Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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Why $x^4+x^3+x^2+x+1$ has root order 1 or 5 in $GF(2)$?

Why $x^4+x^3+x^2+x+1$ has root order 1 or 5 in $GF(2)$? For example: $P(x)=x^4+x^3+x^2+x+1$ is irreducible over $GF(2)$. Note, that $(x^5-1)=(x-1)(x^4+x^3+x^2+x+1)$. Those, any root of $P(x)=0$ has ...
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202 views

General bound on a polynomial's root with the largest norm

Is there a general bound on a polynomial's root with the largest norm? When Rouche's theorem is used, it still seems that the polynomial's root with the largest norm still needs to be found if we ...
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Solving polynomial matrix equations over finite fields

The concrete problem is this: Find triplets of distinct matrices $(A,B,C)$ of dimension $6\times 6$ over the field $\mathbb{F}_{2^2}$ such that: $A^2B=AB^2$ $C^2A=CA^2$ $B^3C=BC^3$ However, I'm ...
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If a polynomial has a rational root is it automatically reducible?

This seems obvious, but I just can't crack it. Let K be a field and $F(X) \in K[X]$ be a polynomial. Does $F(a)=0$ for some $a\in K$ imply that F(X) is reducible. Clearly, by the fundamental theorem ...
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Solution Set of a Polynomial System

Consider the polynomial System $F(x)-c=0,$ where $F:\mathbb{C}^n \rightarrow \mathbb{C}^n.$ Is it true that for almost all values of $c\in \mathbb{C}^n,$ the polynomial system will only have isolated ...
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Is $f:M_n(\mathbb{C})\longrightarrow M_n(\mathbb{C})$ continuous?

I want to know whether this is absurd question or reasonable to ask: Let $f:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be given by $f(A)= B$, where $B$ is a diagonal matrix having the same eigenvalues as ...
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Algorithms for deciding whether a function over a finite ring is polynomial or not?

Let $R$ be a finite ring, and $f$ be a function from $R$ to $R.$ Suppose I want to know whether $f$ can be represented as a polynomial or not? Are there any good algorithms for finding this out?
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Repeated roots for polynomial in $\overline{ \mathbb F_{p}}$

Let the $ \mathbb F_{p}$ denote the finite field of $\mathbb Z/ p \mathbb Z$ and $\overline{ \mathbb F_{p}}$ its algebraic closure. Now let $f(x)=X^p- b \in \overline{ \mathbb F_{p}}[x]$. I want to ...
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measure of image of a polynomial map

For $1\leq k \leq m,$ let $f_k :\mathbb{C}^n \rightarrow \mathbb{C}$ be a multivariate polynomial map. With $\mathbf{x} \equiv (x_1,\ldots,x_n)$, consider the map $F(\mathbf{x}) \equiv ...
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Can properties of a polynomial over $\mathbb{Q}$ be carried over to properties over $\mathbb{R}$?

The following question arose while trying to generalize some combinatorial statements from $\mathbb{Z}$ to $\mathbb{R}$. Suppose I have a multivariate homogenous polynomial $f$ with coefficients in ...
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Polynomial expansion of complex numbers

Given $A$ and $B$ are complex numbers. I want to request anyone who might know any formulas for expanding this following expression. $$ |A-B|^{2n}$$ where $n$ is an integer. The one that I commonly ...
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209 views

Polynomial System has only isolated solutions

How does one show that the polynomial system $F(x)=0,$ where $F:\mathbb{C}^n \rightarrow \mathbb{C}^n,$ has only isolated roots? As an example, let ...
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How to change degree of elements in a field

I'm working through an assignment and need to write several elements as polyomials of degree <= 2. ($x^2+1$)($x+1$) within the field $\mathbb Z$$_3$[x] / ($x^3 + 2x + 1$) And am unsure ...
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Did anyone ever build a mechanical device to take fifth roots, or solve general quintics?

This question is from a post from John Baez's blog on, among other things, geometrical constructions. I was hoping someone here might know the answer. In his post, Baez writes that Nowadays we ...
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127 views

How do I divide Laurent polynomials?

I have an example from a paper (listed below) that I cannot figure out. I can divide normal polynomials, but the alternative ways to divide Laurent polynomials is beyond me at the moment. The paper ...
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129 views

How to express $\frac{x^3+4x^2-1}{(x^2+1)^2}$ as a polynomial plus a proper fraction, using long division?

I'm trying to express $$\dfrac{x^3+4x^2-1}{(x^2+1)^2}$$ as a polynomial plus a proper fraction, using long division but I don't know how to do that. It'd be cool if you can solve this. Thanks.
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Can the ideal $(X_1, X_2, \dots, X_n) $ be generated by fewer polynomials over the field $K[X_1, X_2, \dots, X_n]$?

My algebra teacher asked whether the ideal $(X_1, X_2, \dots, X_n) $ can be generated by fewer polynomials over the field $K[X_1, X_2, \dots, X_n]$. My intuition tells me that it can't, so I tried to ...
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theory about polynomial, how can I resolve this exercise?

This is my first exercise on polynomal, can u explain me, step by step how can I resolve it? I'm good with theory about $Z_n$ and I know something about polynomials, but I haven't clear view and I ...
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polynomial factorisation in $\mathbb R[X]$

I read that every polynomial in $\mathbb R[X]$ factorizes in a product of linear and quadratic polynomials. Do we have a result stating that every polyonomial of degree $\geq 3$ has at least a real ...
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Construction of polynomials with non-commutative elements.

I have a simple set of polynomials which I know how to construct for each integer $n$, but I havn't been able to write them down in terms of concrete sums and products. For $n\in\mathbb N_+$, we have ...
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Finding a cubic equation and a straight line equation that intersects 3 times?

I need some leads and guidance on my homework question: Find a cubic equation of the form, $y = ax^3 + bx^2 +cx + d$ and a straight line equation $y = mx + k$ (m is non-zero) such that the ...
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About the exchange of $\sum$ and LM

Given $f_i,g_i\in k[x_1,\cdots,x_n],1\leq i\leq s$, fix a monomial order on $k[x_1,\cdots,x_n]$, I was wondering whether there is an effective criterion to judge if this ...
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Derivations in a ring. What applications do they have outside algebra?

INTRODUCTION Let $R$ be any ring, possibly without unity. We call a function $d:R\longrightarrow R$ a derivation on $R$ if it satisfies the following conditions. $(1)$ It is an endomorphism of the ...
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Which polynomial equations of higher degree will have a solution formula?

A polynomial equation of degree greater than four will in general have no solution formula. But what are some typical cases one should be aware of as a practical person in which there are solutions?
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Roots of $\frac{z^n-1}{z-1}=b$

I am trying to calculate the inverse Laplace transform of a probability distribution, and while I don't believe I can get a closed form expression, I would like to get an idea of the general shape of ...
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174 views

Polynomial and Integrability

Let $\mu$ be a probability measure on $X \subseteq \mathbb{R}$. Consider a polynomial function $p_d: \mathbb{R} \rightarrow \mathbb{R}$ of degree $d \in \mathbb{Z}^+$. I would like to know if the ...
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Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
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Why don't they teach Fundamental Theorem of Algebra in High School? [closed]

I am currently in AP Calculus BC and one more year to go, I have heard about Fundamental Theorem of Algebra several times, and with the resources that is out there today I tried to search and study ...
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What is $\frac{x-1}{x^{q/p}-1}$ equal to?

Given two positive integers $p, q$ with $p>q$ and $(p,q)=1$. Can we write $\frac{x-1}{x^{q/p}-1}$ into some thing like $x^{1-q/p}+x^{1-2q/p}+\cdots+1$? I guess the answer is no, but is there a ...
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250 views

How to find root of Polynomials?

I'm studing polynomials; I have this exercise: Find the irreducible factors of the polynomial $x^4-2x^2-3 \in \mathbb{Z}_5[x]$ I think in this way: I need to find root of the polynomial. A root ...
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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 ...
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Exercise on irreducible polynomials

I know that this is not the right place for questions like that, but I need someone that explain me step-by-step how can I resolve this exercise (I've exam in the next days): Write as products of ...
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Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$

I have a question, I think it concerns with field theory. Why the polynomial $$x^{p^n}-x+1$$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$? Thanks in advance. It bothers me for ...
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Formula for Legendre polynomials by use of Cauchy's Integral Formula (From _Visual Complex Analysis_)

I decided to look through Tristan Needham's Complex Analysis book since it's usually mentioned with great praise. Just doing some exercises, I got stuck on #4 of Chapter 9). Here $P_n(z)$ denotes the ...
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141 views

A simple question about polynomial theorem

Suppose we have a polynomial with degree $n$ and all the coefficient are integers, with the leading coefficient $+1$ or $-1$. Are the roots only either integer or complex number?
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Solving simultaneous equations with 3 parts

I keep trying to solve this problem, but i keep on getting crazy answers, i think i am right up to a certain point and then doing something wrong, the question is to solvie this : $$ \begin{align*} ...
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How to find the best (integer) polynomial equation?

This is a continuation of How to find curve equation from data? I asked earlier. I am looking for both a formula and the method to find the best (integer) polynomial that fits my data. I still ...
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Field of fractions of $R[X]$

Let $R$ be a domain and let $Q$ be its field of fractions. Show that the field of fractions of $R[X]$ is isomorphic to $Q(X)$. By the way, I don't know exactly what $Q(X)$ is. It means $Q[X]$? Or ...
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Difficulty solving exponential equation

the question is little silly, but I have difficulty in math so, how to solve this exponential equation ? $$\ 4^x - 4^1 + 2^x - 2^1 = \frac{5}{16}.$$ How do I manage to solve it ?
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Recurrence equation for central trinomial coefficients

I've come across the following exercise: Give a recurrence equation for the central coefficients $(a_n)$, where for all $n$, $a_n$ is the coefficient of $X^n$ in $(1+X+X^2)^n$. Here's what I've ...
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Minimal generating sets for homogeneous polynomial ideal in two variables

This question is (somehow) related to System of generator of a homogenous ideal Let $A$ be the ring $A={\mathbb R}[X,Y]$, and let $m \geq 1$. Let $$ {\cal S}_m=\lbrace X^m, X^{m-1}Y,X^{m-2}Y^2, ...
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An instance of Eisenstein's criterion

I am trying to prove the following assertion: Given a prime number $p\in \mathbb{Z}$ , let $$f =\sum_{i=0}^{2n+1}{a_ix^i}\in \mathbb{Z}[x]$$ which is a polynomial of odd degree. Furthermore, we ...
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Polynomial equation

Is it possible to find polynomials with rational coefficients $P(x),Q(x)$ such that $Px^3+Px^2+Qx+2Q=1$? I have trying in vain to find one by inspection, but that might just be me.
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Discontinuity phenomenon in polynomial optimization

Let $Q(x,y)$ be a polynomial in the two variables $x$ and $y$, with real coefficients. Let $a<b$; for any fixed $x$ the polynomial $Q(x,.)$ has a minimum $m(x)$ when $y$ varies in $[a,b]$. We know ...
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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 ...
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Are polynomials over a finite field generally functions from that field or…?

Take $g(X) = X^3 + X^2 - X + 1$ and let it be over $\mathbb{Z}_3$, what is usually the domain of $g$? Grazie. Edit: What I want to do is come up with an intelligent way t think of all the ...
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Polynomial interpolation of the residues of a rational function

Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
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Let K/F be a finite extension, given a polynomial in K[x] find another so that their product is in F[x]

Let $K$ be a finite extension of a field $F$, and let $f(x)$ be in $K[x]$. Prove that there is a nonzero polynomial $g(x)$ in $K[x]$ such that $f(x)g(x)$ is in $F[x]$. Should I do this by induction ...
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Verifying properties of a discrete valuation.

EDIT: Don't think about this. The problem statement is flawed. (See comments) Let $K$ be a field, and let $K(T)$ be the quotient field of polynomials over $K$. Then I define $v(f/g) = \deg(f) - ...
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Multiply polynomials under field $Z_3$

As part of my journey in understanding finite fields , I have a little problem with multiplying polynomials . Given: $(x^2+x-1)(x^2-x-1)$ , a normal multiplication would be : $(x^2+x-1)(x^2-x-1)$ = ...