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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Does there exist a ring automorphism on $\mathbb Q[x]$ of order $2$?

Does there exist a non-identity ring automorphism $\psi$ on $\mathbb Q[x]$ such that $\psi \circ \psi$ is the identity automorphism ?
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1answer
9 views

For every $n \in \mathbb Z^+$ , does there exist a ring automorphism $\psi$ on $\mathbb C[x]$ of order $n$?

For every $n \in \mathbb Z^+$ , does there exist a ring automorphism $\psi$ on $\mathbb C[x]$ such that $\psi^n=\psi \circ... n $ times is the identity automorphism but $\psi^r$ is not identity for ...
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5 views

Proving that the products GCDs of the coefficients of two polynomials is equal to the GCD of their product's coefficients?

Assume that $p(x)=a_nx^n+\dots+a_0$ and $q(x)=b_nx^n+\dots+b_0$ where the coefficients are integers. Let $y$ be the gcd of $a_n,\dots,a_0$ and let $z$ be the gcd of $b_n,\dots,b_0$. How does one prove ...
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When is this sum of perfect powers bounded

For any positive integers $n,d$, let $$ A_d(n)=\frac{\sum_{k=1}^n k^{2d}}{n(n+1)(2n+1)} $$ It is easy to see (and well-known) that for fixed $d$, $A_d(.)$ is a polynomial of degree $2d-2$. Is it ...
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32 views

Polynomial Evaluation

My question is to some extent related to cryptography, but I'd like the mathematicians answer my question, please (as their answers are usualy more clearer than cryptographers). Consider I have a ...
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Proving collinearity of the roots without finding the roots.

I was solving the polynomial $2z^3-(3-3i)z^2-(1+i)=0$ and found that they were in fact collinear! My question is, is there a way to prove that they are collinear without explicitly finding the roots? ...
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How to find the roots $a, b, c, d$ of $f(x)=9x^4+9x^3+2x^2-14x+2=0,$ if $a=2b$?

I'm working on Uspensky's theory of equations and I have to solve this equation using Vieta's formulae (but I just can't solve the system of equations :( ) : $$f(x)=9x^4+9x^3+2x^2-14x+2, $$ if $a=2b. ...
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Problem: Minimum of set of maximum correlations of trig. poly. coef vector with space of all trig. poly. coef vectors

I have a problem that I can't seem to get started with. Let $\mathbf{c} = \{c_k\in \mathbb{C}\}_{k\le|N|}, c_{-k} = -\bar{c}_{k}, \|\mathbf{c}\|_2 = 1$ be the normalised vector of coefficients of a ...
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2answers
35 views

Find the value of the polynomial at a given point

f is a polynomial of degree $1007$ ,if $f(k) = 2^k$ for $0\le k \le 1007$ find the value of $f(2015)$ The solution should be $f(2015) = 2^{2014}$ and the polynomial $p(x) = \sum_{k=0}^{1007}{x ...
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0answers
26 views

Polynomial change of basis

We got asked to solve this problem: Express the polynomial $f(x) = (1 + x)^6, f \in \mathbb{Z}[x]$, in the basis $(1 + x^2)$. I don't really understand how a polynomial change of basis ...
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1answer
31 views

$f$ be a ring automorphism on $R[x]$ such that $f(u)=u , \forall u \in R$ , then is it true that $f(x)=ax+b $ for some $a,b \in R$?

Let $R$ be a ring and $f:R[x] \to R[x]$ be a ring automorphism such that $f(u)=u , \forall u \in R$ , then is it true that $f(x)=ax+b $ for some $a,b \in R$ ?
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1answer
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N-th roots equation

I am facing the following equation and I do not have any idea about how to solve it. $\frac{(n^c-1)^a}{n^{ac}}$ = $\frac{1}{2}$. I am free to choose c (any constant). a on the other hand can be any ...
2
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0answers
27 views

About Runge's method

I have been reading about some Diophantine equations (like Runge's theorem and Cassel's theorem) and in the text says that these theorems are solved using Runge's method, but it doesn't say what ...
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The $k$_th coefficient of the polynomial :$f_n(z)f_m(jz)+f_m(z)f_n(jz) $

Let $j=e^{2i\pi/3}$ ( $i$ is the complex number $i^2=-1$), and let : $$f_n(z)=(1+z)^n$$ Question Is there an expression (without using sums) of the $k$_th coefficient of the following polynomial ...
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1answer
72 views

When is $20q^4-40q^3+30q^2-10q$ a square for positive integer $q$?

For what $q$ is the following polynomial a square? $$ \begin{align} &20q^4-40q^3+30q^2-10q\\ =\:&10q(q - 1)(2q^2 - 2q + 1) &q\in\mathbb N \end{align} $$ I know of two single cases, ...
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1answer
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Squaring Polynomial over $\Bbb F_2[X]$ Is Equivalent to Squaring Argument

Thanks to some assistance below, I can now show that if $g(X) \in \Bbb F_2[X]$ then $g(X)^2 = g(X^2)$. Is there some more direct way to prove this special case (not that the original proof is ...
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0answers
27 views

the polynomial $(X^2+2)^n+5(X^{2n-1}+10X^n+5)$ is irrducible in $Z[X]$

Prove that for any postive integer $n$,the polynomial $$(X^2+2)^n+5(X^{2n-1}+10X^n+5)$$ is irrducible in $Z[X]$ I have try use Eisenstein's criterion and can't it
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Sylvester's dialectic method

Nevermind. I have got it I am willing to study Sylvester's Dialectic Method regarding polynomial. Today in one book I have come to know about it but could not find it. In MSE I tried my best to find ...
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1answer
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How to prove that $x^9-9x^7+27x^5-30x^3+9x-1$ is irreducible in $\mathbb{Q}[x]$?

The problem says: Given the irreducible polynomial $x^3-3x-1$ with root $2\cos(\pi/9)$, prove that $2\cos(\pi/27)$ is a root of a monic irreducible polynomial of degree 9 over $\mathbb{Q}$, and hence ...
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1answer
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About uniqueness of interest yield

I am not sure this belong to this site, in case I will post it elsewhere. Let $P$ be the price of a bond, let $C_k$ the promised cash flow in year $k$. Then we define the interest yield $y$ as the ...
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1answer
31 views

What is the lowest degree of a polynomial with integer coefficients which has a specific root.

What is the lowest possible degree of a polynomial $p(x)$ with integer coefficients which has one root $$x = \sum_{n=1}^k\sqrt{a_n}$$ where $1 \lt a_1 \lt \cdots \lt a_k$ are non-square integers?
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To find solutions of function and to check its roots

I did this question and found out to be answer 1,2,4 correct .But i am quite unsure .Thanks
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2answers
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Variety of an ideal

Let $F$ be a field, and $\underline{x}$ be the $n$-tuple $(x_1, x_2, ... , x_n) \in F^n$. Also, denote $F[x_1, x_2, ... , x_n]$ by $F[\underline{x}]$. Let $J$ be an ideal of ...
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1answer
24 views

Finding primes such that a given polynomial is irreducible modulo $p$

Let $f \in \mathbb{Z}[x]$ be irreducible, and let $\bar{f} \in \mathbb{F}_{p}[x]$ be the image of $f$ in the polynomial ring over the finite field with $p$ elements. Is there a general procedure, ...
3
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1answer
42 views

Sum of the roots of an equation.

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. I used ...
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Find an equation in $x$ and $k$

Find an equation in $x$ and $k$ if, $$6u-8v+2=k^2$$ $$u^{2}=1+2v^{2}$$ $$v=2xy$$ $$u=x^2+2xy-y^2$$ Since we have 4 equations, we can eliminate 3 variables. But somehow, I'm not able to find an ...
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1answer
50 views

Zeros in the polynomial ring $\mathbb{R} [X,Y]$

I know that for $p(X) \in \mathbb{R} [X]$, $a$ is a zero of $p(X) \iff (X-a)|p(X)$. But what would the statement be for $p(X) \in \mathbb{R}[X,Y]$? This question comes from an example in my ...
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1answer
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Determine the integer roots of the polynomial $P(P(x))$

Let $P(x)$ be a polynomial of degree $n\ge 5$ having real coefficients and $n$ distinct integer roots,so that$P(0)=0$,Determine the integer roots of the polynomial $P(P(x))$ Where I am: Let ...
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Continued Fraction summation representation

I have a rational fraction of the form: $$s=\frac{p_0+p_1x+p_2x^2+\cdots+p_Mx^M}{1+q_1x+q_2x^2+\cdots+q_Mx^M} $$ The paper I am reading converts this to the form: $$s = ...
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1answer
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Projective Transformations on Polynomials

Let P be a polynomial in $\Bbb{RP}^2$. (It doesn't have to be in 2 dimensions, but assume it for clarity and simplicity) For example: $P=3x^2+6y^2-7xy+10y-1$ Now, let's perform a projective ...
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1answer
26 views

Newton polynomials

Consider the family of symmetric polynomials $\sum^n_{i=1} x_i^k\in\mathbf{Z}[x_1,\ldots,x_n]$. By the fundamental theorem on symmetric polynomials there is a unique Newton poylnomial ...
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1answer
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Determining the multiplicative group of a ring of polynomials

Let us say that we have the polynomial ring R[x]. Would it be possible to determine the order of the multiplicative group of R[x] modulo a polynomial f?
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1answer
43 views

Francis Galton's surname problem

I am reading a little bit about this problem and am somewhat confused in some of the justifications provided in my readings. So the problem is that we want to find the probability of the extinction ...
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21 views

Proving that a rational function with two variables is continuous

for some reason i'm struggling with this very basic propsition: Let $f:\mathbb R \times \mathbb R \to \mathbb R$ be a rational function. for that matter we can every assume $f(x,y)=\frac{1}{p(x,y)}$ ...
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1answer
45 views

Proving infinitely many primes (or none) for a given polynomial, e.g. $n^4+4$

I've recently started self-studying through Niven's Introduction to the Theory of Numbers and had questions on a few of the problems. In particular, I'm not sure how to show that $n^4+4$ is composite ...
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Suppose there exist infinite subsets $X_1, . . , X_n$ of F such that f($x_1, . . , x_n$) = 0 for all ($x_1, .. , x_n$) ∈ $X_1 × · · · × X_n$.

Let f($t_1, . . . , t_n$) be a polynomial over a field F. Suppose there exist infinite subsets $X_1, . . . , X_n$ of F such that f($x_1, . . . , x_n$) = 0 for all ($x_1, . . . , x_n$) ∈ $X_1 × · · · × ...
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1answer
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A Practical Guide to Splines (De Boor) - Proof of Leibniz formula

In De Boor's A Practical Guide to Splines (1978) Leibniz' formula is defined as follows (p.5): If $f = gh$, i.e. $f(x) = g(x)h(x)$ for all x, then $$ [\tau_i, ..., \tau_{i+k}]f = ...
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1answer
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Polynomial Function and Polynomials.

I've got a doubt about a ring of polynomial functions. The problem starting doing this exercise of Fraleigh (The 30). Here I had to show that $P_F$ isn't necessarily isomorphic to $F[x]$. It's easy, ...
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3answers
172 views

Polynomial division challenge

Let $x,y,n \in \mathbb{Z} \geq 3$, Find $A,B$ such that $$x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1}= A(x^2+xy+y^2)+B$$ What is the best method to approach this?
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General Discriminant Formula for Multivariate Polynomials over Reals.

Consider \begin{eqnarray} b'(I_{s}c-A)b>0 \end{eqnarray} where $A$ is a symmetric, positive definite s by s square and I is the identity and c is a constant. Solutions are to be found using $b ...
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1answer
10 views

Tchebychev's polynomial and vector spaces

This polynomial is defined by: $T_n(x)=cos(narccos(x)) \forall x \in [-1,1]$ I could prove a recurrence relation: $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ But i couldn't deduce from this that Tn is a ...
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1answer
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Polynomial maximization

If $x^4+ax^3+3x^2+bx+1 \ge 0$ for all real $x$ where $a,b \in R$. Find the maximum value of $(a^2+b^2)$. I tried setting up ...
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1answer
40 views

Is a polynomial that vanishes a nonempty open subset of $\mathbb{K}^n$, $\mathbb{K} \subseteq \mathbb{C}$, necessarily zero?

Let be $\mathbb K$ a subfield of $\mathbb C$ and consider $\mathbb K^n$ with the Euclidean topology. If $p \in \mathbb K[x_{1},...,x_{n}]$ vanishes on a nonempty open subset on $\mathbb K^n$, is it ...
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1answer
27 views

How is integer polynomial factorization reduced to factorization over a finite field?

I've read on Wikipedia that the problem of factoring polynomials over $\mathbb Z$ can be reduced to factoring polynomials over some finite field, but I can't find any information on how this is done. ...
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3answers
37 views

Turn iterative function into polynomial.

So, I have an iterative function that looks something like this. $$f(x_n) = (x_n + 0.08) \cdot 0.98$$ e.g. So if $n = 2$ and $x$ started at $0$, then the equation would be equal to $(((0 + 0.8) ...
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1answer
32 views

Let $f(x)=(x^2-1)^n$. Prove that for $r=0,1, … ,n$, $f^{(r)}(x)$ is a polynomial with value $0$ at no fewer than $r$ distinct points on $(-1,1)$.

Let $f(x)=(x^2-1)^n$. Prove that for $r=0,1, ... ,n$, $f^{(r)}(x)$ is a polynomial whose value is $0$ at no fewer than $r$ distinct points on $(-1,1)$. In other words, prove that $f^{(n)}(x)$. I ...
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2answers
66 views

Prove: p-mq | f(m) where 'm' is any integer

How to prove that $p-mq \mid f(m)$ where $m$ is any integer, $f(x) = A_0 + A_1 x + A_2 x^2 + ... + A_{n-1} x^{n-1} + A_n x^n$, $f(x)∈ ℤ[x]$, $p/q$ is a zero for $f(x)$ and $p$ and $q$ are coprime ...
3
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0answers
28 views

Find the degree of the polynomials in the following groups

Let $f(x) = x^4 + 6x^3 + 15x^2 + 10x + 1$ and $g(x) = 2x^2 + 15x + 1$. Consider $f$ and $g$ as polynomials with coefficients in (a) $\mathbb Q$, (b) $\mathbb F_2$, (c) $\mathbb F_3$, and (d) $\mathbb ...
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Set of polynomials of degree less or equal than $n$ is equicontinuous (or compact) over every interval $[a,b]$ using Arzela-Ascoli theorem

Define $\Pi=\{\text{polynomials of degree }\le n \text{ over } [a,b]\}$ with fixed $n$. Norm is $\|f\|=\sup_{x\in D(f)}|f(x)| $ I am trying to proof that this set is equicontinuous using ...
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1answer
17 views

Find all solutions to the congruence relation

Let $p$ be a prime and $d$ is a divisor of $p-1$ Let $a$ be an integer that is not divisible by $p$, and suppose $a$ has order $d \pmod p$. List all solutions to $x^d -1 \equiv 0\pmod p$ My attempt ...