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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Divisibility in $\mathbb F_p[x]$

For any prime $p$ and nonzero $a\in \mathbb{F}_p$ prove that for all positive integers $n$, $x^{p^n}-x+na$ is divisible by $x^p-x+a$ in $\mathbb{F}_p[x]$. Can anyone give me some hints to proceed?
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2answers
60 views

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$ I tried to to solve it from the right side: ...
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2answers
129 views

Bernstein polynomial looks like this: $B_i^n={{n}\choose{i}}x^i(1-x)^{n-i}$.Find it's $r$'th derivative.

Bernstein polynomials are defined like this $B_i^n={{n}\choose{i}}x^i(1-x)^{n-i}$.I need to prove that $r$'th derivative of it is equal to: ...
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1answer
73 views

Show a polynomial is irreducible

I'm working through the proof of Hasse's theorem and I think I need to show that the polynomial $x^4 - 2ax^2 - 8bx + a^2$ is irreducible over $\mathbb{F}_p$, where $a$, $b$ are integers and $p$ is ...
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2answers
49 views

Frobenius maps exist/do not exist for integers?

Does there exist an infinite ring $R$ such that $(x+y)^b=x^b+y^b$, and similarly for $2$ other odd primes $a,c$; in which $\Bbb{Z}$ can be embedded as a ring? I have no idea where to begin. Maybe ...
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1answer
39 views

Does there exist a infinite ring in which there are $3$ Frobenius homs?

Does there exist an infinite ring $R$ such that $(x + y)^b = x^b + y^b$, and similarly for $2$ other odd primes $a,c$? Or what's the best that can be done?
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1answer
94 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 $x^7-12$ ...
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2answers
44 views

Expansion Coefficient needed

This is probably something very easy, but wth... my mind is totally stuck right now. I need to find the coefficient of $x^{11}$ of the expansion $(x^2 + 2\frac yx)^{10}$ Well I know that the answer ...
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2answers
104 views

Definition of residue in context of the polynomial ring of $\mathbb{Z}[x]/(f)$ where $f = x^4+x^3+x^2+x+1$

I'm doing Artin 11.5.1 and I got stuck on the definition of a residue. More specifically, the question asks: Let $f = x^4+x^3+x^2+x+1$ and let $\alpha$ be the residue of x in ...
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1answer
266 views

Linear Algebra: Finding the matrix representation with respect to standard basis

I would appreciate some help with a linear algebra practice question, I'm studying for my final and I am stuck, this is a screenshot of the question: Are my answers correct? a) $P_{2}$: $ ...
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3answers
195 views

Strategy for dividing multivariable polynomials leaving no remainder. Decide whether polynomial $f $ lies in ideal $I=\langle f_1, …, f_n\rangle$.

I'm looking for some general strategy to divide polynomials leaving no remainder after division using the canonical multivariable polynomial division algorithm where we divide some polynomial $f$ by a ...
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1answer
26 views

Simultaneous irreducibility of minimal polynomials

Let $F$ be a field. Let $u,v$ be elements in an algebraic extension of $F$ with minimal polynomials $f$ and $g$ respectively. Prove that $g$ is irreducible over $F(u)$ if and only if $f$ is ...
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2answers
83 views

How can I show that there are only finitely many solutions for the following system?

$x^2+yz=x$ $y^2+zx=y$ $z^2+xy=z$ I could not do anything to find the solutions. Please give some hints.
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5answers
102 views

system of equations solving for positive $a,b,c$

i need help i need to find positive number $a,b,c$ solving this system of equations? $$(1-a)(1-b)(1-c)=abc$$ $$a+b+c=1$$ I found that $0<a,b,c<1$ and I try to solve it by try $(1-a)=a$, ...
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3answers
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Simultaneous solution(s) to $a^2+4b^2+4ab=0$ and $a^2+4b^2+32+16a-8b=0$?

Could you tell me just how should I solve this system: $$ a^2+4b^2+4ab=0\\ a^2+4b^2+32+16a-8b=0 $$ I can't remember the proceeding and it's driving me crazy. Thanks a lot
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1answer
44 views

coefficients of product of polynomials proof

If $f(x)=\sum_{i=0}^{n}a_{i}x^{i}$ and $g(x)=\sum_{i=0}^{m}b_{i}x^{i}$ are two polynomials of degree n and m respectively their product is defined by $f(x)+g(x)=\sum_{i=0}^{m+n}c_{i}x^{i}$ where ...
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4answers
243 views

How do I divide a polynomial of a very high degree by a polynomial of degree $2$?

I'm preparing for an entrance exam and got stuck on a question. Let $f(x)$ be a polynomial of degree greater than $1$. If $f(x)$ is divided by $x-a$, then $f(a)$ is the remainder. Q1) Let $f(x) ...
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1answer
106 views

FFT and Convolution

Suppose we are asked to find convolution of two polynomials, actually not necessarily convolution it is product of two polynomials. Is there two different formula for multiplication via FFT? One ...
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1answer
40 views

What more can we say about $\mathbb{Z}[x]/(x+1)$?

Related to this question here Adjoining elements to $\mathbb{Z}$ given a set of generators: I want to determine the structure of $R'$ obtained by adjoining $\alpha$ to $\mathbb{Z}$ with generators ...
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2answers
35 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: ...
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2answers
235 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 ...
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0answers
81 views

This might be interesting. The ring $R[x_1, \dots]$ (Non-specific)

Let $R$ be a commutative ring and define $\mathcal{R} = R \oplus \bigoplus_{i=1}^{\infty} x_i R[x_1, \dots, x_i]$. Then $\mathcal{R}$ is an $R$-algebra of polynomials in any finite number of ...
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2answers
101 views

Is it a Fermat polynomial?

A Fermat polynomial is a polynomial which can be written as the sum of squares of two polynomials with integer coefficients. Let $f(x)$ be a Fermat polynomial such that $f(0)=1000$. Prove that ...
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1answer
40 views

Generation of “random” multilinear polynomials for testing non-negativity algorithm

Multilinear polynomial is a multivariate polynomial where the exponent is zero or one. My instructor suggests to test my non-negativity algorithm with ...
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1answer
37 views

Factors of $x^n+1$ over $\mathbb{Z}[x]$

Is there any equivalent to $x^n-1 = \prod\limits_{d|n} \phi_d$ where $\phi_d$ is the $d$th cyclotomic polynomial but for $x^n+1$? Even better, can we generalize any further?
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1answer
55 views

Symmetric and anti-symmetric systems of polynomial equations

Can I present a set of polynomial equations as a sum of symmetric and anti-symmetric systems?
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1answer
176 views

Adjoining elements to $\mathbb{Z}$ given a set of generators

I looked this up on here, but I couldn't find anything that explained it clearly enough for me. I'm doing problems in Artin, in particular 11.5.4, which asks: Determine the structure of $R'$ ...
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1answer
151 views

Proving set of polynomials of degree less than $n$ is closed

$P_n$ is a subspace of $C[0,1]$ where the norm is defined as $\|f-g\| = \sup |f-g|$ where $x$ is restricted to $[0,1]$. In addition the coefficients are reals restricted to the domain $[0,1]$. How ...
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2answers
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Finding roots of the fourth degree polynomial: $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$.

My son is taking algebra and I'm a little rusty. Not using a calculator or the internet, how would you find the roots of $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. Please list step by step. Thanks, Brian
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2answers
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Finding the discriminant and roots of a polynomial

How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here $\Delta$ is the ...
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2answers
762 views

Intersection points of two polynomials

How to prove that two distinct polynomial functions of degree m and n, respectively,the graphs intersect in at most $max(m,n)$ points.
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1answer
131 views

Proof polynomial is always divisible by number

Given $f(x) \in \mathbb{Z} [x] $ a polinomyal, that evaluated in any $a \in \mathbb{N} $, results allways in a multiple of 101 or a multiple of 107 (both prime numbers). Prove then, that $f(x)$ it's ...
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0answers
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How to decide whether a multivariable polynomial lies in an ideal before doing division

How to decide whether multivariable polynomial $f$ lies in ideal $I = \langle f_1, f_2 \rangle$ before doing division to find $a_1, a_2 \in k[X,Y]$? I've solved an exercise stated like this: ...
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1answer
99 views

conditions for the existence of complex roots:

find the necessary conditions under which the following polynomial will have non-real roots: $P(x)=Ax^3+Bx^2+x-D$ where $A>0$ and $D>0$. well if it has a+ib and a-ib as conjugate root then the ...
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2answers
118 views

Prove that a polynomial is irreducible over $\mathbb Z$

How do you prove that a polynomial is irreducible, and in this case that $(x - a)(x - b) - 1$ is irreducible over $\mathbb Z$? EDIT: $a≠b$
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1answer
400 views

Expansion of polynomial raised to high power

Is there an easy way to expand something like (x + x^2 + x^3)^6 ? Thanks in advance!
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1answer
60 views

isomorphic quotient rings of polynomial ring and Hilbert functions

Let $k$ be a field, $R=k[x_1,\cdots,x_n]$ and $I,J$ homogeneous ideals of $R$. Denote by $H_I(s), H_J(s)$ the Hilbert functions of $I,J$ respectively. If $R/I, R/J$ are isomorphic as graded rings, ...
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1answer
162 views

Computable Criteria to check whether a given basis is a Gröbner Basis

In an upcoming exam we have to do Gröbnber-Basis computation with Buchberger's algorithm. A typical example looks like this: $$ \langle f_1,f_2 \rangle $$ Then I compute the S-Polynomial ...
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4answers
473 views

First examples in Galois theory

I'm studying Field Theory and after studying theorems and problems about extensions, splitting fields, etc... I'm starting with the first theorems of the Galois Theory itself. In order to see if I ...
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2answers
371 views

Discriminant of a trinomial

Let $b,c \in \mathbb{Z} $ and let $n \in \mathbb{N} $, $n \ge 2. $ Let $f(x) = x^{n} -bx+c$. Prove that $$\hbox{disc} (f(x)) = n^{n }c^{ n-1}-(n-1)^{n-1 }b^{n }.$$ Here $\hbox{disc} (f(x)) = ...
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1answer
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Inversion in factor rings

I have this polynomials: $f = x^{4} + 3x^{3} + x^2 + 3 \in \mathbb{Z}_{5}[x]$, $g = x + 2 \in \mathbb{Z}_{5}[x]$ Does g + (f) have inversion in ring $(\mathbb{Z}_{5}[x]/(f),+,.)$ ? I should found ...
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2answers
798 views

Use a given zero to write P(x) as a product of linear and irreducible quadratic factors

The polynomial in question is: $x^4 - 8x^3 - 19x^2 + 288x - 612$ and the zero is $4 - i$. What I don't understand is how to go from the given zero to factorizing, especially as it's imaginary. ...
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3answers
95 views

Additional solutions to quadratic equations which don't match the formula answer.

I'm hoping for link to some resource which can explain why the following is true. $$ x^2 + 104x - 896 = 0 $$ Using the quadratic formula we pull a = 1, b = 104, c = 896. Putting that into the ...
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1answer
155 views

What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
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0answers
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How find this polynomial such $(f,f'(x))=h(x)$,but $f(x)$ the geometric and algebraic multiplicities is one

Let $h(x)$ be a polynomial of non-zero degree over a field $F$. I am wondering whether there must always exist a Frobenius Polynomial $f \in F[x]$ satisfying the following: ...
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5answers
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Is there a polynomial, in terms of $x^4$ and $x$, whose graph is not below the graph of the function $y=x^3$

I tried several coefficients before $x^4$ and $x$, but I didn't get it unless adding a constant term.
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Question about zero-divisors , rings and polynomials.

Let $i,n,m$ be positive integers. For every nonnegative integer $k<i+1$ , let $a_k$ be elements of a ring $A$ that satisfies : $1)$ The ring $A$ is isomorphic to ${\Bbb R}^{n}\times{\Bbb ...
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2answers
49 views

Polynomials and roots?

The degree of all roots have to add up to the degree of the polynomial. Intuitively this makes sense, but could someone formally explain why?
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77 views

Is the ideal generated by $X^4+1$ maximal in $\mathbb{R}[X]$?

I've tried to check whether $\mathbb R[X]/(X^{4}+1)$ is a field, and because $\mathbb R[X]$ is an euclidian ring I just have to prove that $X^{4}+1$ is irreducible over $\mathbb{R}$. I think this is ...
2
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1answer
88 views

Sylvester matrix and GCD degree

How to prove that the degree of a $\gcd$ of two polynomials is equal to the dimension of the null space of the Sylvester matrix? I know that any linear combination of the rows of $S(u,v)$ is a linear ...