This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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3
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0answers
13 views

Derivative of a characteristic polynomial at an eigenvalue

Let $p(\lambda)$ be the characteristic polynomial of an $n\times n$ matrix $A$. We know that the roots of $p(\lambda)$ are the eigenvalues of $A$, hence the sum of the roots of the polynomial (taking ...
3
votes
5answers
93 views

Factorize a third degree polynomial

It's my first time posting here so I'm not used to describing my problem in mathematics. I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but ...
0
votes
0answers
13 views

What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$. What are ...
0
votes
4answers
62 views

How do I find the solution(s) to my second-degree equation?

$$f(x) = x^2 - 3x$$ My attempt : $$ \begin{align} x^2-3x &= 4\\ x(x-3) &= 4\\ x-3 &= 4 \\ x &= 7\\ \end{align} $$ I managed to solve one part of this problem but that one part is ...
5
votes
3answers
55 views

Prove that for any polynomial $P(x)$ there exist polynomials $F(x)$ and $G(x)$ such that $F\left(G(x) \right)-G\left(F(x) \right)=P(x)$

Prove that for any polynomial $P(x)$ there exist polynomials $F(x)$ and $G(x)$ such that $\forall x \in \mathbb R:$ $$F\left(G(x) \right)-G\left(F(x) \right)=P(x)$$ My work so far: Let $G(x)=x+1$...
2
votes
1answer
25 views

Is it possible to express the inverse of a polynomial as a series?

Is it possible to express the multiplicative inverse of a nth order polynomial i.e. \begin{equation} \frac{1}{\left[\sum_{i=0}^na_ix^i\right]^2} \end{equation} as a series using binomial theorem or ...
1
vote
0answers
35 views

Is there a special name for polynomials related by Möbius tranformation of the variable?

If we take a general polynomial with complex coefficients: $$C_n z^n+C_{n-1}z^{n-1}+\dots+C_1z+C_0$$ We can apply a general Möbius tranformation to the variable: $$z=\frac{aw+b}{cw+d},~~~~a,b,c,d \...
9
votes
1answer
103 views

Only five solvable quintic equations of the form $x^5+ax^2+b=0$? What are their solutions?

According to Wikipedia there is only five solvable quintic equations of the form $x^5+ax^2+b=0,~~a,b \in \mathbb{Q}$ (up to a scaling constant $s$). $$x^5-2s^3x^2-\frac{s^5}{5}=0 $$ $$ x^5-100s^3x^2-...
0
votes
0answers
45 views

Proof of Ramanujan's famous cubic identity

Ramanujan found that given a polynomial $y=x^3+ax^2+bx+x$, one can find $\sqrt[3]{u+x_1}+\sqrt[3]{u+x_2}+\sqrt[3]{u+x_3}=\sqrt[3]{3\sqrt[6]{d}+w}$ where $$d=\frac {4(a^2-3b)^3-(2a^3-9ab+27c)^2}{27}$$$$...
1
vote
2answers
25 views

How to evaluate GF(256) element

I wonder is there any easy way to evaluate element of GF(256) meaning that assume I would like to know what is alpha ^(32) or alpha^200 in polynomial form? given the primitive polynomial is D^8+D^4+D^...
2
votes
2answers
191 views

Maple help needed

Consider the multivariable polynomial $$g(x,y,z,w)=a_1xyw+a_2xy^2+a_3xyz+a_4x^3z+a_5z^3+a_6y^2z+a_7w^4\;,$$ where $a_1,\cdots, a_7$ are constants. I would like to use Maple to extract the coefficients ...
1
vote
1answer
32 views

Interpolation for $f(n),n\in\mathbb{Z}$: Does it converge?

Assume a function $f(n)$ which is defined for $n\in\mathbb{Z}$. For each period $[n,n+1]$ the function could be interpolated with a polynomial of degree $m$. The polynomials should be built in a way ...
5
votes
2answers
68 views

When is a 5th degree polynomial with at least 1 non-real root solvable by radicals?

Let $f(X)$ be an irreducible polynomial of degree 5 with coefficents in the field of rational numbers $\mathbb{Q}$. Assume that $f$ has at least one non-real root in the complex field $\mathbb{C}$. ...
2
votes
2answers
94 views

Functional equation $P(X)=P(1-X)$ for polynomials

I have encountered the following problem : Find all polynomials $P$ such as $P(X)=P(1-X)$ on $\mathbb{C}$ and then $\mathbb{R}$. I have found that on $\mathbb{C}$ such polynomials have an even ...
1
vote
2answers
829 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
0
votes
3answers
39 views

Meaning of Vector Space over $\mathbb{R}$ being a Subspace of $\mathbb{R^R}$

$\mathscr{P(\mathbb{R})}$ is the set of all polynomials with coefficients in $\mathbb{R}$. How are below sentences related and why? (1) $\mathscr{P(\mathbb{R})}$ is a vector space over $\mathbb{R}...
2
votes
0answers
40 views

Factoring $x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$ over $\mathbb{Q}$

For a quntic polynomial to be reducible to the following form over $\mathbb{Q}$: $$x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$$ We need to match the coefficients ($a=B$ obviously, so we ...
0
votes
0answers
9 views

Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form $$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...
0
votes
2answers
31 views

primitive polynomials and their factorisation

A polynomial with integer coefficients is called primitive if its coefficients are relatively prime. For example, $$3{x^2} + 7x + 9$$ is primitive while $$10{x^2} + 5x + 15$$ is not. (a) Prove that ...
2
votes
1answer
99 views

How to decide if a polynomial is symmetric?

First, is the following: $$f=\frac{3}{5}(x_1^5 + x_2^5 + x_3^5 + x_4^5)-\frac{7}{12}(x_1^2x_2^2 - x_1^2x_3^2-x_1^2x_4^2-x_2^2x_3^2-x_2^2x_4^2-x_3^2x_4^2)$$ a symmetric polynomial? And, if yes, how do ...
2
votes
2answers
33 views

Are a uniformly random polynomial's roots are distributed uniformly in the field?

Assume we have a $\mathbb{F}_p$, where $p$ is a large prime (e.g. 128-bit value). We define all polynomials over the field, and pick a polynomial,$P(x)$, of degree $d$, where the polynomials' ...
1
vote
1answer
25 views

Given a polynomial with integer coefficients and prime independent term, show that any root has absolute value greater than 1.

I was looking at exercises about algebraic structures, and in ring theory I stumbled upon this problem. Given $p$ a prime number and $f(x)=\pm p + a_{1}x+\cdots+x^{n} \in \mathbb{Z}[x]$ so that $\...
0
votes
1answer
50 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
0
votes
1answer
42 views

Coefficients of a Polynomial Approximation in Minimax Sense

I am working on a Uniform Random Number Generator using a IEEE paper, and I got stuck with the coefficients for a Piecewise Polynomial Approximation using Horner's Rule : $$ y = ((C_d x + C_{d-1})x +...
2
votes
0answers
37 views

Series expansion of inverse polynomial

Suppose an nth order polynomial $P_n(x)$ with real and distict roots $d_1,d_2,\dots,d_n$, which has the factorization \begin{equation} P_n(x)=(x^2-d_1^2)(x^2-d_2^2)\cdots(x^2-d_n^2).\end{equation} ...
0
votes
1answer
27 views

Brumer quintic polynomials - is there a general formula for roots?

There exist a family of quintic polynomials, called Brumer's polynomials (or Kondo-Brumer), which have the form: $$x^5+(a-3)x^4+(-a+b+3)x^3+(a^2-a-1-2b)x^2+bx+a,~~~a,b \in \mathbb{Q}$$ According to ...
0
votes
3answers
28 views

How to find other basis of polynomials of degree three or less? [on hold]

How can i find a basis of polynomials of degree three or less, which is other than $\{1,t,t^2,t^3\}$ ?
3
votes
3answers
44 views

construct polynomial from other polynomials

If I have a polynomial, P, with root $a$ and a polynomial, Q, with root $b$, is there a way to construct polynomial R such that $a+b$ is a root of R? Here's a concrete example. a = $\sqrt2$. $P(x) = ...
2
votes
1answer
43 views

Does there exist an algebraic solvability algorithm?

I was ruminating over quintics and got curious about the following idea. Consider a quintic equation: $$ Q(x) a_0 + a_1 x + a_2 x^2 + ... a_5 x^5$$ Such that the solutions to $$ Q(x) = 0 $$ Are ...
-1
votes
2answers
40 views

If $A$ is diagonizable then $p(A)$ is diagonalizable

Show that if a matrix $A$ of size $n \times n$ is diagonalizable, then $p(A)$ is diagonalizable for each polynomial $p$.
4
votes
2answers
86 views

'Strange' trigonometric roots of $x^5-4x^4+2x^3+5x^2-2x-1$ - could someone explain?

This quintic equation has $5$ real roots: $$x^5-4x^4+2x^3+5x^2-2x-1=0 \tag{1}$$ The roots are, from left to right: $$x_1=\frac{\cos \frac{19}{22} \pi}{\cos \frac{1}{22} \pi}$$ $$x_2=\frac{\cos \...
9
votes
0answers
189 views

Bounds on derivative of real positive coefficient polynomial satisfying certain properties

While thinking about this question of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying: (I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and $\...
3
votes
1answer
29 views

$p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $A \to p(A)$ surjective on $M(n,\mathbb R)$?

Let $p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $f: M(n,\mathbb R) \to M(n, \mathbb R)$ defined as $f(A)=p(A) , \forall A \in M(n,\mathbb R)$...
0
votes
1answer
37 views

5th degree polynomial with positive leading coefficient

I'm guessing C or D because odd degree polynomials which aren't linear extend one way to infinity and the other way to negative infinity? So what's the relevance of the a>0? As x approaches infinity ...
3
votes
1answer
39 views

$p(x) \in \mathbb R[x]$ be non-constant polynomial , $n>1$ , the function $A \to p(A)$ is surjective on $M(n, \mathbb C)$?

Let $p(x) \in \mathbb R[x]$ be a non-constant polynomial and $n>1$ , then is it true that the function $f:M(n,\mathbb C) \to M(n, \mathbb C)$ defined as $f(A)=p(A) , \forall A \in M(n, \mathbb C)...
0
votes
0answers
26 views

Bernoulli Polynomials from Apostol's calculus book

Question 35 from book of calculus, volume 1 Apostol, in chapter "The relation between integration and differentiation". Define Bernoulli polynomials as: $P_0(x)=1$, $P'_n(x)=nP_{n-1}(x)$, $\int_0^...
0
votes
1answer
36 views

A solvable quintic with the root $x=(\sqrt[5]{p}+\sqrt[5]{q})^5$ - what are the other roots?

I derived a two parameter quintic equation with the root: $$x=(\sqrt[5]{p}+\sqrt[5]{q})^5,~~~~~p,q \in \mathbb{Q}$$ $$\color{blue}{x^5}-5(p+q)\color{blue}{x^4}+5(2p^2-121pq+2q^2)\color{blue}{x^3}...
1
vote
2answers
53 views

How to find the roots of this 4th order polynomial?

Can someone explain how to factor/find roots to this 4th order polynomial: $$ s^4 + 14s^3 +45s^2 +650s + 1800 = 0 $$ It's such a nightmare. I've been stuck for hours, any help would be appreciated :)...
5
votes
2answers
123 views

Polynomials generating the same $p$-adic fields

I wonder if the following fact is true: Pick $l\in \mathbb N$ a number and let $f,g\in \mathbb Z_p[x]$ be monic polynomials with coefficients in the ring of $p$-adic integers such that $f\equiv g \...
0
votes
3answers
27 views

Find the algebric form of the zeros(roots) of the following polynomial: $\left(\:z^2+iz+2\right)\left(z^3-8i\right)$

Good morning to everyone. I don't know how to find the zeros(roots) of the following polynomial function: $$\left(\:z^2+iz+2\right)\left(z^3-8i\right)$$. What I've tried: The zero(root) of the second ...
0
votes
1answer
20 views

Conditions for Non-negativity

Let's consider $A$ to be a square symmetric matrix whose entries are non-negative real numbers that sums to one. Even more, we shall consider its diagonal elements to be equal to zero. The question is:...
1
vote
3answers
112 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
3
votes
1answer
105 views

Monic polynomial reducible in rationals

Let $P(x)\in \mathbb{Z}[x], Q(x),R(x)\in \mathbb{Q}[x]$, and all three polynomials are monic. Suppose $P(x)=Q(x)R(x)$. Is it true that $Q(x),R(x)\in\mathbb{Z}[x]$? Gauss's Lemma says that since $P(x)$...
1
vote
2answers
443 views

Proving Gauss' polynomial theorem

Let $P \in \mathbb{Z}[x], P(x) = \displaystyle\sum\limits_{j=0}^n a_j x^j, a_n \neq 0$ and $a_0 \neq 0$; if $p/q$ is a root of P (with p and q coprimes) then $p|a_0$ and $q|a_n$ I've managed to prove ...
1
vote
2answers
55 views

$f(x+a)$ irreducibility means $f(x)$ irreducibility

Let $a~\in~\mathbb{Z}$ and let $f(x)~\in~\mathbb{Z}\left[x\right]$. Suppose that $f(x+a)$ is irreducible over $\mathbb{Z}$. Prove that $f(x)$ is irreducible over $\mathbb{Z}$. My idea is: $f(x)=u(x)*...
0
votes
1answer
50 views

Principal ideal of $\mathbb{C}[Z,\bar{Z}]$

Let $I$ be an ideal of $\mathbb{C}[Z,\bar{Z}]$. How to prove that $I$ is principal in $\mathbb{C}[Z,\bar{Z}]$ ? It exists some simple criterion to say that an ideal will be principal or not?
6
votes
2answers
280 views

Why Rational Root Theorem only works with integers

Why does the rational root theorem only work when the polynomial has integer coefficients?
0
votes
0answers
23 views

Factoring polynomials when roots are external to the ring

I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question. It is clear to ...
-1
votes
2answers
37 views

Polynomial ring, ideals and Spec

Morning everyone, I want some hint about this. i) Determine all ideals of $\frac{\Bbb{R[X]}}{<X^3-1>}$ where $R$ is real set ii)Is $\frac{R[X]}{<X^3-1>}$ integral Domain iii)...
3
votes
2answers
95 views

What kind of $n^{th}$ order polynomials are solvable by a square matrix with integer entries?

Consider a polynomial (monic for simplicity): $$x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ Here we assume the roots are complex numbers. $a_k$ are integers. Now consider the corresponding matrix ...