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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1answer
50 views

Universal Equation For $x$

For: $ax+b=0\;;\; x= \frac{-b}{a}\;\;\;\;\;$ and for:$$ Ax^2 +bx+c = 0\;; \;x = \frac{-b\pm\sqrt {b^2-4ac}}{2a}$$ And for $$ Ax^3+bx^2+cx+d =0$$ Is there a constant transformation from equations ...
1
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1answer
17 views

Finding the square root of a ring element modulo a polynomial

Suppose I am working with polynomials in $\mathbb{Z}_5$. Let $P(x)$ be some irreducible quadratic. We know that the remainders modulo $P(x)$ will form a ring of remainders. Now suppose I wanted to ...
0
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0answers
18 views

Question about $F(x,y)=m$

Let $F(x,y)$ be a homogeneous polynomial of degree $\ge3$ with mutually prime coefficients, then we consider the problem $$F(x,y)=m\tag1$$ such that $m$ is an integer, we set $f(x):=F(x,1)$ then ...
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2answers
45 views

Power of a polynomial in Galois field

Let $f(x) \in GF[q](X)$, where $q = p^m$ and $p$ prime. Is the following true? $$f^{p^m}(X) = f(X^{p^m}).$$ I tried to prove the assertion above and got stuck at the following: $$ \begin{align} ...
-2
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1answer
22 views

Real polynomials from repunits to repunits ( Putnam 2007 A4) [closed]

Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$
9
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1answer
66 views

Is it possible to study the properties of sequences by studying the family of polynomials generated with the elements as coefficients?

Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows: $p_0 = a_0$ $p_1 = a_0x+a_1$ $p_2 = a_0x^2+a_1x+a_2$ $p_n = ...
4
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1answer
35 views

The biggest possible degree of a polynomial given a condition

Let $P(x) \in R[x]$ be a polynomial with real coefficients such that $$(\forall n \in \mathbb N)(\exists q \in \mathbb Q)(P(q)=n)$$ What's the biggest possible value of $\deg P$? ($\deg P$ is the ...
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1answer
28 views

Interesting 4th order factoring question

$$ A = \frac{(4\cdot2^4 + 1)(4\cdot4^4 + 1)(4\cdot6^4 + 1)}{(4\cdot1^4 + 1)(4\cdot3^4 + 1)(4\cdot7^4 + 1)}$$ What is the value of $ \dfrac{113A}{61}$ ? So i tried factoring this ...
6
votes
1answer
87 views

How to tell if a system of polynomial equations has no real solutions

I have a system of $3n + 3$ polynomial equations in $6n$ variables, where $n$ is probably going to be less than about $5$. I can compute its Groebner basis and I see that it does not contain $\{1\}$, ...
0
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2answers
41 views

How to factor a third degree polynomial once you know one root?

Suppose $p(x) = 9x^3 - 30x^{2} + 29x - 8 $. If we wish to solve $p(x) = 0$, then we can observe that $x=1$ is a root of $p$. Then, we can write $(x-1)( \dots ) = 0$. How does one find the expression ...
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0answers
27 views

Polynomial Path [duplicate]

Let $x(t)$ and $y(t)$ be real polynomials in $t$. Show that there is always a polynomial relation $f(x,y)=0$. This question is taken from Artin, Algebra, Chapter 3 Vector Spaces. I have no idea how ...
1
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1answer
29 views

Polynomial problem involving divisibility, prime numbers, monotony

Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio ...
2
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0answers
40 views

Monotonic roots

Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
0
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3answers
31 views

find a and b using the information given

I have been presented with the following question : The polynomial $$f(x) = x^3 - 2x^2 +ax + b$$ satisfies the following : a) It is divisible (x-1) b) it leaves a remainder of -24 when divided ...
3
votes
1answer
41 views

Solution to a simple system of quadratic equations

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations: $$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$ for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial ...
1
vote
1answer
39 views

Why is $\mathbb{F}_5[x]$ a Jacobson ring? [closed]

As the question title suggests, why is $\mathbb{F}_5[x]$ a Jacobson ring?
1
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1answer
34 views

Polynomial having all integral coefficients $P_n(a)=b$ and $P_n(b)=c$ and $P_n(c)=a$

Let $a,b,$ and $c$ denote three distinct integers, and let $P_n$ a polynomial having all integral coefficients. Show that it is impossible that $P_n(a)=b$ and $P_n(b)=c$ and $P_n(c)=a$. I started ...
0
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1answer
41 views

Exercise: Evaluate a polynomial function such as $P(x)=2x^3-3x^2+7x-2$ at a surd such as $x=1+2\sqrt{3}$.

Exercise: Given polynomial function $P(x)=2x^3-3x^2+7x-2$ evaluate $P(x)$ at the surd $x=1+2\sqrt{3}$.
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0answers
54 views

What does the irreducible polynomial over $\mathbb{Q}$ and $\mathbb{Q}(i)$ mean?

So here'as the problem Find the monic irreducible polynomial $g(x) \in \mathbb{Q}(x)$ for $i+ \sqrt{3}$ over $\mathbb{Q}(i)$ Erm...huh? Okay, so a minimal polynomial $m$ is what I need, and ...
0
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3answers
86 views

If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, prove that $x^{16}>15$.

"If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, prove that $x^{16}>15$." The above problem came on a local question paper. I tried to solve it by factorizing and sum of G.P. , But I was unable to ...
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3answers
40 views

Is it true that a polynomial is reducible over a field only if the polynomial has a zero in the field?

I am doing some practice problems for abstract algebra and have come across this idea in a couple places, but it seems fundamentally wrong. For example, in order to prove that $f(x) = x^2 + x + 1$ is ...
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2answers
22 views

The polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over the field $K$ has no zero divisors

Show that the polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over a field $K$ has no zero divisors (except the zero polynomial). When revising some Linear Algebra topics, I got stuck with this ...
1
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0answers
40 views

Image of element not square of any element, maximal ideal, field is quadratic extension?

This is a followup to my question here. Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us ...
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votes
2answers
33 views

Basis of polynomial [closed]

I tried to do this but no result. Can anyone please explain me and make me understand this) Let $a \in \Bbb R - \{0\}$, and consider the family of polynomials $$B_a=\{x^2,\ (x - a)^2,\ x^2(x - a),\ ...
3
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0answers
39 views

Bijective mapping between face polytopes of permutohedra and partitions of integers

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
1
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1answer
27 views

Solving a multivariate polynomial system involving the power sums

I would like to know if there is a way to solve or simplify the system of equations given by: $$ x_1^1+x_2^1+\cdots x_n^1 = c_1\\ x_1^2+x_2^2+\cdots x_n^2 = c_2\\ \vdots\\ x_1^n+x_2^n+\cdots x_n^n = ...
0
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0answers
20 views

prove that specific polynom is a sum of squares of polynoms [duplicate]

Given polynomial P(x) with real coefficients and condition: $P(x) \ge 0 $ for every x, prove that P(x) can be represented as sum of squares of polynomials with real coefficients. I understand, that ...
0
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1answer
19 views

Residue at infinity of $zp'(z)/p(z)$

Suppose a polynomial $p\in\mathbb{C}[x]$ of degree $m$ with complex roots $b_1,\ldots,b_m$. Then $$ f(z):=z\frac{p'(z)}{p(z)}=\frac{z}{z-b_1}+\ldots+\frac{z}{z-b_m}.$$ I want to compute ...
0
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2answers
48 views

Show that the set of polynomials with 1 as a root form a linear subspace

Let $\mathbb{C}(x)$ be the vector space $\mathbb{C}$ of polynomials $p\left(x\right)$ in one variable $x$ with coefficients in $\mathbb{C}$. Is the set $p(x) \in \mathbb{C}\left(x\right)$ such that ...
-3
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0answers
18 views

How to find the coordinate matrix?

I want the coordinate matrix of the conjugated (T*)linear transformation in the R2 real polynomial spaces with scalar product and T linear transformation. The matrix must be in standard basis of ...
5
votes
1answer
81 views

Putnam 1985 B-1 Polynomial Problem

Problem: Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial $$p(x)=(x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)$$ has exactly ...
0
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0answers
11 views

Product of heights of factors smaller than length of a polynomial with integer coefficients

I have the following question. Given a (univariate) polynomial with integer coefficients, I want to prove, if true, that the product of heights of its (irreducible) factors is smaller or equal to its ...
3
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0answers
49 views

The GCD of a Univariate Integer-Valued Polynomial over a Set

Let $\mathcal{I}[X]$ denote the subring of $\mathbb{Q}[X]$ consisting of all integer-valued polynomials (i.e., $f(X)\in \mathbb{Q}[X]$ such that $f(k)\in\mathbb{Z}$ for all $k\in\mathbb{Z}$). For ...
3
votes
2answers
35 views

The content of a polynomial vs the ideal of its values

Let $f(x) = \sum_i a_i x^i$ be a degree $d$ polynomial over some ring $A$. Define the content of $f$ to be the ideal: $$c(f) = (a_0,\dots,a_d).$$ One can ask for the relation of the above ideal to the ...
4
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1answer
57 views

Does $c(f) = \gcd(\{ f(n) | n \in \mathbb{Z} \})$?

Consider $\sum_{i = 0}^n a_i x^i \in \mathbb{Z}[x]$. Recall that the content of a polynomial is the gcd of its coefficients. I'm wondering whether the content is equal to $\gcd ( \{ \sum_{i = 0}^n a_i ...
3
votes
2answers
215 views

a general continued fraction satisfying $\frac{(i+\Theta\sqrt{z})^m}{(i-\Theta\sqrt{z})^m}=\frac{(ik+\sqrt{z})^{m+1}}{(ik-\sqrt{z})^{m+1}}$

Given any natural number $m\gt2$, let $z$,$k$ be complex numbers, where $i=\sqrt{-1}$ and consider the general continued fraction $$\Theta(k,z,m)=\cfrac{(m+1)}{km+\cfrac{z(0m-1)(2m+1)} ...
2
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1answer
36 views

When you divide the polynomial $A(x)$ by $(x-1)(x+2)$, what remainder will you end up with?

When you divide the polynomial $A(x)$ by $x-1$, you get a remainder of $10$. When you divide $A(x)$ by $x+2$ you get remainder $0$. When you divide $A(x)$ by $(x-1)(x+2)$ what remainder will you end ...
2
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0answers
34 views

A function that is locally a quotient of polynomials but not globally [duplicate]

Let $X =\{ x_1x_4=x_2x_3\;, (x_2,x_4) \neq (0,0)\} \subset \mathbb{C^4}$, i.e. not both of $x_2,x_4$ are zero. Define a function $\phi$ on $X$ by $\phi(x)=\left\{\begin{matrix} \frac{x_1}{x_2} ...
1
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2answers
23 views

Are the “weights” inside a neural network actually “terms” for a polynomial?

This just hit me today. I am not too experienced with math or neural networks, but I am trying to find out about them in my own way so I can some day understand them well. So I was thinking about how ...
0
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0answers
59 views

Don't understand what the roots of $t^6-8$ are

I am ultimately asked to find the splitting field of the said polynomial over $\mathbb{Q}$. So I must find the roots first and honestly $t^6-8=(t^2-2)(t^4+2t^2+4)$ So at least two of my solutions ...
2
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0answers
20 views

Mathematical development with Polynom modulo n

I have to implement a method seen in an article, and I'm stuck with some mathematical development. The article is on iEEE Xplore, so I'll try to be as specific as I can. It's about pairing-based ...
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0answers
32 views

Number of ways to represent a number as a sum of K numbers in subset S

Let the set $S = \{ 1 , 2 , 4 , 5 , 10\}$ Now I want to find the number of ways to represent $x$ as sum of $k$ numbers of the set $S$ (a number can be included any number of times). If $x = 10$ and ...
0
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2answers
34 views

Zeroes of the polynomial $f(x)$ over the field $F$ of order 256.

Let $F$ be a field with 256 elements and $f \in F[x]$be a polynomial with all roots in $F$. Then (1) $f \neq x^{15} -1$. (2) $f \neq x^{63} - 1$ (3) $f \neq x^2 + x + 1$ (4) if $f$ ...
2
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1answer
40 views

$f(x)$ is a quadratic polynomial with leading coefficient $1$, $|f(x)| \leq 8 \: \forall \: x \in [1,9]$ find $f(x)$

$f(x)$ is a polynomial of the form ($b,c$ are real numbers) $$f(x) = x^2+bx+c$$ such that $|f(x)| \leq 8 \: \forall \: x \in [1,9]$. Find all $f(x)$ satisfying the given condition. I found ...
1
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1answer
46 views

Degree of the field extension

I need to determine the degree of the field extension $\mathbb{Q}(\sqrt{(2+\sqrt{2})(3+\sqrt{3}}))/\mathbb{Q}$. I've determined that the minimal polynomial of $\sqrt{(2+\sqrt{2})(3+\sqrt{3}})$ is ...
1
vote
3answers
81 views

Congruence $16^{(x^ 2+x+1)} \equiv 4 \mod 11$

Given the congruence $16^{x^2+x+1}≡ 4 \mod 11$ I'm not necessarily sure how to approach this problem if someone can help me head in the right direction since 16 is not a primitive root of mod 11 I ...
0
votes
1answer
36 views

Algebraic Long-Division, where the divifing $x$ has a co-efficient

I know how to do algebraic long division when dividing by $(x+a)$ however when a co-efficient is added to the $x$ what do you do? e.g $\frac{x^{3}-4x^{2}+12}{3x-4}$ I recall you having to times by ...
2
votes
1answer
21 views

real affine varieties are hypersurfaces

In $\mathbb{R}^n$, let X be a Zariski-closed set. then $X=\mathbb{V}(f)$ for some polynomial $f$. Elementary formulation: let $X \subset \mathbb{R}^n$ be the set of common zeroes of some ...
0
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1answer
8 views

Given this discrete non-linear set of values how can I get an equation for it?

I want to generate a equation in the form f(x) = {...} for this discrete data below. As X doubles Y halves but its a bit more complicated. Using an online Polynomial Interpolation calculator I got: ...
0
votes
1answer
69 views

Approximation of a polynomial with fractional power

I have a polynomial I need to find the roots of, the major difficulty is that this polynomial has fractional exponents. I have made an approximation and I would like to have some idea of the error I ...