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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On the location of the roots of a polynomial

Consider the following two polynomials \begin{align} p(s)&:=s^n+\alpha_{n-1}s^{n-1}+\cdots+\alpha_1s+\alpha_0,\\ q(s)&:=s^{n-1}+\alpha_{n-1}s^{n-2}+\cdots+\alpha_2 s+\alpha_1, \end{align} ...
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2answers
39 views

Proving a polynomial is prime in $ \mathbb{R}[t]$?

I had gotten this question on a recent assignment, and am confused on how to approach it. Would I need to use Gauss's Lemma? Prove that $t^2 + 1$ is a prime in $ \mathbb{R}[t]$. Prove that no ...
2
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1answer
64 views

Average number of linear factors in a monic polynomial of degree $n$ over $\mathbb{F}_p$

Let $p$ be a prime and $P_n$ the set of all monic polynomials with coefficients in $\mathbb{F}_p.$ I am interested in the average number of linear factors of polynomials in $P_n.$ In an exercise in ...
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2answers
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What can one conclude for a poynomial with the property that p(x)=p(ix)?

From a theorem in the theory of cyclotomic polynomials I deduced that a special polynomial $p(x)$ of even degree $n$ has the property $$p(x)=p(ix)$$ with $i$ being the complex unit. What can one ...
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Degrees of spaces of polynomials

Let $I$ be an ideal in $K[x_1,\dots,x_n]$ where $K$ is a char $0$ field. Let $Z(I)$ be a set of discrete points whose cardinality is exponential in $n$ and spanning $n$ dimensions. Let $P$ be the ...
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1answer
46 views

How do you 'rotate' a polynomial?

I have a polynomial equation: $$y=(-5 \times 10^{-6} \times x^3)+(0.0004 \times x^2)+(0.0582 \times x)-0.4397$$ Is it possible to "rotate" this polynomial curve (maintaining the shape) around the ...
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1answer
40 views

A sufficient condition to ensure a polynomial to be zero

Let $p_i(x)$, $p(x)$ be real coefficient polynomials. Suppose that $$\sum_{i=0}^{n-1}x^ip_i(x^{in})=p(x^n), (x-1)\mid p(x).$$ Show that $p_i(x)=0$, $1\leq i\leq n-1$. I could only show that ...
2
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2answers
59 views

How do I find out if a polynomial is irreducible?

I have this polynomial: $f(x)=x^4+x^3-4x^2-5x-5$. How can I find out if this polynomial is irreducible over the field $Q$ of rational numbers? I know about mod p irreducibility test but it fails in ...
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4answers
66 views

Algebraic division with two variables? $\frac{a^3 + b^3}{a+b}$

I know there's a formula for this, but I would like to know how to do algebraic division the long way - would appreciate if you can guide me along. How can I use long division for $$\frac{a^3 + ...
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1answer
41 views

how to generate rook polynomial

I've encountered rook polynomials. I just can't seem to understand how to generate them by hand for small examples such as 3x3 boards. Take for instance: $$\begin{matrix} 1 & 1 & 0 \\ 1 ...
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2answers
28 views

Prove coefficients of polynomial are elementary symmetric polynomials

I want to show that for the $k$-th elementary symmetric polynomial $s_k:=\sum_{i_1\lt\cdots\lt i_k}X_{i_1}\cdots X_{i_k}\in R[X_1,\ldots,X_n]$ a monic polynomial that factors $\prod_{i=1}^n ...
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1answer
55 views

Factor of determinant with identical row

How the following fact applies to determinants (I came across it while solving problems): Consider A is a nxn matrix, the elements of which are real (or complex) polynomials in x. If r rows of the ...
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2answers
25 views

Can all cubic/quartic polynomials be expressed in a form with only one x term?

Quadratic expressions $ax^2+bx+c$ can all be expressed in a form with only one x term: $$a(x+\frac{b}{2a})^2+c-\frac {b^2}{4a}$$ Is the same true for all cubic or quartic expressions? Is there a ...
7
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1answer
55 views

how can I find the smallest integer n such that a polynomial divides x^n-1

I have a simple question.. Assume that I have an arbitrary polynomial $f$ in $F_q[x]$. Is there a practical way to find the smallest integer $n$ for which $f$ divides $x^n-1$ ? A small example ...
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1answer
30 views

How to change the gradient of polynomial?

I have a polynomial equation: $$y=(-2 \times 10^{-10} \times x^5)+(1 \times 10^{-7} \times x^4)-(2 \times 10^{-5} \times x^3)+(0.0018 \times x^2)-(0.0156 \times x)-0.164$$ I want to be able to ...
4
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1answer
35 views

Finding Basis for a Radical of an Ideal

I am to find a basis of the following ideal: $$\sqrt{<x^5-2x^4+2x^2-x, \quad x^5-x^4-2x^3+2x^2+x-1>}$$ Truth be told, I'm not entirely confident of my solution. I will present it and then ask ...
7
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87 views

System of 4 quartic equations

\begin{align*}a &=\sqrt{4+\sqrt{5+a}},\\ b &=\sqrt{4-\sqrt{5+b}},\\ c &=\sqrt{4+\sqrt{5-c}},\\ d &=\sqrt{4-\sqrt{5-d}}.\end{align*} Compute $abcd$. I set up each as a quartic and got ...
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2answers
40 views

Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$

Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$. I'm not entirely sure how to tackle the "infinitely many elements ...
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1answer
29 views

How many polynomials in $Z_{p}[x]$ have degree n or less?

For your reference, $Z_{p}[x]$ refers to the set of all polynomials with coefficients integer mod p. To me it seems like this and the degree (power) of the two polynomials are unrelated. What ...
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0answers
35 views

Degree of the field extension for $x^4+3$

I want to find a splitting field for : $x^4+3 \in \mathbb{Q}[x]$ and the degree of that extension , I would be thankful if you verify/correct what I have tried for this problem : First I find the ...
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Give the idempotent generators of the four binary QR codes C1 , C2 , C3 , C4 , of length 7.

I'm having trouble on some homework. This is the last problem and I can't figure it out. Can anyone help or point me in the right direction? Thanks! For each code Ci , 1 ≤ i ≤ 4, from part (a), give ...
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1answer
25 views

Self-Studying Algebraic Geometry: Finding $f$ in $I(V(J))$

Problem 4.1.2 in Ideals, Varieties, and Algorithms asks: Let $J=\langle x^2+y^2-1,y-1\rangle$. Find $f \in I(V(J))$ such that $f \notin J.$ I started by trying to get an idea of what $V(J)$ is- ...
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1answer
14 views

separable polynomial

How to show that if $K$ is a field of characteristic $p$ with $p$ prime and if $f(X)\in K[X]$ is an irreducible and inseparable polynomials, therefore there exist a $d\in\mathbb N, d>0$ such ...
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1answer
34 views

Questions concerning $\mathbb Z_3[x]/(x^3+2x-1)$

Is the automorphism group of $\mathbb Z_3[x]/(x^3+2x-1)$ cyclic ? Is $\mathbb Z_3[x]/(x^3+2x-1)$ separable ?
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Abitrary derivatives of lagrange basis functions

The lagrange basis functions are given by \begin{align} \phi_k(x) =\prod_{j\not = k} \frac{x-x_j}{x_k-x_j} \end{align} I try to reproduce the numerical results of a paper. In this paper, the ...
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0answers
18 views

Does it make sense to solve polynimals using pertubation theory?

I was curisious to see if pertubation theory could be used to solve polynimals. For example \begin{align} x^5+b(\epsilon x)+1=0 \end{align} I started with the following expansion \begin{align} x = ...
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1answer
43 views

Computing Resultant of Two Polynomials

Consider these two polynomials: $$f=x^2y+3xy-1$$ $$g=6x^2+y^2-4$$ I need to compute their resultant, denoted in my textbook as $h=Res(f,g,x)$. Here's where I need help: setting up the Sylvester ...
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7 views

Generator polynomials of idempotent binary QR code

I'm doing some homework and I ran into a question that I just do not know how to do. I have to give the generator polynomials of the four binary quadratic residual codes of length 7. Quite honestly I ...
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1answer
29 views

Generator matrix of a binary cyclic code

I need to find the Generator and Parity check matrix of a binary cyclic [9,2] code. If I calculated right, the Generator polynomial is x^7 + x^6 + x^4 + x^3 + x + 1 and the check polynomial is x^2 - x ...
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4answers
57 views

Show that $p(A)\vec{v}=p(\lambda)\vec{v}$ when $A\vec{v}=\lambda\vec{v}$

Suppose $p(x)=c_0+c_1x+...+c_kx^k$. Let $\lambda \in \mathbb{R}$, $\vec{v}\in \mathbb{R}^n$, $A_{n\times n}$, such that $A\vec{v}=\lambda\vec{v}$. Show that for any $p(x)$, ...
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Finding a generator polynomial of all binary cyclic codes

I need to find the generator polynomials of all binary cyclic codes of length 7 that contain the vector (1, 1, 1, 0, 0, 1, 0). From what I know of generator polynomials of a cyclic code divides x^n - ...
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1answer
27 views

Functions in $\mathbb {R}[X] $

For the ring of polynomials over the reals, which can be considered an infinite-dimensional vector space with infinite monomial basis, is the following true: Any analytic function $f$, which is ...
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60 views

Prove $g(a) = b, g(b) = c, g(c) = a$

Let $$f(x) = x^3 - 3x^2 + 1$$ $$g(x) = 1 - \frac{1}{x}$$ Suppose $a>b>c$ are the roots of $f(x) = 0$. Show that $g(a) = b, g(b) = c, g(c) = a$. (Singapore-Cambridge GCSE A ...
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1answer
55 views

Milnor's proof of the fundamental theorem of algebra (Topology from the Differentiable Viewpoint)

I am studying the proof of the fundamental theorem of algebra out of John Milnor's book Topology from the Differentiable Viewpoint, located on page 8 here: ...
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1answer
25 views

Radical expression for roots of unity

Can somebody point out a reference to the nested radical formula of the complex roots of unity when $n = 2^N$, i.e. in solving $x^n=1$ ?
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On multilinear polynomials

Let $S=\{s_1,s_2,\dots,s_N\}$ be a sequence of points such that $s_i\in\Bbb R^n$. Let $f\in \Bbb R[x_1,\dots,x_n]$ be a multilinear polynomial. Let $S[0]\subseteq S$ and $S[1]=S\backslash S[0]$. ...
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$L^2$ inequality for derivatives of polynomials on triangles

I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let $p: \mathbb{R}^2 \to \mathbb{R}$ ...
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34 views

Is $f=t^5 + t^4 + 1$ reducible or irreducible over the field of $Z_2$ integers modulo 2.

I need to find out whether $f=t^5 + t^4 + 1$ is reducible or irreducible over the field of $Z_2$ integers modulo 2. I approached the question by substituting 0 and 1 into the function and got answers ...
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1answer
44 views

Simpson's rule over cubic splines

I'm helping a friend of mine to do her homework, but i need help understanding some results (sorry but i took numeric methods class a looooong time ago) So, the task is to fit a cubic spline over ...
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1answer
36 views

Symmetric polynomials, group of permutations

Could somebody give me a clue, related to the possible solution of the problem? Let's denote a polynomial $f(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})=x_{1} x_{2} x_{3} +x_{2} x_{3} x_{4} + x_{3} x_{4} ...
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1answer
67 views

Irreducibility over $\mathbb{C}$ of symmetric polynomials

Problem. Find all elementary symmetric polynomials that are irreducible over $\mathbb{C}$. My attempt. It's easy to see that if we have polynomial $f(x_1, \dots, x_n)$ and it can be reduced to ...
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4answers
84 views

Is $x^8+1$ irreducible?

How to decide that $f(x)=x^8+1$ is irreducible or not in the following fields: 1) $F=\mathbb R$ 2) $F=\mathbb Q$ I can't use Eisenstein's criterion. So the only possibility is computing the complex ...
2
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1answer
37 views

Most efficient way to find polynomial roots

Given a polynomial: $$z^7+10z^6+42z^5+96z^4+129z^3+102z^2+44z+8$$ find it's roots. I started off by using Horner's method (I believe one of the roots has to be $1$, so that's my starting point) but ...
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1answer
27 views

Simplifying the polynomial for integration

Hi, I am trying to find the length of the function above from $x = 1$ to $x = 2$. I applied the length formula but I was not able to simplify it past $\sqrt{x^6 + \frac 12 + \frac 1{16x^6}}$. Can ...
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41 views

How do I go with proving that the coefficient of each terms of $\prod^{k=n}_{1}{1-x^k}$ is either 1,-1 or 0?

How do I go with proving that the coefficient of each terms of $\prod^{n}_{k=1}({1-x^k})$ is either 1,-1 or 0 for n that is sufficiently large? Also, is there any pattern in terms of the 1,-1 and 0s?
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0answers
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Find all local minimums of polinomial function of two variables

I am intrigued by the task of numerically finding all local minimums of a polynomial $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ in the interval $[0,1]\times [0,1]$ ...
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1answer
18 views

Sequence of quadratic polynomials

Let $P_n$ be a sequence of Quadratic polynomials on $[0,1]$ such that $\lim_{n \rightarrow \infty}P_n(a_i) = b_i$ for $i = 1,2,3$ where $b_i$ are real numbers. Then 1) $P_n$ converges pointwise in ...
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1answer
11 views

common factors of multilinear polynomial

Say $F,G\in\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ are two multilinear polynomial. If $F$ and $G$ vanish at a common set of coordinantes $(a_{i1},a_{i2},\dots,a_{in-1},a_{in})\in\Bbb R^n$ for ...
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1answer
26 views

What is $s_3$ and $s_4$ for $x$

$\sum_{i=0}^n i^k = s_k(n)$, $s_k$ polynomial from degree $k+1$ I have already shown for $s_2(x) = \frac{x(x+1)(2x+1)}6$ How from the sum and $s_2(x)$ can be shown for $s_3(x)$ and $s_4(x)$ ...
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2answers
52 views

Finding a polynomial satisfying the equation

For $$ f: x^6+3x^4-4 \\ g: x^5-x^4+5x^3-5x^2+6x-6 $$ how do I find a polynomial $a \in \mathbb{Q}[x]_{(\deg f-\deg \gcd(f,g))}$ so that a polynomial $b \in \mathbb{Q}[x]$ exists when ...