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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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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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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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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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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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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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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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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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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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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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 ...
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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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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 ...
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Expression for polynomial

I wonder if it is possible to find a closed form expression for following sequence: \begin{equation*} C_1=1 \end{equation*} \begin{equation*} C_2=x^2+\frac 32 \end{equation*} \begin{equation*} ...
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Qusetion about Zero divisors in a polynomial rings

Let $x^4-16$ be an element of the polynomial ring $E= \mathbb{Z}[x]$ and use the bar notation to denote passage to the quotient ring $\mathbb{Z}[x]/(x^4-16)$. Prove that $\bar{(x-2)}$ and ...
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Modified version of Eisenstein's irreducibility criterion

I have an assignment to extend/modify (and of course prove it) Eisenstein's criterion as follows: Let $f(x)=\sum a_ix^i\in\mathbb{Z}[x]$ with $n\ge 2$ and let $p$ be a prime such that $p\mid a_i$ for ...
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34 views

General formula for iterated cumulative sum

Consider the sequence $S_0$ consisting of ones: $$ 1,1,1,1,1,1,\ldots $$ Now compute the cumulative sum of this sequence, and call the resulting sequence $S_1$: $$ 1,2,3,4,5,6,\ldots $$ Proceed ...
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Determining polynomial from roots of another polynomial

I am working on an exercize and I know how to more bruteforcely solve it through pure algebra in its simplest form, but it's such a massive mess to demonstrate so I would like to see if there is ...
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Product of Polynomials in Several Variables?

Let $p$ and $q$ be the polynomials $\mathbb R$ given by: $$p(x)=\sum_{j=0}^m a_j x^j\quad \textrm{and}\quad q(x)=\sum_{j=0}^n b_j x^j.$$ We know that $$p(x)\cdot q(x)=\sum_{j=0}^{m+n} ...
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Separability of $X^{q^d}-X\in\mathbb F_q[X]$ for $d\in\mathbb N$

I know that I can verify that $f:=X^{q^d}-X\in\mathbb F_q$ and $f'$ (the formal derivative) are coprime in order to establish $f$'s separability. Is there an easier way, in particular one that does ...
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19 views

Transposing formula, possibly polynomial

I'm working on a game which I would like to follow behaviour of an already existing game. Unfortunately they have an odd way of calculating a players the xp(x) requirement for a level(y). When y is 1 ...
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Limits that require polynom actions?

i have encountered this example one day in the exam and i could not solve it. The tip that professor gave me was x^3-2x-4 / x-2 But yet i could not understand it, nor did i know how to start it. ...
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Product of Sums: Show that the following is a Polynomial by converting it into standard form. [duplicate]

$$\prod_{k=0}^n (1+x^{2^k})$$ The given expression simplifies to $(1+x)(1 + x^2)...(1 + x^{2^n})$ I am not able to proceed further. How do I express this in Summation form?
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If $c_{n} > 0$ then $\sum_{0}^{n}c_{k}x^{k} > 0$ for some $x \in \mathbb{R}$?

Let $n \geq 1$ be an integer and let $c_{0}, \dots, c_{n} \in \mathbb{R}$. If $c_{n} > 0,$ is there necessarily an $x \in \mathbb{R}$ such that $$\sum_{0}^{n}c_{k}x^{k} > 0?$$ I just realized ...
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The root of a monic polynomial with algebraic coefficients.

Let α be a complex number that satisfies α3 + βα2 + γα + δ = 0 β, γ, and δ satisfy cubics with rational coefficients. For example, β satisfies β3 + aβ2 + bβ + c = 0. However, it is not stated that ...
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Finding length and width from depth using factors of a cubic equation?

So I have this application question: A pool designer is creating a pool with dimensions of length width and depth that must have specific relationships amongst their scale. Because the design ...
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21 views

Polynomial to polynomial function on (in)finite field [on hold]

Let K be a field. Prove that a transformation K[x]->(polynomial functions K->K) is injective if and only if K is an infinite field. How do I approach it? It's probably a very simple problem cause ...
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Etingof Problem 5.1, “Field embeddings”

Recall that $k(y_1, \dots, y_m)$ denotes the field of rational functions of $y_1, \dots, y_m$ over a field $k$. Let $f : k[x_1, \dots, x_n] \to k(y_1, \dots, y_m)$ be an injective $k$-algebra ...
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Proof of the Computability of Polynomials

In studying properties of polynomial functions I have read that they are computable. The usage of the word read implies that I cannot prove this statement, and withhold using learned for this reason. ...
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Why are the following two statements equivalent?

I was reading some time series material and I came across this: the condition for causality is that $1 - \phi_1 z - ... - \phi_pz^p \neq 0$ for all $|z| \leq 1$ ,i.e. the zeros / roots of the ...
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Confused about Finite fields and polynomials

I'm asked to give a polynomial that has a root over a finite field but not a root over R. My understanding is that the finite field is contained in R (more restrictive) so how can there be a root in ...
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48 views

Show $ x^4 + x^2 + 1 $ has no integer roots, but that it has a root modulo 3 and factorize it

Show that the following polynomial $ x^4 + x^2 + 1 $ has no integer roots, but that it has roots modulo 3, and factorize it over $ℤ_3$. I'm not sure how to go about this problem. Thank you for your ...
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Solution to $x^\alpha + p x = q$?

I was wondering if there was any tricks, similar in spirit to the Vieta's substitution, that would apply the equation $$ x^\alpha + p x = q, $$ where $p,q$ and $\alpha$ are real constants. In ...
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if $x^3-x\in\mathbb{Z}$ and $x^4-x\in\mathbb{Z}$ for some $x\in\mathbb{R}$, then $x\in\mathbb{Z}$.

Assume that $x^3-x\in\mathbb{Z}$ and $x^4-x\in\mathbb{Z}$ for some $x\in\mathbb{R}$. Prove that $x\in\mathbb{Z}$. my attempt: Let $a=x^3-x$ and consider polynomial $X^3-X-a$, then $x$ is a root of it ...
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Find the roots of the polynomial $ 3X^3 -32X^2+73X +28$ using Vieta's relations

They are asking me to find the roots of the polynomial $ 3X^3 -32X^2+73X +28$ using Vieta's relations. They also tell me that $x_1 - x_2 = 3$. I have tried to use first Vieta's relation($x_1 + x_2 + ...
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Approximate a positive multivariate function with a sum of squares of polynomials?

I am constructing approximation to a multivariate function which I know is positive. My question is the following: Let $f(x)$ be a multivariate positive and continuous function. Can we approximate ...
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Characteristic polynomial of recurrence relation $\lambda^4 + \lambda^3 - 9\lambda^2 + 11\lambda -4 = 0$

The characteristic polynomial of this recurrence relation is $$λ^4 + λ^3 - 9λ^2 + 11λ- 4 = 0$$ or $$(λ − 1)^3 \cdot (λ + 4) = 0.$$ So the solution is of the form $a_n = α({-4})^n + β~n^2 + γ~n + ...
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Meaning of derivatives

I need to know the meaning of the higher order derivatives of a polynomial. Let make an example. Assume we have a polynomial of degree n. Then $$ f(x)=a_0+a_1x+a_2x^2+\ldots+a_nx^n $$ We know that ...
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unique real - integer polynomial

If $ f(x) = x^{10} + 2x^9 - 2x^8 - 2x^7 + x^6 + 3x^2 + 6x + 2014 $ so can anyone here proof $f(\sqrt[2]{2} -1) = 2017$ Please do it with hands not by computer help or calculator help
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1answer
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Why multivaraite positive polynomials cannot be written as sum of squares?

It is wellknown that a positive univariate polynomial $p(x)>0$ for all $x\in R$, can be written as a sum of squares: $p(x) = \sum_{i=1}^n q_i^2(x)$, and I found references saying (without any ...
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Induction, Polynomials, Proof [closed]

Assume that there is a polynomial $p$ of degree 5 such that $$ \sum_{n=0}^N n^4 \;\; =\;\; p(N). $$ Find $p$ and prove that the formula you propose is correct.
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In $R[x]$, $f=g \iff f(x)=g(x), \forall x \in R$

Let $R$ be an integral domain and $R[x]$ the polynomial ring over $R$. Let $f,g \in R[x]$ such that $\max(\deg f, \deg g)< \#R$. Show that $f=g \iff f(x)= g(x), \forall x \in R$. $\bf Attempt:$ ...
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About roots of multivariable complex polynomials.

We have a function $f : \mathbb{C}^2 \rightarrow \mathbb{C}$ such that, $f(z_1,z_2) = \prod_{i} (z_1 - a_i) = A(z_2-b)(z_2-c) $ where $a_i$ are known to be real. Now say $T$ is an operator which ...
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Nonnegative vs SOS

Consider the polynomial $f(x_1, \cdots, x_n)$, I want to characterize $f$ being nonnegative, i.e., $f\geq 0$. For $n=1$, this is equivalent to saying that $f$ is SOS (sum of square). However, in ...
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26 views

Polynomials/Trinomial Word Problem

I have no idea how to do this: The product of two consecutive odd integers is 143. Find their sum. We're learning about factoring quadratics, trinomials, polynomials, etc. I haven't seen this ...
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33 views

How to find a subset that contains all linearly independent polynomials?

I found that a set S is linearly independent. How can I find a subset A of S that contains all linearly independent polynomials? My set S consists of the following polynomial vectors in P3: pv1 = ...
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44 views

Solving for $z^2 = x^2 -xy + y^2$

Recently, I came across the following solution to finding integer solutions for $z^2 = x^2 - xy + y^2$: $x = k(-n^2 -2mn)$ $y = k(m^2 - n^2)$ $z = k(mn + m^2 + n^2)$ I've been scratching my head ...
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About multivariable quadratic polynomials

Say one has a polynomial function $f : \mathbb{C}^n \rightarrow \mathbb{C}$ such that it is quadratic in any of its variables $z_i$ (for $i \in \{ 1,2,..,n\}$). Then it follows that for any $i$ one ...
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Find the inverse function of $y = g(x) = 6 x^3 + 7$: $g^{-1}(y) =?$

The question states, Find the inverse function of $y = g(x) = 6 x^3 + 7$, $g^{-1}(y) =?$ I have tried setting the equation to $y$ and then solving for $x.$ This resulted in the answer ...