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
1answer
734 views

How many irreducible factors does $x^n-1$ have over finite field?

The polynomial $x^n-1$ is needed to be factorized into irreducibles over finite field $\mathrm{F}_q$. How many are them? I guess the question is about of number of cyclotomic cosets. Let $p$ be the ...
5
votes
2answers
894 views

When is a cyclotomic polynomial over a finite field a minimal polynomial?

When is the cyclotomic polynomial $f(x)$ over a finite field $\mathrm{F}_q$ also the minimal polynomial of some element $\alpha \in \mathrm{F}_q$?
0
votes
1answer
82 views

Is it true there is no general solution in radical form of symetric groups $S_n$ ($n>4$)? [closed]

Since there is a general relationship by which a $S_5$ symmetric group can degenerate into lower order symmetric groups, isn’t that enough evidence that general quintics are solvable in radicals? ...
1
vote
2answers
314 views

Getting rid of the denominator of a polynomial

I'm tutoring a high school precalculus student; our current topic is the roots of higher order polynomials. The problem we're solving is: Find a polynomial with the roots $\frac23$, -1, and $(3 + ...
3
votes
4answers
563 views

Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly $4$ distinct real roots

Problem : Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly four distinct real roots, then which of the following options are correct : (a) $f(x) =0$ has all three real roots ...
2
votes
1answer
81 views

Composition of polynomials - is it a simple group?

I wouldn't be surprised if this can be found maybe even on Wikipedia but I'm not a native English speaker and unfortunately couldn't find this myself. So for a set of polynomials $F = \left\{ \, f(x) ...
1
vote
2answers
52 views

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ …

Problem : If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ then the value of $(1-\alpha_1)(1-\alpha_2)(1-\alpha_3)(1-\alpha_4)$ is (a) 2$\sqrt{3}$ ...
0
votes
1answer
68 views

work out the value of a - b from the identity $ax+18=2(x-b)$

How do I solve the following question? You are given the algebraic identity: $ax+18=2(x-b)$ Work out the values of $a-b$
5
votes
1answer
3k views

How to get the roots of a quartic function when given a quadratic factor

We have the function $$x^4 + 4x^3 - 17x^2 -24x + 36 = 0.$$ $x^2 -x - 6$ is a factor of this function. Find all the roots of the polynomial. So we have $(x-3)(x+2)$, and since it is a quartic we ...
13
votes
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, ...
0
votes
1answer
35 views

Inverse function without the original function

I am going through this paper, 'Certifiable Quantum Dice Or, True Random Number Generation Secure Against Quantum Adversaries' by Vazirani and Vidick. In 'Our results' section on the page 2, it says: ...
1
vote
2answers
124 views

Simple Polynomial Interpolation Problem

Simple polynomial interpolation in two dimensions is not always possible. For example, suppose that the following data are to be represented by a polynomial of first degree in $x$ and $y$, ...
3
votes
1answer
73 views

Mutual dependency of polynomial expressions

Suppose you are given the values of $m$ polynomial expressions in $n$ variables. That is we know that $P_1(x_1,x_2,...,x_n)=a_1,P_2(x_1,x_2,...,x_n)=a_2,...,P_m(x_1,x_2,...,x_n)=a_m$ for some ...
1
vote
1answer
71 views

Boolean algebra generated by value sets of polynomials over $\mathbb{N}$

Update For each polynomial $P \in \mathbb{N}[X]$, let $S_P = \{ P(n) \mid n \in \mathbb{N}\}$. Does the Boolean algebra generated by the subsets $S_P$ of $\mathcal{P}(\mathbb{N})$ such that $P$ is ...
0
votes
1answer
1k views

Calculation of Chebyshev coefficients

The Chebyshev polynomials can be defined recursively as: $T_0(x)=1$; $T_1(x)=x$; $T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$ The coefficients of these polynomails for a function, $\space f(x)$, under ...
2
votes
3answers
163 views

nitpicking the definition of a polynomial function

A textbook I'm reading says that $f(x)=0$ is NOT a polynomial function, yet $g(x)=8$ IS a polynomial function since $g(x)=8=8x^0$ which satisfies the non-negative integer degree requirement. Yet, ...
5
votes
1answer
80 views

Finding a polynomial $g(x)$ such that $ g(x)g(x-1)=g(x^2)$

Find all polynomials $g(x)$ with real coefficients with the property $$g(x)g(x-1)=g(x^2).$$ My try: I found $$g(x)=(x^2+x+1)^n$$ satisfies the condition; maybe there are other solution? If so, how to ...
1
vote
1answer
90 views

Find the rank of a matrix representing $p$ distinct polynomials of maximum degree $n$

I can't find the Reduced Row Echelon Form to find the number of pivots because I don't have numbers to work with. I know an upper bound for the rank is the smaller amongst $p$ and $n+1$. Any tips on ...
1
vote
0answers
106 views

Analytical solution(root) for a tenth order polynomial?

is it possible to develop an analytical solution (root) for such a polynomial: $f(x)=\left(x^{10}-c_1^2\right)*\left(c_2-x\right)^2-0.2*\left(x^2-1\right)*c_1^2$ with $c_1$ and $c_2 >0$. Numerical ...
4
votes
2answers
212 views

showing every ideal of some quotient ring is principal.

Let $\mathbb F$ be a field and $A=\mathbb F[t]/(t^2)$, where $(t^2)$ is the ideal of $\mathbb F[t]$ (This quotient ring is not an integral domain as you know), and I write an element of $A$ by ...
-1
votes
1answer
38 views

Linear transformation / Polynomial Question

$T:P_{3}\rightarrow P_{3}$ defined by $T(p(t))=tp'(t)+p(0)$ is a linear transformation. Determine whether $T$ is invertible. If yes, find $T^{-1}(q(t))$, where $q(t)$ is a polynomial of degree at ...
2
votes
1answer
172 views

For a linear operator,T, show the existence of a polynomial, p, such that p(T) is a nilpotent operator

I am faced with the following (homework) problem. Given an operator $T : C^{n} \rightarrow C^{n}$, show that there exists a polynomial $p(z) \in C$ of degree at most $n - 1$ such that $p(T)$ is a ...
5
votes
3answers
2k views

Prove that $x^3 + x^2 = 1$ has no rational solutions?

Is this enough for a proof?: $$x^3+x^2 = 1$$ I would factor and get: $x^2(x+1) = 1$ I would show that $x = \sqrt1$, which is irrational but then do I have to show more? $x+1=1$ which gives me $x=0$ ...
0
votes
1answer
63 views

How find $\lim\limits_{x\to+\infty}\sum_{j=0}^k (-1)^j \binom{k}{j} \sqrt[n] {f(x+k-j)}$

Let $k,n \in\mathbb N$ and let $f(x)=a_{0}x^{kn}+a_{1}x^{kn-1}+\cdots+a_{kn}$. Find the limit $$\lim_{x\to+\infty}\sum_{j=0}^k (-1)^j \binom{k}{j} \sqrt[n] {f(x+k-j)}$$ I think this uses Taylor's ...
0
votes
1answer
268 views

Minimization of Sum of Squares Error Function

Given that $y(x,{\bf w}) = w_0 + w_1x + w_2x^2 + \ldots + w_mx^m = \sum_{j=0}^{m} w_jx^j$ and there exists an error function defined as $E({\bf w})=\frac{1}{2} \sum_{n=1}^{N} \{y(x_n, w)-t_n\}^2$ ...
1
vote
0answers
186 views

Determine generator over $GF(2^4)$

Working in $GF(2^4$) Field generated modulus $x^4+x^3+x^2+x+1$. Find a generator of $F$. What I have figured out so far - $16$ polynomials to consider. If $b$ is generator then start with $b = x + ...
2
votes
1answer
54 views

A curious class of polynomials

In connection with some calculations involving generating functions I have encounetered the following family of polynomials $$ p_{k,N}(x) = \sum_{0<n_1<n_2<\ldots<n_k<N} ...
3
votes
0answers
67 views

Roots of the derivative as symmetric (?) functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
0
votes
1answer
60 views

Can this polynomial transformation produce new symmetry?

I've got a polynomial transformation on $\mathbb{R}^6$, and I have a conjecture about it, but I'm having a hard time proving it. The transformation looks like this: $ u:= abcde + abc + abe + ade + ...
0
votes
0answers
134 views

Approximations other than taylor series and pade approximation

I have a function which has the following form: $$ f(x)=K_1 \coth (Q_1 Q_2 \sqrt{x})^2 + \frac{1}{x}\left[K_2 + K_3 \coth(Q_1 Q_3\sqrt{x})\sqrt{x}\right]$$ and I want to find $x$ for $f(x)=1$. I'm ...
4
votes
2answers
1k views

Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros

I need a proof to show that the inequality $m < n$ leads to a contradiction and $P_n$ has $n$ distinct real roots, all of which lie in the open interval $(-1, 1)$.
0
votes
1answer
32 views

Finding a polynomial $k$ respects $e^{-\epsilon k}\leq \delta$

I am given $e^{-\epsilon k}$ and my goal is to find a polynomial $k$ (in $\epsilon$ and $\delta$) such that $e^{-\epsilon k}\leq \delta $ where $\epsilon,\delta,k>0$. The exercise shows that ...
1
vote
1answer
68 views

An algebraically closed field with characteristic $p>0$

I want to know about an algebraically closed field that is not of characteristic $0$. I really don't know about infinite fields with characteristic $p$ so I will appreciate your comments.
0
votes
1answer
39 views

On the condition of $ac$ when factoring a quadratic trinomial where $a\neq1$

I am currently reading the section "Factoring quadratic trinomials when a ≠ 1 using the grouping method" this site. I don't understand the last part of this paragraph: Where did the condition for ...
4
votes
1answer
91 views

Determinant of a circulant matrix as Chebyshev-like recurrence

It is while studying the Hückel Method of Physical Chemistry that I came across the following recurrence relation: \begin{align*} U_n(x)=xU_{n-1}(x)-U_{n-2}(x)+(-1)^{n-1}(4+2x) \end{align*} Where ...
0
votes
1answer
67 views

Polynomial (third degree)

A third degree polynomial $p(x)=0$ when $x=1$ and $x=3$. We also learn that $p(x) \geq 0 $ when $x \geq 1$ and $p(2) =2$. Determine $p(x)$. How should I proceed? I presume no calculus is needed.
1
vote
1answer
42 views

Taylor polynomial help

About Taylor polynomials: I have $$f(x)=(x-4)^9$$ And I need to find the 10th order taylor polynomial about $x=4$ Now, I tried solving it: All but the last term is equal to $0$ since for example ...
4
votes
2answers
113 views

What is the field of definition of an invariant ideal?

Let $K/k$ be a finitely generated field extension, such that $k=K^G$ for some (possibly infinite) set $G$ of automorphisms of $K$. Now, consider the extension of polynomial rings $$ ...
3
votes
2answers
75 views

Is it necessarily true that if a polynomial is irreducible in $\mathbb Z_n$ ($n$ is prime) then it is irreducible in $\mathbb{Q}$?

I have played around with a couple examples and I've consistently seen a pattern where the polynomials that are irreducible in $\mathbb{Z}_n$ are irreducible in $\mathbb{Q}$. Can anyone challenge this ...
2
votes
2answers
1k views

Solving Quartic Equation

Could someone please explain how to solve this : $x^4+3x^3-6x^2+16x+56=0$ - not the answer only, but a step-by-step solution.
4
votes
1answer
87 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
2
votes
1answer
59 views

Using roots of irreducible polynomials to rewrite products.

Suppose $F$ is a field, and $p(x)\in F[x]$ is irreducible, of degree $n$, with a root $\alpha$. "$F(\alpha)$ is closed under multiplication since $\alpha^n,\alpha^{n+1},\ldots $ can be written as ...
0
votes
1answer
60 views

Rolles Theorem Simple and multiple zeros

I have this problem with Legendre polinomials Use Rolle's Theorem to show that Pn cannot have multiple zeros in the open interval (-1, 1). In other words, any zeros of Pn which lie in (-1, 1) must be ...
0
votes
1answer
173 views

Find the irreducible factors of $f(x)=x^4-5x^2+6$ over $\mathbb{Q}$ and over $\mathbb{R}$ individually.

Find the irreducible factors of $f(x)=x^4-5x^2+6$ over $\mathbb{Q}$ and over $\mathbb{R}$, individually. I have a bit of a start, but need some help.
0
votes
1answer
74 views

Similarity of polynomials

I'm looking for a method that takes two simple polynomials (cubic) and gives a value of how visually-similar they are to each other on some specified domain. How could I do this?
0
votes
1answer
153 views

Justify $\gcd$ of $f(x) = x^3 - 6x^2 + x + 4$ and $g(x) = x^5 - 6x +1$

Let $f(x) = x^3 - 6x^2 + x + 4$ and $g(x) = x^5 - 6x +1$. Using Euclidean algorithm I find $\gcd[f(x), g(x)] = 1$. How could I JUSTIFY that $h(x) = 1$ is the ACTUAL $\gcd$ of $f(x)$ and $g(x)$? ...
2
votes
2answers
134 views

Find the greatest common divisor (gcd) of $f(x) = x^2 + 1$ and $g(x) = x^6 + x^3 + x + 1$

Find the greatest common divisor (gcd) of $f(x) = x^2 + 1$ and $g(x) = x^6 + x^3 + x + 1$. Since $x^6 + x^3 + x + 1 = (x^2 + 1)(x^4 - x^2 + x + 1)$, $\mathrm{gcd}[f(x),g(x)] = x^2 + 1$. My question ...
0
votes
4answers
74 views

Galois field and polynomials?

Show there are only two polynomials of degree 3 over $\mathbb{F}_2$ such that it is irreducible and all other degree 3 polynomials can be reduced. So $x^2 = x$ and $x = -x$ I cant think of anything ...
3
votes
0answers
69 views

$f \in \mathbb{C}[x_1,\ldots,x_n]$ and its zeroes [duplicate]

Given a polynomial $f \in \mathbb{C}[x_1,\ldots,x_n]$, then how can I prove that $f(a_1,\ldots,a_n) = 0$ implies $f$ is a summation of factors of $x_i - a_i$ for $i \in \{1,\ldots, n\}?$ This is not ...
2
votes
1answer
87 views

Compound quadratic problem

The first issue I have is that I am not sure why this is called a 'compound quadratic problem', but anyway to proceed: Suppose that $x-y=14$ and $$(x+y)(x^2+y^2)(x^4+y^4)=a(x^b-y^b)$$ where $a$ and ...