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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1answer
23 views

For any $n \in \mathbb{Z^+}$ Not extis $P(x) \in \mathbb{R}[x]$ with coefficients in $B$ and all roots of $P(x)$ in $A$

Problem: Let $A=\{a_1,a_2,..,a_m\}$ and $B=\{b_1,b_2,...,b_p\}$ where $a_1,a_2,...,a_m,b_1,b_2,...,b_p \in \mathbb{R}$ Prove that , the following statements is bad : for any $n \in ...
57
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3answers
962 views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
1
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1answer
17 views

Limit at $\infty$ of a polynomial multiplied by a negative exponential

I am trying to show $\int_0^{\infty} x^2 e^{-2 x} dx = 1/4 $ Integration by parts gets the indefinite integral $$\int x^2 e^{-2 x} dx = \frac{-1}{4} e^{-2 x} (2 x^2+2 x+1)+constant$$ In order to ...
13
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4answers
2k views

showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

I studied the cyclotomic extension using Fraleigh's text. To prove that Galois group of the $n$th cyclotomic extension has order $\phi(n)$( $\phi$ is the Euler's phi function.), the writer assumed, ...
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0answers
25 views

How can I find the closure of $P[a,b]$ [closed]

Let $P[a,b]$ the space of all polynomials on the interval $[a,b]$ clearly $P[a,b]$ is a subspace of $C[a,b]$ but how can find the closure of $P[a,b]$ , In special case $[0,1]$ .
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1answer
30 views

Question on direct sum of vector spaces

I have the following linear algebra question on direct sums: I am given the vector spaces: $ V = R_4[x] $ $ W = span\{x^4-x^2,3x^4-x^3+1 \}$ I am asked to find the complement to the direct sum i.e. ...
2
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1answer
35 views

Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ [duplicate]

Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ I've only managed to show that the free coefficient of any unit in $A$ is a unit in $\mathbb Z$.
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1answer
32 views

Help with finding the basis of a polynomial vector space

I know the definitions of a basis and spanning, but I can't figure how to apply the concept to these two problems. $$Let\:S\:=\:\left\{t^2-t+1,\:t+1,\:t^2+1\right\}\:and\:v\:=\:4t^2-\:2t\:+\:3$$ ...
-3
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1answer
44 views

Mathematical Expressions [closed]

What do you think about the template in this wikipedia article? http://en.wikipedia.org/wiki/Expression_(mathematics) There are variables but no exponents and roots in arithmetic expressions? I ...
2
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4answers
57 views

Solve the following integral: $ \int \frac{x^2}{x^2+x-2} dx $

Solve the integral: $ \int \frac{x^2}{x^2+x-2} dx $ I was hoping that writing it in the form $ \int 1 - \frac{x-2}{x^2+x-2} dx $ would help but I'm still not getting anywhere. In the example it was ...
0
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0answers
15 views

Are there Karnaugh maps over other algebras?

Karnaugh maps are a useful way to minimize or factorize polynomial expressions in Boolean algebra by considering the smallest combinations of logical "subcomponents" of an expression, whose sum is ...
1
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2answers
40 views

$x^3+ (5m+1)x+ 5n+1$ is irreducible over $\Bbb Z$

How to prove that the polynomial: $x^3+ (5m+1)x+ 5n+1$ is irreducible over the set of integers for any integers $m$ and $n$? I was trying to put $x= y+p$ for some integer $p$ so that I could apply ...
-1
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1answer
28 views

Lagrange Interpolation Polynomial Degree N [on hold]

I want a Lagrange Interpolation formulae/code/online calculator which determines a apolynomail of degree n passing through given points.
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2answers
54 views

How should you go about simplying cubic polynomial: $y(x) = x^3+12x^2+21x+10$

Claim: $$y(x) = x^3+12x^2+21x+10$$ Can be factored into $$(x+1)^2(x+10)$$ But what is the quickest way to see this?
0
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1answer
30 views

What are the steps to function design?

So I'm trying to write a program, and I want to use math functions to help it. In this example, I'm trying to change the color of a line based on the position of each pixel on the line. Anyway, I ...
5
votes
1answer
863 views

Understanding how Prime Polynomials are applied to LFSRs?

In doing some research on LFSRs I understand that a primitive polynomial can determine what taps to be used to create an LFSR that has as many bits as the degree of the polynomial that will cycle ...
2
votes
2answers
84 views

The unit group of $\mathbb{Q}[x, y]/(x^2+y^2+1)$

During some calculations, I encountered with the problem of calculating the unit group of the $\mathbb{Q}$-algebra $\mathbb{Q}[x, y]/(x^2+y^2+1)$. I believe it is the unit group of the field of ...
0
votes
2answers
41 views

Same roots, same polynomial? How to prove characteristic polynomial of $AB = BA$?

I'm giving a (simple) proof that the characteristic polynomial of $AB$ = characteristic polynomial of $BA$ (without using the fact that $AB$ and $BA$ are similar). $det(AB) = det(A)det(B) = ...
2
votes
2answers
92 views

Calculate determinant of Vandermonde using specified steps.

$V_n(a_1,a_2\dots, a_n)$ is a $n\times n$ Vandermonde matrix = $$\left|\begin{array}[cccc] 11&a_1&\cdots&a^{n-1}_1\\ 1&a_2&\cdots&a^{n-1}_2\\ ...
1
vote
1answer
34 views

How to prove whether $x^{2}+y^{2}+1$ is irreducible over $\mathbb{C}$ or not?

Let's consider a 2-variable polynomial $f(x, y)= x^{2}+y^{2}+1$. It can be established that it's irreducible over $\mathbb{R}$. For example, if it's irreducible over $\mathbb{R}$ as a polynomial of ...
6
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4answers
212 views

How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials?

How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials? The link I've given to the theorem uses elaborate wordings using 'rings', ...
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0answers
30 views

Lagrange interpolation given a list of points

I have to calculate a value in which I use Lagrange interpolate to calculate numerator and denominator individually. On dividing the interpolated numerator and denominator I don't get the required ...
0
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0answers
7 views

A question related Kharitonov's Theorem for Hurwitz stable interval polynomials

Definition: An interval polynomial is the family of all polynomials $$p(s)= a_0 + a_1 s^1 + a_2 s^2 + ... + a_n s^n\tag{1}$$ where each coefficient $a_i \in \mathbb{R}$ can take any value in the ...
4
votes
4answers
111 views

$tr(A)=tr(A^{2})= \ldots = tr(A^{n})=0$ implies $A$ nilpotency

Let's consider a $n \times n$ matrix and the sequence of traces $tr(A)=tr(A^{2})= \ldots = tr(A^{n})=0$. How to prove that $A$ is a nilpotent matrix (a matrix so that $A^{k} \times u = 0$ for all $u ...
0
votes
1answer
32 views

Factoring and solving a cubic polynomial

When can we not use synthetic division to solve for a cubic polynomial? For example we can use synthetic division to solve $-t^3 -4t^2 +20t +48$. When I can't use synthetic division what are my other ...
-1
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0answers
33 views

polynomial generation via lagrange interpolation [closed]

I am calculating large numbers from lagrange interpolation. However their they are not as accurate as i want. Any solution to the problem
3
votes
1answer
54 views

How to remove duplicate roots from a polynomial?

Given a polynomial equation (with real coefficients of any degree with any number of repeating roots), let say $x^5 + 6x^4 - 18x^3 - 10x^2 + 45x - 24 = 0$, ... (A) it can be written as $(x-1)^2 ...
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0answers
28 views

Polynomial optimization and AM-GM inequality

I want to maximize the function $f(\mathbf{x},\mathbf{y}) = \sum \limits_{k=1}^{K}p_k(\mathbf{x})q_k(\mathbf{y})$, where $0 < p_k(\mathbf{x}) \leq \delta_k$ and $0 < q_k(\mathbf{y}) \leq ...
0
votes
0answers
17 views

lagrange interpolation doubt [closed]

I had initial values of a function at given input. $$\begin{align} f(3) &= 2.000000 \\ f(4) &= 1.77778 \\ f(5) &= 1.656250 \\ \end{align}$$ On using polynomial interpolation I wasn't ...
1
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1answer
25 views

Generate polynomial

Given solutions of a n degree polynomial , how can we find the polynomial. Eg. : I have solutions like : For x = 3, answer = 2 ...
0
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1answer
28 views

What is the difference between the largest and smallest possible positive roots?

I am faced with the following question: What is the difference between the largest and the smallest possible positive roots of $4x^5 + 3x^3 -5x^2 + 7x - 12$? Now, my first attempt was to try ...
1
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0answers
30 views

Proof verification: any n-th order complex polynomial has at most n distinct roots

Here is a proof I came up with in the exam I just took. But I suspect there may be some issues since I think it seems too simple. Proof Let $p_n(x)$ denote a complex polynomial of order $n$ ...
4
votes
1answer
383 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
1
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1answer
22 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
2
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0answers
28 views

Polynomials and Divisibility Rule.

The question is this - If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^3)+x^2g(x^6)$ is divisible by $x^2+x+1$, then which of the following are true? 1. $f(1)=g(1)$ ...
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0answers
14 views

HCF by Division method of Polynomials [closed]

Find step by step Highest Common Factor(HCF) by division method of these two polynomials i.e 2X^3-5X^2+4 and X^3-16X+24.
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2answers
31 views

Puzzled question of remainder of cubic polynomial

Let $f(x)$ be a cubic polynomial. If $f(x)$ is devided by $x+2$, the remainder is $-10$. If f(x) is divide by (x-1), the remainder is 20. (a) If $f(x)$ is divided by $x^2+x-2$, find the remainder. ...
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2answers
20 views

Question about finding remainder of polynomials [closed]

A. Find the remainder when $x^{2014}$ is devided by $x+1$. B. Find the remainder when $4^{2014}+3$ is devided by $5$.
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0answers
52 views

Polynomial division, multivariable, indeterminates

Trying to understand something in the proof of Nullstellensatz, if we have a polynomial $p(x_1,...,x_n,t) \in k[x_1,...,x_n,t]$ with $f(t)$ divides $p(a_1,...,a_n,t)$ for all fixed $(a_1,...,a_n) ...
0
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2answers
428 views

Finding the minimal polynomial in this field extension of $\mathbb Q$?

I have a field extension $K = \mathbb Q[x]/(x^2 - 5)$ of $\mathbb Q$, and an element $a = \bar x \in K$. I need to find the minimal polynomial of $a$ over $\mathbb Q$. I have worked out that ...
-1
votes
2answers
43 views

Prove that a linear transformation $T$ over linear space of real polynomials of $deg \leq n$ only has one eigenvalue=1 [closed]

Let $V$ be the linear space of polynomials p(x) of degree $\leq n$. If $p\in V$ define $q = T(p)$ to mean $q(t) = p(t+1)$, for all real $t$. Prove $T$ has only the eigenvalue 1. What are the ...
0
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0answers
51 views

Find the sum of the coefficients in front of the even degrees of x in the normal form of a polynomial

Find the sum of the coefficients in front of the even degrees of x in the normal form of a polynomial $$(x^6 + x + 1)^{2015} + (x^6 + x - 1)^{2015}$$ I am familiar with the binomial theorem , ...
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0answers
17 views

Stuck on polynomial equation in optimization problem

I've been trying to solve an optimization problem, but I am completely stock on one step. I had the following Langrangian: $$\nabla\mathcal{L}(x,\lambda)= e\frac{\sum_{t\in I}e^t \Delta P(t)( x^t ...
0
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1answer
30 views

little question about a notation of polynomials

In algebra we often consider the ring of polynomials $K[x]$ with coefficients in a field $K$ for example. If you write out a polynomial $p\in K[x]$, sometimes I see different things: ...
3
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1answer
34 views

Bounding $x^2+6x$ between consecutive cubes when solving $y^3=x^2+6x$

I am familiar with the method of bounding a polynomial between consecutive squares to prove it is not a square. For example, this method can prove $y^2=x^2+x+1$ has no solutions since ...
7
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2answers
463 views

When does a polynomial have only nonnegative real roots

Is there a general criterion to determine whether a polynomial with rational coefficients has the property that all of its roots are real and nonnegative?
1
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1answer
29 views

N-th roots equation

I am facing the following equation and I do not have any idea about how to solve it. $\dfrac{(n^c-1)^a}{n^{ac}}$ = $\dfrac{1}{2}$. I am free to choose $c$ (any constant). $a$ on the other hand can be ...
1
vote
1answer
13 views

Linear forms which vanish on commutators

In some exercise, $E$ denotes the vector space $\mathbb{R}[X]$ and $\mathcal{L}(E)$ the algebra of endomorphisms of $E$. I am asked to determine all the linear forms $T \, : \, \mathcal{L}(E) \, ...
0
votes
3answers
73 views

Show that $\mathbb{F}[x^2,y^2,xy]$ is not polynomial

$\mathbb{F}[x^2,y^2,xy]$ is the polynomials in two variables whose terms all have even degrees. Of course, this generating set $x^2,y^2,xy$ is not algebraically independent, but I need to show that no ...
0
votes
1answer
18 views

Divisibility in a certain ring and divisibility in integers.

Divisibility in the ring $\mathbb{Z}[x,y]$ implies divisibility in $\mathbb{Z}$ ? Let $P(x,y)=Q(x,y)\cdot R(x,y)$ with $P,Q,R$ polynomials with integer coefficients, evaluating in $(x,y)=(a,b)$ with ...