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.

learn more… | top users | synonyms

2
votes
2answers
206 views

Is this an irreducible polynomial?

I have this polynomial: $x^8+x^4+x^3+x+1$ and I would like to know if it is irreducible over $\mathbb{F}_q$ with $q=2^8$. My book gives me it is irreducible but matlab says it is not irreducible.
1
vote
2answers
146 views

Linear Transformations on Function Spaces

So I'm working on homework for my Introductory Linear Algebra Course. The text is Gareth Williams Linear Algebra with Application - 7th Edition. I'm currently working section 4.7: Kernel, Range and ...
2
votes
3answers
127 views

question related to radical sign

My question is- Let $p(x)= \sqrt{x + 2 + 3\sqrt{2x-5}} - \sqrt{x - 2 + \sqrt{2x-5}}$. Then $p(2012)^6$ equals? Any solution for this question would be greatly appreciated. Thank you,
0
votes
2answers
161 views

Polynomials with roots modulo every integer

In the paper Polynomials with roots modulo every integer, in section $2$, the authors say: Here we assumed implicitly that the greatest common divisor of the coefficients of the polynomial $P(x) = ...
2
votes
1answer
76 views

Evaluate a certain derivative

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let $$ h_j(x)=\prod_{i\neq j,i=1,\dots, n} ...
10
votes
1answer
248 views

Find three polynomials whose squares sum up to $x^4 + y^4 + x^2 + y^2$

Prove that $$p(x,y) = x^4 + y^4 + x^2 + y^2$$ can be written as a sum of squares of three polynomials over $x,y$ for real numbers.
1
vote
1answer
161 views

Polynomial remainder theorem for bivariate polynomials.

Suppose $p(x,y)$ is a bivariate polynomial of degree at most $N$ in $x$ and $y$. If $p(x_0,y) = 0$, for all y, then is it true that $$ p(x,y) = (x-x_0)q(x,y)?$$ I know this holds in the ...
5
votes
2answers
280 views

How do I prove that this polynomial is irreducible?

How do I prove that $x^4+1$ is an irreducible polynomial over $\mathbb Q$? I've already tried the Eisenstein criterion which gives to me any results to solve this question, I need help here. Thanks
0
votes
3answers
6k views

How to find the imaginary roots of polynomials

I'm looking for a simple way to calculate roots in the complex numbers. I'm having the polynomial $2x^2-x+2$ I know that the result is $1/4-$($\sqrt{15}/4)i$. However I'm not aware of an easy way to ...
1
vote
2answers
619 views

Isomorphism of two field extensions

Let $\alpha$ and $\beta$ be algebraic elements of an extension $L$ of $K$. Is it always true that if they have the same minimum polyomial then $K(\alpha)$ is isomorphic to $K(\beta)$? I think it is ...
9
votes
1answer
546 views

What is $f(x)$ divided by $(x-a)$?

This is an exercise from Spivak's Calculus: Prove that for any polynomial function $f$, and any number $a$, there is a polynomial function $g$, and a number $b$, such that $f(x)=(x-a)g(x)+b$ for ...
1
vote
3answers
90 views

Factorization of polynomial

I was just asked to factor $x^3+1$. I came to $(x^2-x+1)(x+1)$ after a while, and I was wondering, whether there is a good method to quicky factor such of polynomials.
6
votes
2answers
2k views

roots of a cubic polynomial

Consider a cubic polynomial of the form $$f(x)=a_3x^3+a_2x^2+a_1x+a_0$$ where the coefficients are non-zero reals. Conditions for which this equation has three real simple roots are well-known. What ...
1
vote
2answers
1k views

Factoring polynomials of degree 6 in 2 ways.

Let $P(x)$ be an integer polynomial of degree $6$ that is irreducible over the integers. $P(x) = x^6 + (A+a) x^5 + (B+ aA+ b) x^4 + (C+aB+bA +c) x^3 + (aC +bB +cA) x^2 + (bC+cB) x + cC = x^6 + ...
2
votes
0answers
109 views

Spectrum of Transition Matrix for Random Walk

Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. ...
1
vote
1answer
223 views

Polynomial division, Reed Solomon error correcting codes

I am trying to implement qr-codes (2d "barcodes"). The part with generating error correction codes is difficult to me, and I've found a tutorial on the www that helps me understanding the math behind ...
4
votes
1answer
92 views

Discriminant ideal in $\mathbb{Z}$

Given a monic polynomial $f \in \mathbb{Z}[X]$, I would like to consider the ideal $$(f, f')_{\mathbb{Z}[X]} \cap \mathbb{Z}$$ in $\mathbb{Z}$. In particular: is it true that this is generated by the ...
5
votes
2answers
136 views

In an ideal, pairwise non-coprime implies globally non-coprime?

Let $R$ be a polynomial ring $R=k[X_1,X_2, \ldots ,X_n]$. Let $I$ be an ideal of $R$ such that any two elements of $I$ have a non-constant gcd. Does it follow that there is a non-constant $D$ dividing ...
0
votes
2answers
107 views

irreducible polynomials of deg n in $\mathbb F_p$

Is it possible to find out a general formula for one polynomial of degree $n$ that is irreducible over $\mathbb F_p$ ?
0
votes
1answer
48 views

What is the zero set of an exponential polynomial on a torus

Let $a,b,c\in\mathbb C$, and define $$f(x,y)=ae^{i(x+y)}-b(e^{ix}+e^{iy})+c$$ for $x,y\in[-\pi,\pi]$. For a "generic" triple $a,b,c$ the set $\{f(x,y)=0\}$ consists of two points, but occasionally ...
4
votes
1answer
159 views

Showing a polynomial not reducible.

How do I show that $f(x)=1+2x+\cdots+(p-1){x}^{p-2}$ is not reducible on $\mathbb{Q}$, where $p$ is prime.
1
vote
1answer
83 views

Bernoulli polynomials, Apostol

Define Bernoulli polynomials as: $P_0(x)=1$, $P'_n(x)=nP_{n-1}(x)$, $\int_0^1P_n(x)=0$ if $n\geq1$ Need to prove that for $n\geq2$ we have $$\sum_{r=1}^{k-1} r^n= \frac{P_{n+1}(k)-P_{n+1}(0)}{n+1}$$ ...
2
votes
1answer
173 views

Factoring polynomials of degree 6 over extension fields.

Let $f(x)$ be a polynomial with integer coefficients that is irreducible over the integers and has degree 6. Let $L$ be the splitting field of $F$. Then we can ask, whether there exist intermediate ...
1
vote
3answers
194 views

$r(r-1)^2=1$, how to solve this polynomial analytically?

I found this characteristic equation $r(r-1)^2=1$ in my homework Euler differential equation problem (page 669, again the book), $x^{3} y'''+xy'-y=0$, when I misread the problem as ...
1
vote
1answer
24 views

How to verify that $(x_1 + … + x_n)^2$ represents $MOD_3$

I have question about Computational Comlexity, the following statement can be found in $AC^0$ Circuits Cannot Compute PARITY. Each $n$-varibale polynomial over $\mathbb{Z}_3$ defines a function from ...
2
votes
1answer
533 views

Automatic calculation of the intersection of discrete curves

first of all, let me apologize for a poor math-english translation, I'll try my very best. I have the following situation: I have over 16.000 data files which I generated from a biometric ...
2
votes
2answers
98 views

Mutiple root of a polynomial modulo $p$

In my lecture notes of algebraic number theory they are dealing with the polynomial $$f=X^3+X+1, $$ and they say that If f has multiple factors modulo a prime $p > 3$, then $f$ and $f' = ...
1
vote
1answer
154 views

Sequences with induction and proving. Polynomial and rational functions

$1.$We define a sequence of rational number {$a_n$} by putting $$a_1 =3,\;\text{ and}\;\; a_{n+1} = 4 - \frac{2}{a_n} \text{ for all}\; n \in \mathbb{R}.\;\text{ Put}\;\; \alpha = 2 + \sqrt{2}.$$ ...
1
vote
2answers
731 views

Ideals in Polynomial Rings

$I=\langle x^2,2x,4\rangle$ is an ideal of $\Bbb Z[x]$. Prove that $I$ is not a principal ideal and find the size of $\Bbb Z[x]/I$. Using the theorem that ideals are principal iff the generator is ...
8
votes
1answer
197 views

Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?

Consider a modified version of Collatz sequence: $C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$ Let $F_n$ be the ...
1
vote
2answers
93 views

Induction (concerning $1+z+\dots+z^n$) and follow up question

I am doing a review of stuff from earlier in the semester and I can't prove this by induction: Use induction on $n$ to verify that $1+x+\cdots+z^n= \frac{1-z^{n+1}}{1-z}$ (for $z\not=1)$. Use this ...
4
votes
1answer
540 views

Why don't they teach Fundamental Theorem of Algebra in High School? [closed]

I am currently in AP Calculus BC and one more year to go, I have heard about Fundamental Theorem of Algebra several times, and with the resources that is out there today I tried to search and study ...
0
votes
2answers
184 views

Find a rational function $f: \mathbb R \rightarrow \mathbb R$ with range $f(\mathbb R)=[-1,1]$

Find a rational function $f: \mathbb R \rightarrow \mathbb R$ with range $f(\mathbb R)=[-1,1]$ (Thus $f(x)=\frac{P(x)}{Q(x)}$ for all $x \in \mathbb R$ for suitable polynomials P and Q, where Q has ...
5
votes
1answer
101 views

Irreducible polynomials over $F_q$ with exponents of the form $q^k - 1$.

Let $q$ be some prime power. Is there an explicit family of irreducible polynomials in $F_q[X]$ of the form $\sum_j a_j X^{q^j - 1}$? Thanks!
2
votes
1answer
37 views

Sequence of polynomials (Q2)

Define $P_0(x) = 0$ and for $n > 0, \ P_n(x) = (x \ + \ P_{n-1}^2(x)) / 2$ and $Q_n(x) = P_n(x) - P_{n-1}(x)$. Are all the coefficients of the polynomials $Q_n(x)$ nonnegative?
0
votes
3answers
52 views

$x^3-\alpha \in \Bbb Q(\alpha)[x]$ is irreducible

Given $\alpha\in \Bbb C$ trascendental , and such that $|\alpha|=1$ (I don't know if this is necesary but I need only this case). Then I have to prove that the polynomial $x^3-\alpha \in \Bbb ...
0
votes
3answers
53 views

Determining a polynomial from its crossings with another polynomial

Is it possible to determine the coefficients of two polynomials, if we are given 2n different points at which they cross each other ? In other words, If $f(p) = \sum_{i}\alpha_{i}p^{i}$ and $g(p) = ...
3
votes
2answers
468 views

Factoring in Z3[x]

I need to factor $x^6+x^4+x^2+1$ into irreducible parts in $Z_3[x]$. Obviously this polynomial reduces to $(x^4+1)(x^2+1)$ which is irreducible in $Z[x]$, but I'm not sure how to confirm that it's ...
7
votes
2answers
723 views

Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?

Given the elementary symmetric polynomials $e_k(X_1,X_2,...,X_N)$ generated via $$ \prod_{k=1}^{N} (t+X_k) = e_0t^N + e_1t^{N-1} + \cdots + e_N. $$ How can one get the monomial symmetric functions ...
2
votes
0answers
93 views

How to find the root of a polynomial

I don't know how to solve the following equation: $x^5-h_1x^4-h_2x^3-h_5=0$, where $h_1=\beta_1+\beta_2$, $ h_2=\beta_1+\beta_2-\beta_1\beta_2-\frac{\beta_1\beta_2}{\beta_1+\beta_2}$, ...
1
vote
0answers
151 views

Factoring large polynomials

I'm asked questions like to find the zeros, and multplicities of the $\mathbb{C}$-polynomials $$f(z) =z^6 +4z^2 - 1 \hspace{10mm} , |z| < 1$$ or worse yet $$f(z)=z^{87}+36z^{57}+71z^4+z^3-z+1 ...
3
votes
2answers
105 views

$F[x]/\langle f(x) \rangle$ has $q^n$ elements

today I have a problem which I see in book Abstract Algebra of David. This is problem: Let $F$ be a finite field of order q and let $f(x)$ be a polynomial in $F(x)$ of degree $n\geq 1$. Prove that ...
2
votes
2answers
268 views

Basis for this $\mathbb{P}_3$ subspace.

Just had an exam where the last question was: Find a basis for the subset of $\mathbb{P}_3$ where $p(1) = 0$ for all $p$. I answered $\{t,t^2-1,t^3-1\}$, but I'm not entirely confident in the ...
4
votes
1answer
662 views

the discriminant of the cyclotomic $\Phi_p(x)$

I'm very bad in computations of this kind :/. I don't know tricks for computing the discriminant of a polynomial, only the definition and using the resultant, but it's very complicated to do only with ...
3
votes
1answer
127 views

Diophantine equations and Groebner bases

I'm trying to teach myself the basics of algebraic geometry and have run into something that I don't understand. I know that the problem of deciding whether a Diophantine equation $P(\vec{x}) = 0$ ...
1
vote
1answer
420 views

Piece-wise linear interpolating polynomials

Somebody please help me to obtain piece-wise interpolating polynomials for the function $f(x)$ defined by the below data: $x=1$, $f(x)=3$; $x=2, f(x)=3$; $x=4, f(x)=21$; $x=8, f(x)=73$ I know the ...
2
votes
0answers
443 views

Interpolating polynomial with Chebyshev nodes

I am interested in constructing an polynomial that interpolates some known arbitrary function $f(x)$ over the domain $x \in [0,70]$. I want the polynomial to have degree 14 and so need 15 points. ...
1
vote
2answers
109 views

find $x,y,z$ such that $x+yz=M$, $y+zx=N$, $z+xy=K$

somebody asked me this. I don't know whether it is interesting but I hope someone can solve it. find $x,y,z$ such that $x+yz=M$, $y+zx=N$, $z+xy=K$, where $M,N,K$ are constants.
1
vote
1answer
81 views

counting the real zeros of a polynomial and proving that it's irreducible over $\Bbb Q$

Let's consider the polynomial $$ f\left( x \right) = \left( {x^2 + 2} \right)\prod\limits_{i = - k}^k {\left( {x - 2i} \right) + 2 \in {\Bbb Q}\left[ x \right]} $$ . Let's suppose that $ p = 2k + ...
7
votes
1answer
120 views

“Real part” of a number field

Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...