for unknown polynomial using 50 points of (x,y)

If we have $50$ points of known $(x,y)$ provided from a certain polynomial, ( note; this polynomial is unknown and we do not know it's degree). So is there any way to find the exact equation for this ...
2
votes
1answer
41 views

Finding the maximum of sum of coefficients of a polynomial

Suppose $p(x)=ax^2+bx+c$ is a quadratic polynomial with real coefficients and $|p(x)| \leq 1$ for all values of $x$ in the range $[0,1]$. Prove that maximum possible value of $|a|+|b|+|c|$ is $17$. ...
0
votes
2answers
955 views

Finding the minimal polynomial in this field extension of $\mathbb Q$?

I have a field extension \begin{equation*} K = \mathbb Q[x]/(x^2 - 5) \end{equation*} of $\mathbb Q$, and an element $a = \bar x \in K$. I need to find the minimal polynomial of $a$ over $\mathbb Q$...
1
vote
2answers
75 views

$x^9 - 2x^7 + 1 > 0$

$x^9 - 2x^7 + 1 > 0$ Solve in real numbers. How would I do this without a graphing calculator or any graphing application? I only see a $(x-1)$ root and nothing else, can't really factor an ...
6
votes
4answers
445 views

Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$ for a 4-th degree monic polynomial

If $f(x)=x^4+ax^3+bx^2+cx+d$. Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$. This problem has troubled me a lot.The more I try to solve it,it becomes lengthier. My problem is that there are four ...
1
vote
1answer
40 views

Expression of coefficients of a product of Dirichlet polynomials

Suppose we have two Dirichlet polynomials: $$ f_1(s) = \sum_{n=1}^{m} \frac{a_n}{n^s} \\ f_2(s) = \sum_{n=1}^{m} \frac{b_n}{n^s} $$ Their product will also be a Dirichlet polynomial: $$ f_1(s)f_2(s)...
14
votes
1answer
188 views

If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
0
votes
1answer
97 views

Laguerre theorem

I'm looking for a proof of the theorem 7, page 6, of this document : http://www.nipne.ro/rjp/2013_58_9-10/1428_1435.pdf Theorem 7 (E. Laguerre) Let $f \in \mathbb{R}[x]$ be a polynomial of degree ...
3
votes
2answers
98 views

Is $x^4+nx+1$ irreducible?

Consider the polynomial $\xi= x^4+nx+1\in \mathbb Z[x]$. Show that if $n=\pm2$ then $\xi$ is reducible and that $n\neq\pm2$ implies $\xi$ is irreducible. I got the answer by writing the ...
5
votes
2answers
87 views

When can variables simply be variables?

This may seem a somewhat strange question, but I've been tying myself in knots about it recently. When constructing a polynomial ring, you must formally define a polynomial as an ordered ω-tuple, ...
2
votes
1answer
67 views

Linearly Independent Linear Transformations

I am currently studying some theories of single linear transformations. I feels like I understant 99% of it, but there is still one thing that I have not been able to resolve. My book explains it by ...
3
votes
1answer
31 views

Are there no polynomials in $\mathbb{C[x]}:f^2 − Xf = −X^2 + 1$?

Are there no polynomials in $\mathbb{C[x]}:f^2 − Xf = −X^2 + 1$? What I did: $$ f^2 − Xf = −X^2 + 1 \iff f^2=Xf-X^2+1 $$ $\deg(f)=n \rightarrow \deg(f^2)=2n$, $\deg(Xf)=n+1$ and $\deg(-X^2+1)$=2 So $...
0
votes
1answer
378 views

Find the number of roots of a polynomial using Rouche's Theorem

Use Rouche's theorem to find the number of roots of the polynomial $z^5+3z^2+1$ in the anulus $1<|z|<2$. I am looking for a solution to this problem. My thoughts: This is a topic that ...
2
votes
4answers
61 views

Finding horizontal tangents to a function.

Find the points at which the line tangent to the following function is horizontal $$q(x)=(x+3)^4(2x-1)^7$$ Every time I've gotten to the point of finding $x$ the numbers are all irrationally too ...
14
votes
3answers
1k views

Why is the zero polynomial not assigned a degree?

Yesterday, I read in my textbook, We assign degree to every polynomial and even a non-zero constant is assigned a degree $0$ but $0$ itself is not assigned a degree. Why is that? Why we don't ...
1
vote
3answers
46 views

Finding quadratic factors

Show that $(x-√3)$ and $(x+√3)$ are factors of $x^4+x^3-x^2-3x-6$. Hence write down one quadratic factor of $x^4+x^3-x^2-3x-6$, and find a second quadratic factor of this polynomial. My attempt: $f(...
5
votes
1answer
528 views

Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})[X]$

Is it possible to determine how many irreducible factors has $X^p-1$ in the polynomial ring $(\mathbb{Z}/q \mathbb{Z})[X]$ has and maybe even the degrees of the irreducible factors? (Here $p,q$ are ...
0
votes
1answer
28 views

How to find all monomials $\left\{\left.x^n\in P_m\right|T(x^n)=0\right\}$ and which are in $\text{ker }T$?

Let $P_5$ be the set of one variable polynomials with real coefficients, whose degree are $\leq5$. Let $T$ be a linear transformation $\left\{\left.T:P_m\rightarrow\mathbb{R}\right|f(x)\mapsto\int_{-1}...
5
votes
6answers
699 views

Polynomial with a prime number as a root

Is it possible to prove that this equation is false: $$ \sum_{i=0}^n a_i p^i = 0 $$ with following conditions: $a_i \in [-1;1]$; [Might $a\in\{-1,1\}$ have been intended here?] $p$ is a prime ...
2
votes
2answers
82 views

Polynomial maps on indeterminate of vector space of polynomials

I am studying polynomial rings and would like to get an idea of what it takes to study the problems of transforming polynomial forms by performing polynomial map on indeterminate of the polynomials. ...
2
votes
2answers
44 views

Find the possible values of a in the cubic equation.

Given that $(x-a)$ is a factor of $x^3-ax^2+2x^2-5x-3$, find the possible values of the constant $a$. I believe you first have to find the $a$ in the cubic equation then the other $a$ in $(x-a)$, but ...
2
votes
1answer
25 views

Let $p$ be a prime number $\ge2$ and $u = \cos\left(\frac 2p\pi\right)+i\sin\left(\frac2p\pi\right) \in \mathbb C$. Prove that …

Let $p$ be a prime number $\ge2$ and $u = \cos\left(\frac 2p\pi\right)+i\sin\left(\frac2p\pi\right) \in \mathbb C$. Prove that $u$ is root of $f(x)=x^{p-1}+x^{p}+...+x+1$. I know that $f(x)$ is ...
0
votes
2answers
72 views

Showing that an equation has a root in an interval

Show that the equation $x^4 - 7x^3 + 1 = 0$ has a root in the interval $[0,1]$. How would I go about working this out in steps?
0
votes
3answers
49 views

Find $a\in\mathbb{R}$ such that $x_{1,2,3}\in\mathbb{Z}$

Consider $a\in\mathbb{R}$ and $x^3-x+a=0$ with $x_{1,2,3}\in\mathbb{C}$. We need to find $a\in\mathbb{R}$ such that $x_{1,2,3}\in\mathbb{Z}$. It seems be equivalent with to find a such that $f=x^3-x+a$...
0
votes
1answer
179 views

finding roots when polynomial does not equal zero

I was trying to solve this polynomial $$x(3-x^2)=1$$ I worked for the term $(3-x^2)$, I thought that this term cannot be $0$, thus $$3-x^2 >0$$ $x< \sqrt{3}$, $x<-\sqrt{3}$ is rejected ...