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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Is my understanding of this corollary correct?

The following is a theorem/corollary pair in an introductory abstract algebra course. Theorem: $f(x)\equiv g(x) $ mod $p(x)$ if and only if $[f(x)]=[g(x)]$, where $[h(x)]=h(x)$ mod $p(x)$. ...
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2answers
64 views

polynomials root finding [closed]

Is every root of a polynomial of positive integer degree n, and with a rational coefficients is considered algebraic number? and how one can find some roots to this polynomial ...
0
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0answers
77 views

Centripetal Catmull–Rom spline

What is "t" in this short and simple example below? There are 4 points Pn[xn,yn] in 2D space: A[1,6] B[3,1] ...
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0answers
8 views

Representation of polynomial order in CFD codes

I currently working on a CFD code over a cubic grid. Now, the number of elements used in the simulation is decomposed among the number of processors. Each of those processors (a section of the cube) ...
2
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1answer
104 views

Irreducible Polynomials over Finite Fields [closed]

How would I show that $p(x)=x^5+x^2+1$ is an irreducible polynomial over $\Bbb Z_2=\{0,1\}$.
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0answers
25 views

Explain why $I$ is a function from $P$ to $P$ and determine whether it is one-to-one and onto.

The question and the solution are:( uploaded a photo so it is easier to see the formulas) So I am confused about the formula of p(x). P is the set of polynomial of x. OK, but why it makes p(x) = ...
1
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1answer
32 views

Regularity of a quotient ring of the polynomial ring in three indeterminates

Let $I=(f)$ be a prime ideal in $R=\mathbb{C}[x,y,z]$, so $f$ is an irreducible polynomial, and further assume that $f$ is of the following form: $f=z^n+c_{n-1}z^{n-1}+\ldots+c_1z+c_0$, where ...
2
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1answer
42 views

Constant term of noncommutative $(X+Y+(XY)^{-1})^n$

As the title reads I am trying to find a formula for the constant term of the above noncommutative polynomal expression, $$[1](X+Y+(XY)^{-1})^{3n}\quad \bigg(\in \mathbb{C}\langle X^{\pm 1},Y^{\pm ...
3
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4answers
45 views

Simplifying quartic complex function in terms of $\cos nx$

$$z= \cos(x)+i\sin(x)\\ 3z^4 -z^3+2z^2-z+3$$ How would you simplify this in terms of $\cos(nx)$?
12
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1answer
173 views

Polynomials $f$ with integer coefficients such that $f(x) \geq 0$ on $[-2,2]$ and $f(x) \leq \frac{1}{1+x}$ on $(-1,2]$

Find polynomials with integer coefficients $f\in\mathbb{Z}[x]$ such that $f(x)\ge 0$ on $x\in[-2,2]$ and $\frac{1}{1+x}\ge f(x)$ on $x\in(-1,2]$. I guess only such polynomial is just $0$, but it ...
0
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1answer
25 views

minimal value polynomials with integer coefficients

Let $D$ be the set of polynomials of integer coefficients $f\in\mathbb{Z}[x]$ such that $f(x)\ge 0$ at $x\in[-2,2]$, where the zero polynomial $f=0$ is excluded. Can I find a finite "minimal" set ...
0
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0answers
20 views

Barbeau's Polynomials: Quadratic Polynomials, 1.2.2

I've verified $(a)$ by expanding the $RHS$. I've partially verified $(b)$ doing the following: $$\begin{eqnarray*} ...
0
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3answers
66 views

Having the roots of a polynomial, is it possible to go back and find a polynomial that have exactly these roots?

This might be very silly. But I've been wondering if it's possible to assume $n$ numbers as roots of $p(x)$ and find a polynomial that have these roots. I've made a table with some polynomials ...
1
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1answer
37 views

Interpretation of summations in regards to combinatorics

I've been studying for a final in combinatorics and ran into 3 different summations that have me stumped. 1) interpret the equation in terms of counting words. (Hint: $e^a$$e^b$$e^c$) $$e^{3x} = ...
1
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3answers
52 views

How to find the remainder of polynomial division?

Im trying to solve this problem but I do not understand what the question is asking: Let $n\ge 2$ be an integer and $ p_n(x) $ be the polynomial: $$ p_n(x) = (x-1)+(x-2)+\cdots+(x-n) $$ What is the ...
1
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4answers
147 views

Show a polynomial is irreducible mod 29

Is there an easy way to see that the polynomial $x^2 + 3x + 10$ is irreducible modulo 29 without having to go through each element 0,1,..,28 and check for roots?
4
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1answer
44 views

Trigonometric root of a polynomial

If $4\cos^2 \left(\dfrac{k\pi}{j}\right)$ is the greatest root of the equation $$x^3-7x^2+14x-7=0$$ where $\gcd(k,j)=1$ Evaluate $k+j$ I tried factorizing the equation but it wasn't ...
0
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1answer
20 views

Find the coordinate matrix of a polynomial with respect to a non-standard basis

I'm stuck on this question here: Find the coordinate matrix of $2-4x-3x^2$ with respect to $B = {2, x^2-1, 1-2x-x^2}$ I did the following: $a(2) + b(x^2 - 1) + c(1-2x-x^2) = 2-4x-3x^2$ But now I'm ...
3
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3answers
110 views

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. My attempt: Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$ As $f\mid g, g$ all the roots of $f$ are roots of $g$. This means I ...
3
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1answer
54 views

Working in $\mathbb Q[x]$. Two polynomials are coprime if their gcd is a constant?

When are two polynomials coprime? Is it when their gcd is a constant? If we divide $x^3-7x-5$ by $x-4$, we get: $$x^3-7x-5=(x-4)(x^2+4x+9)+31$$ So, is $31$ their gcd, but since $31$ is not monic ...
0
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3answers
82 views

What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots…

I found this question in a maths-group in Facebook- What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots........ I think we do not count repeated roots ...
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3answers
39 views

Factor this polynomial into linear factors with coefficients in $F = \mathbb{Q}(2^{1/3}, i\sqrt{3})$

The polynomial is this: $x^3 -2$ Okay, so first I can create my field extension. I can easily extend the field to $2^{1/3}$. And I know the elements of the extension of $\mathbb{Q}(2^{1/3})$ can be ...
9
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2answers
424 views

Application of Taylor's Theorem in Number Theory

I'm working through Alan Baker's book A Concise Introduction to the Theory of Numbers, and there's an assertion in there that confuses me. Here's the quote: It is easily seen that no polynomial ...
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1answer
26 views

a question about field theory and polynomials

Hello all I was given this question in my field theory class on which I would certainly appreciate the help: I am given a field F of characteristic p ($ ch(F) > 0 $) and this polynomial $ f(x) = ...
2
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1answer
33 views

How do I derive this polygonal function from sample values?

I have 4 parameters with 16 sample data points each. When I plot them, I get this: The curves lead me to suspect that all these of 64 data point are derived from one polygonal function with 4 ...
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1answer
32 views

Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$.

Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$. I tried this problem two ways, substituting $x=1,-1$ in $f(x)$ to find the remainder, and by long ...
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0answers
13 views

Show that $G$ is a Groebner bases of $I$ if division of $f$ on $G$ is zero for all $f\in I$.

Let $I=\langle g_1,g_2,\dots, g_t\rangle$ be an ideal in $k[x_1,\dots,x_n]$ with $k$ a field. Let $G=\{g_1,\dots,g_t\}$ be a bases for $I$. Show that if the remainder of $f$ on division by $G$ is $0$ ...
0
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1answer
14 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
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2answers
30 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 $f(6)$?
4
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1answer
65 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
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1answer
29 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 = ...
2
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0answers
19 views

Find eigenvalues of a matrix using Perron–Frobenius theorem

I have to find the largest eigenvalue of a matrix containing only positive entries: $$\left( \begin{array}{ccc} e^{a} & 1 & e^{-a} \\ 1 & 1 & 1 \\ e^{-a} & 1 & e^{a} ...
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1answer
24 views

What is Horner's Method of synthetic division… [closed]

What is Horner's method of synthetic division. Is there is any proof of this method??? How does it works?
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0answers
21 views

How do I evaluate this expression? Product inside sum.

Let $a_0,a_1,\ldots,a_n,b_0,b_1,\ldots,b_n \in \Bbb{C}, n\geq 1$ such that $a_i\neq a_j \forall i\neq j$. Prove that the polynomial: $$ f=\sum_{k=0}^n b_k \left(\prod_{\substack{0\leq j \leq ...
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2answers
25 views

What does $p(x)=p(1-x)$ in $P=\{p(x)\in {{R}_{4}}[x]|p(x)=p(1-x)\}$ mean? [closed]

What does $p(x)=p(1-x)$ in $P=\{p(x)\in {{R}_{4}}[x]|p(x)=p(1-x)\}$ mean?
0
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2answers
45 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 ...
1
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1answer
21 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 ...
4
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2answers
96 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 ...
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0answers
32 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 + ...
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1answer
30 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?
2
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2answers
56 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.
3
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3answers
51 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 ...
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0answers
40 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 ...
4
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6answers
66 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 ...
3
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3answers
36 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 ...
0
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1answer
20 views

Series expansions of inverse polynomials

Suppose one is given a strictly monotonous polynomial, $$f(x) = \sum_{n=0}^N a_n x^n$$ So that for a given $y$ there exists a single real $x=f^{-1}(y)$. It would be nice* to be able to calculate the ...
0
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1answer
12 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 ...
1
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1answer
36 views

Methods for factorising polynomials into irreducibles over finite fields

I was given a problem recently, part of whose solution was to factorise $x^{15}+1$ in $\mathbb F_2[x]$. It turns out that the factorisation is $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
1
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3answers
68 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?
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1answer
37 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.