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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1answer
30 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 ...
-1
votes
1answer
25 views

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. ...
3
votes
2answers
226 views

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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2answers
19 views

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 ...
0
votes
1answer
79 views

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 ...
0
votes
1answer
49 views

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 ...
-1
votes
1answer
51 views

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

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 ...
2
votes
2answers
103 views

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 ...
2
votes
1answer
57 views

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. ...
1
vote
2answers
42 views

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 ...
0
votes
2answers
37 views

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 ...
0
votes
1answer
69 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 ...
2
votes
2answers
85 views

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 ...
5
votes
2answers
115 views

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 ...
2
votes
1answer
48 views

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 + ...
2
votes
1answer
87 views

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 ...
0
votes
1answer
51 views

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 + ...
0
votes
3answers
54 views

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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votes
4answers
53 views

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
1
vote
1answer
46 views

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 ...
3
votes
1answer
76 views

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 $f=g \iff f(x)= g(x), \forall x \in R$. $\bf Attempt:$ ...
2
votes
0answers
23 views

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 ...
0
votes
1answer
100 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 ...
0
votes
1answer
244 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 = ...
0
votes
1answer
53 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 ...
1
vote
2answers
39 views

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 ...
1
vote
1answer
63 views

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 ...
0
votes
1answer
42 views

Proving that sum of two form is irreducible

let $f_r$ an homogenous form of degree $r$ and $f_{r+1}$ a form of degree $r-1$ without commun factors, in $k[x_1, \dots, x_n]$. I want to show that the sum $f = f_r + f_{r+1}$ is irreducible. ...
0
votes
3answers
35 views

Root of real and complex polynomial

Let $z \in \mathbb{C}$ be a root of real polynomial $p(x)=\sum_{k=0}^{n} a_k x^k$ ,$a_k\in \mathbb{R} \forall k$. How to proove, that $\overline{z}$ is also a root of given polynomial? Is that true ...
2
votes
1answer
102 views

Which polynomials with binary coefficients evaluate only to 0 or 1 over an extension field?

Consider the polynomial $p(x) = 1+x^5+x^{10}$ with binary coefficients. Consider the multiplicative group of $\mathbb{F}_{16}$, and let $p(x)$ be evaluated at each of these $15$ elements. The only ...
0
votes
1answer
75 views

weierstrass approximation theorem and polynomials

Let $f$ be a continuous function on $[0,1]$. Show that there exists a polynomial $p$ such that $sup |p-f|<\epsilon$, and $p'(0) = p'(1)= 0$. The uniform convergence part comes from Weierstrass ...
2
votes
2answers
190 views

Proof of Weierstrass Theorem from Rudin's-Multiple questions though the proof

I am studying Stone-Weierstrass theorem. I am not understanding well and I have a multiple point that I need some help to have better understanding. In the proof, he set $Q_n(x)=C_n (1-x^2)^2$ ...
0
votes
2answers
67 views

Prove $ \lim_{n->\infty}\sqrt[n]{p(n)}=1 $ for polynomial

Let $$ p(x)=\sum_{k=0}^{d}a_kx^k $$ polynomial such that $$ \forall x>0, p(x)>0 $$ Prove that: $$ \lim_{n->\infty}\sqrt[n]{p(n)}=1 $$
1
vote
1answer
49 views

pointwise convergence of polynomials of degree 2 with 2 real roots to $x^2+1$

In some problem, I thought I need to prove the following subproblem : Let $\{p_n(x)\}_{n\in\mathbb{N}}$ be a sequence of polynomials of degree 2 which converges to $p(x)=x^2+1$ point by point, i.e ...
1
vote
1answer
84 views

Subring generated by $x$ is an integral domain iff it is a field iff the minimal polynomial of $x$ is irreducible

Let $R$ be a ring, $K$ a subfield of $R$, and $x \in R$. Let $F(X)$ be the minimal polynomial of $x$ over $K$. I want to prove that: $K[x]$ is a field $ \iff K[x]$ is an integral domain $\iff ...
0
votes
1answer
30 views

How are the (syntactic) differences in these two definitions of the characteristic polynomial of a square matrix explained?

I’m used to defining the characteristic polynomial of a square matrix as follows: $$p_A(x) = det(xI_n - A)$$ where $A$ is a square matrix over a field and $I$ is an $n \times n$ identity matrix. The ...
1
vote
1answer
113 views

Backward differences Newton Polynomials question

Use Newton's backward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. f (-1/3) if f(-0.75) = -0.07181250, f(-O.5) = -0.02475000, ...
3
votes
1answer
96 views

Proving that $X^4+1$ is reducible over all finite prime fields

In that article, I prove that the polynomial $X^4+1$ is reducible over all finite prime fields of odd characteristic. The proof is based on the fact that for $p$ odd prime, the multiplicative group ...
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votes
0answers
49 views

Different forms of lagrange theorem

I know the proof of the original Lagrange Theorem which states that: Given a group $G$ the order of its subgroup $H$, divide its order. There are two more statements related that I do not know ...
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votes
2answers
29 views

Derivatives of symmetric expressions

So I was bored in math class and came up with this series of related questions, that I cannot answer: Is there a clean expression for $f'(x),$ where $$f(x)=\prod_{i=1}^{n}\dfrac{(x-i)}{(x+i)}?$$ ...
1
vote
3answers
44 views

Why does F(X) divide F'(X)

Let $K$ be a field of characteristic zero or a finite field, let $F(X) \in K[X]$ be a monic irreducible polynomial, and let $F(X)= \prod_{i=1}^n(X-x_i)$ be its decomposition in an extension $K'$ of ...
0
votes
1answer
194 views

finding roots of polynomial in MAGMA or GAP

Please hint me. I want to find the roots of $x^3+(1-2n)x^2-(1+4n)x+8n^2-10n-1$ with Magma or GAP. In Magma: $P<x> := PolynomialRing(Rationals())$; $P<n>:= ...
3
votes
1answer
75 views

On a problem of Schinzel about reducibility of polynomials!

Let $ f(x)=5x^9+6x^8+3x^6+8x^5+9x^3+6x^2+8x+3 $. Prove that $x^nf(x)+12$ is reducible in $\mathbb Z[x]$ for any positive integer $n$. This problem is due to polish mathematician Andrzej Schinzel.
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5answers
305 views

Expressing in terms of symmetric polynomials.

How to express $$a^7+b^7+c^7$$ in terms of symmetric polynomials ${\sigma}_{1}=a+b+c$, ${\sigma}_{2}=ab+bc+ca$ and ${\sigma}_{3}=abc$ ?
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0answers
53 views

Assuming $m \ge 3$, under what conditions is $z^{m-1} = \frac{x^m + y^m}{x + y}$?

One case where this is true is $z=1$: $$(1)^{m-1} = \frac{1^m + 1^m}{1+1}$$ Another case is where $x=y$: $$(x)^{m-1} = \frac{x^m + x^m}{x+x}$$ Assuming that $x,y,z,m$ are positive integers with $z ...
2
votes
1answer
58 views

polynomials such that $P(k)=Q(l)$ for all integer $k$

In a book I have read this problem: Given $P\in \mathbb{R}[X]$, if $P(X)$ takes at every integer, a value which is the $k$-th power of an integer, then $P(X)$ itself is the $k$-th power of a ...
3
votes
1answer
205 views

Bezier curve polynom coefficients

How can I calculate coefficients for bezier polynom? I can do this manually on the paper, but I need to plug this into program, where degree of polynom can be higher than 3 ( more than 4 control ...
0
votes
0answers
68 views

Minimize multivariate (multivariable) polynomial over the integers

I'd like to minimize the following polynomial in 6 variables $h_0,h_1,g_0,g_1,g_2,g_3$: $$ g_3^2\cdot h_0^3\cdot h_1^3 - g_2\cdot g_3\cdot h_0^2\cdot h_1^4 + g_1\cdot g_3\cdot h_0\cdot h_1^5 - ...
0
votes
1answer
40 views

What does $a$ is a root of $f(x)$ $\text{modulo}\; \mathfrak p$ mean?

I have a ring $A$ and a prime ideal $\mathfrak p$ and a polynomial $f(X)\in A[X]$ and also $a\in A$. I have read in several books phrases like "$a$ is a root of $f(x)$ $\text{modulo}\; \mathfrak p$" ...
0
votes
0answers
51 views

Substituting for the variable in a standard taylor series

I'm trying to show that 4/(3x+1)=1/(1+(3/4)(x-1)) so i can then find the taylor series for the function f(x)=4/3x+1 and determne an interval of validity I know 4/3x+1 is similar to 1/(1-x) for which ...