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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Non-negative polynomials decomposition

I'm trying to understand whether the following is true: I have the polynomial $P(x)$ which is non-negative when $-1<x<1$ Is it correct that $P(x)$ could be represented by a sum with ...
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1answer
38 views

Conversion of $F(x) = {1 \choose 0} (2t^2 - 3t + 1) + {3 \choose 1} (4t - 4t^2) + {1 \choose 2} (2t^2 - t)$

I have a textbook with a calculation step that is pretty unclear to me: $$F(x) = {1 \choose 0} (2t^2 - 3t + 1) + {3 \choose 1} (4t - 4t^2) + {1 \choose 2} (2t^2 - t)$$ $$= {-8 \choose 0} t^2 + ...
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1answer
122 views

Problem: when the sum of two squares is a square

Please, I need help to solve the following problem: Let $K$ be a field with characteristic different from $2$ and $3$. Show that the following statement are equivalent: The sum of two ...
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1answer
106 views

Prove that there not real roots with $P(x)=ax^3+bx^2+cx+d, $

let $P(x)=ax^3+bx^2+cx+d,a,b,c,d\in R$, such that $$\min{\{d,b+d\}}>\max{\{|c|,|a+c|\}}$$ show that $P(x)=0$ have no real roots in $[-1,1]$
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Polynomials Question: Proving $a=b=c$.

Question: Let $P_1(x)=ax^2-bx-c, P_2(x)=bx^2-cx-a \text{ and } P_3=cx^2-ax-b$ , where $a,b,c$ are non zero reals. There exists a real $\alpha$ such that ...
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1answer
139 views

Proper map, what's wrong?

"A map $f$ from $\mathbb R^2$ to $\mathbb R^2$ is proper if the full preimage of every compact set under $f$ is compact. Prove that every complex polynomial $f$ regarded as a self-map of the plane of ...
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5answers
221 views

Polynomials - Solutions

How I can find the exact solutions of this polynomial? I can not get to the exact roots of the polynomial ... what methods occupy for this "problem"? $$x^3+3x^2-7x+1=0$$ Thanks for your help.
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1answer
115 views

Can we test whether a polynomial only takes non-negative values on the non-negative orthant?

EDIT: Feel free to replace "non-negative on the non-negative orthant" with "non-negative on a convex set, cone, or any other class of sets that includes the orthant". A popular way to establish that ...
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1answer
68 views

Polynomial whose only values are squares

Given a polynomial $ P \in \Bbb Z [X] $ such that, $ P (x)$ is the square of an integer for all integers x, is $ P $ necessarily of the form $ P (x)= Q (x)^2$ with $ Q \in \Bbb Z [X]$?
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245 views

Do there exist polynomials $f,g$ such that $\mathbb{C}[a,b,c]\le\mathbb{C}[f,g]$ for $a,b,c$ given polynomials?

I want to prove something bigger than the problem in the title and I want to create a lemma that is useful for the solution of the problem. But I am unable to prove (or give a counterexample) the ...
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2answers
540 views

Test to see if a degree $\leq4$ polynomial is factorable

I'm in the middle of a programming project and we'd like to have tests to determine if polynomials in $\mathbb{Z}[x]$ of degrees up to 4 are factorable over $\mathbb{Q}$. A test that computes the ...
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3answers
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A Topology such that the continuous functions are exactly the polynomials

I was wondering which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous if and only if it is a polynomial function $f(x)=a_nx^n+\cdots+a_0$. Obviously, the ...
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1answer
31 views

Prove that $q(a_i)\in \{a_1,…, a_n\}$

Let $p(x)$ and $q(x)$ be polynomials with rational coefficients such that $p(x)$ is irreducible over $\mathbb{Q}$. Let $a_1,..., a_n\in \mathbb{C}$ be the complex roots of $p$, and suppose that ...
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1answer
103 views

Cyclotomic Polynomials over $\mathbb Q$ and reduction modulo $p$.

Let $p$ be prime, and let $\pi : \mathbb Z \to \mathbb Z / (p\mathbb Z)$ be the canonical projection $\pi(z) = z + p\mathbb Z$. Define its extension $\pi : \mathbb Z[x] \to \mathbb Z/(p\mathbb Z)[x]$ ...
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1answer
994 views

Difference between polynomial functions and polynomials and why these two polynomial functions are equal?

Here's an excerpt from abstract algebra book that I'm reading and my question is given later: The difference between a polynomial and a polynomial function is mainly a difference of viewpoint. Given ...
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Can $(\frac{1}{4} t^{2} + \frac{1}{2} t +200) – \frac{2}{3} t$ be a polynomial?

I know that the expression inside the brackets is a polynomial but when the expression outside the brackets is combined with it, can it be considered a polynomial?
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Why are Vandermonde matrices invertible?

A Vandermonde-matrix is a matrix of this form: $$\begin{pmatrix} x_0^0 & \cdots & x_0^n \\ \vdots & \ddots & \vdots \\ x_n^0 & \cdots & x_n^n \end{pmatrix} \in ...
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4answers
95 views

Can x- ( $ \frac x 2 + \frac x 4$ ) be a polynomial?

I was confused whether it is a polynomial or not because most of the polynomials which I see don't have brackets like this.
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2answers
911 views

Product and Sum of Polynomial Roots

The ratio of the sum of the roots of the equation, $8x^3+px^2-2x+1=0 $ to the product of the roots of the equation $5x^3+7x^3-3x+q=0 $ is $3:2$. What is the value $\frac{p-q}{p+q}$? Well I found out ...
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1answer
87 views

What is the semantic of square brackets after the set denoting coefficients of polynomial?

I have the following excerpt: Unless stated otherwise, we assume all polynomials take integer coefficients, i.e. a polynomial $f \in \mathbb{Z}[{\bf y}, x]$ is of the form $$f(y, x) = a_m · x^{d_m} ...
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1answer
31 views

Can a known change in both axes of a polynomial function be used to find the value of the independent variable?

I have a polynomial function of one variable, $d = f(t)$. I have a known change in t which causes a known change in $d$. What I want to find is the total $t$. Is there a way of solving this other than ...
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1answer
124 views

Proof $d(x)=\text{GCD}(f(x),g(x))$

Question1: Proof: If $d(x)|f(x),d(x)|g(x)$, $d(x)$ is an combination of $f(x)$ and $g(x)$, then $d(x)$ is $\text{GCD}(f(x),g(x))$. My proof 1 $d(x)=u(x)f(x)+v(x)g(x)$. ...
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1answer
499 views

Define the Intersection Points of Polynomials

I am facing the following problem. Let’s consider that there are 2 points that are not known. $${(x_0,y_0) (x_1,y_1)}$$ I know that from these 2 unknown points a set of quadratics passes ...
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1answer
90 views

Help with non-linear system of equations

This system of equations $$\begin{align} xy+yz+zx & =3 \\ \\ x^4+y^4+z^4 & =3\end{align}$$ How to solve this system of equations? Any help, Plz. Thank all
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1answer
61 views

how do you find the highest common factor of two multivariate polynomials?

How do you find the highest common factor of two multivariate polynomials? I am happy to get answers that are only useful for polynomials over the real numbers, as that is what I am dealing with.
3
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1answer
105 views

Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?

Let $P(z)$ be a complex non-constant polynomial. Let $G$ be a connected component of open set $\{z : |p(z)|<1 \}$. How to prove that $G$ contains a zero of $P$? I have no idea how to even ...
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1answer
315 views

Showing that a root $x_0$ of a polynomial is bounded by $|x_0|<(n+1)\cdot c_{\rm max}/c_1$

I have doubts about the following problem (Problem 3.21 from Sipser's "Introduction to the Theory of Computation"): Let $c_1 x^n + c_2 x^{n-1} + \cdots + c_n x + c_{n+1}$ be a polynomial with a ...
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1answer
167 views

question about Laguerre polynomials

how to prove that $$L_{n+1}(x)=\frac{1}{n+1}((2n+1-x)L_n(x)-nL_{n-1}(x))$$ I see it on wikipedia but I dont know how they prove it
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0answers
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Simplify $\frac{[m+n-1]!}{[m]![n]!}$ where $[k]=x^k-x^{-k}$ and $[k]!=[2][3]…[k]$.

Adopting the notation $[k] = x^k - x^{-k} $ and $[k]! = [2][3]...[k]$ (note that $[1]$ is omitted), and letting $m,n$ be two integers greater than $1$ such that $n>m$ and $gcd(m,n)=1$, would it be ...
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1answer
87 views

Prove fact about polynomial in uncountable fields

$F$-uncountable field. $I_{i}$-ideal in $F[x_{1},...,x_{n}]$ $F^{n}=\cup_{i=1}^{\infty}V(I_{i})$   $V(I_{i})\subseteq V(I_{i+1})$ Prove that $\exists k, V(I_{k})=F^{n}$ All that I've find is that ...
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Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

The current issue (vol. 120, no. 6) of the American Mathematical Monthly has a proof by probabilistic means that $$\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k} $$ for ...
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1answer
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What does an expression $[x^n](1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1-x^4)^{-1}…$ mean?

I came across the function that describes number of partitions of $n$ (I mean partitions like $5=4+1=3+2=3+1+1$ and so on. There was defined a Cartesian product: ...
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106 views

Factoring any single-variable polynomial in $\mathbb C$

The fundamental theorem of algebra says $$ \forall p(x):\mathbb C \to \mathbb C,\ p(x) = a\prod_{n=0}^m\big(b_nx+c_n\big) $$ where $p(x)$ is a single-variable polynomial, and $\{a;m\}\cup\{\forall ...
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1answer
164 views

Streaming algorithm for polynomial fitting data?

The specific problem I'm trying to solve is: $$h_k(x, n) = \left(\frac{\alpha}{n} + 1 - \alpha\right) \sum_{i=0}^{k} c_ix^i.$$ Given $k$ and a stream of tuples $(x, n, h_k(x, n))$ (where the $x$'s ...
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0answers
79 views

prove that polynomial has root of unity

Prove that $ f=x^n\pm x^m\pm1 $ is either irreducible over rationals or has a root which is a of unity. I tried to see what if $x=|r|e^{i\phi}$ but I have no proper result.
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2answers
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Characterization of polynomial injection from Q to Q?

I want to know if we can find (or characterize) all the polynomials $f(x) \in \mathbb{Q}[x]$ that induces an injection $f : \mathbb{Q} \rightarrow \mathbb{Q}$ by evaluation. Some examples are $x, ...
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1answer
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Help with generating functions.

Background. Let $P_0(y)=2y-3$ and define recursively $$P_{n+1}(y)=4y\cdot P_n'(y)+(5-4y)\cdot P_n(y).$$ I would like to know as many properties of $P_n$ as I can. For example, it can be shown that ...
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1answer
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What is the pattern of this sequence?

I went though this pattern and I think the results might be interesting. It was a long one but I'm only showing the first five (to make things look simpler). $$0,1,a+b,a^2 + b^2 + \frac 32ab , ...
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1answer
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multiplication in GF(256) (AES algorithm)

I'm trying to understand the AES algorithm in order to implement this (on my own) in Java code. In the algorithm all byte values will be presented as the concatenation of its individual bit values (0 ...
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2answers
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If $p(2x+1)=p(x^2)$ for all $x\in\mathbb{R}$, then $p\equiv\text{const.}$

Let $p\in \Bbb{R}[x]$ (polynomial) with $\deg(p)=n$. Suppose that $p(2x+1)=p(x^2)$ for all $x\in\mathbb{R}$. Prove that $p\equiv\text{const.}$
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Formula for the $nm$th cyclotomic polynomial when $(n,m) = 1$

Let $n,m$ be coprime. I want to find a formulae for $\Phi_{n\cdot m, \mathbb Q}$. I conjecture that because $$d \mid nm \implies d \mid n \lor d \mid m,$$ that $$ \Phi_{n\cdot m, \mathbb Q} = ...
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Unital polynomial ring over a non-unital ring

If we have a unital ring $A$, the ring of polynomials over it always has a unit since $$1_{A[x]} = (1_A, 0_A, 0_A, ...)$$ is always a unit. I wonder if there are cases of polynomial rings that ...
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The vector space of polynomials

I was given a theorem: The polynomials (where $f$ and $g$ are complex polynomials of degrees $n$ and $m$) $$f(z), zf(z), \ldots , z^{m−1}f(z), g(z), zg(z), \ldots,z^{n−1}g(z)\tag{7.6.4}$$ ...
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1answer
68 views

If a monic $f\in\overline{K}[x]$ has a power $f^n\in K[x]$, where the characteristic of $K$ doesn't divide $n$, then must $f\in K[x]$?

Suppose you have a monic polynomial $f(x) \in \overline{K}[x]$, and some integer $n>1$, where $\mathrm{char}(K)\nmid n$, and $\big (f(x)\big )^n\in K[x]$. Does it imply $f(x) \in K[x]$? The ...
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247 views

polynomials equality

Why degrees are equal if polynomial are equal? Two polynomial f(x) and g(x) are equal then their degrees are equal. This is a very trivial statement and it shouldn't worry me much but it is. I get ...
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3answers
103 views

Prove that: $f$ is not a constant function on the open unit disc $\Delta (0,1)$

I have come across the following questions from my text book! However, I'm not sure how to go about answering... Any help (or hint) would be greatly appreciated. Thanks. Suppose $F=\mathbb{C}^2$ with ...
2
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2answers
90 views

polynomial congruence equations

Is there a general method to solve the following equation: Finding $f(x)$ to satisfy: $$\left \{ \begin{matrix} f(x) \equiv r_1(x) \pmod{g_1(x)}\\f(x) \equiv r_2(x) \pmod{g_2(x)} \end{matrix} ...
2
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0answers
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Analogue of Helly’s theorem for non-exact interpolation

Let $\overrightarrow{x}=(x_1,x_2, \ldots ,x_n),\overrightarrow{a}=(a_1,a_2, \ldots ,a_n)$ and $\overrightarrow{b}=(b_1,b_2, \ldots ,b_n)$ be vectors in ${\mathbb R}^n$, with $a_k \leq b_k$ for every ...
3
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1answer
133 views

Roots of $z^{2n} + \alpha z^{2n -1} + \beta ^2$

I've been looking at a problem available here. The problem is: Let $n$ be a natural number, and $\alpha$, $\beta$ nonzero reals. Show that the number of roots of $p(z) = z^{2n} + \alpha z^{2n -1} + ...