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Questions tagged [polynomials]

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 and checking for irreducibility.

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1answer
34 views

Irreducible polynomials for that each output is divisible by an integer n

Feel free to delete this question if it has been asked somewhere else before. I've recently stumbled upon this question on the Mathematics StackExchange and I've wondered how the polynomials for ...
4
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1answer
45 views

Is it possible to graph complex zeros of a polynomial?

I am sorry if this question is a complete nonsense, but keep in mind that I am a senior in high school, so my math knowledge is really low. My question is, can you graph complex zeroes on a three ...
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0answers
11 views

Tough test polynomials for (finite precision) complex root finding methods, especially Aberth's method

Today I have implemented Aberth's method for complex polynomial root finding. And I have to say I am enchanted about its astonishing performance and its intriguing simplicity. Before I go on believing ...
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0answers
20 views

Polynomials in the Pancake problem

I noticed something interesting in this table. The columns can be expressed by polynomials of order k. I can't check if it is still a polynom for $k=7$. $$k=0: 1$$ $$k=1: n-1$$ $$k=2: n^2-3n+2$$ $$k=3:...
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0answers
11 views

algebraic field extensions and polynomials

Let $K$ be a field and $f$ be a polynomial, whose coefficients are algebraic over $K$. If we have a factorization $f=gh$ and the highest coefficient of $g$ is algebraic over $K$, is it true that the ...
2
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2answers
20 views

GCD of $f=X^3 +9X^2 +10X +3$ and $g= X^2 -X -2$ in $\mathbb{Z}/5\mathbb{Z}$.

I have to calculate the gcd of $f=X^3 +9X^2 +10X +3$ and $g= X^2 -X -2$ in $\mathbb{Q}[X]$ and $\mathbb{Z}/5\mathbb{Z}$. In $\mathbb{Q}[X]$ I got that $X+1$ is a gcd and therefore $r(X+1)$ since $\...
2
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4answers
41 views

Long division of $\frac{3x^3-x^2-13x-13}{x^2-x-6}$

I'm self-studying from Stroud & Booth's amazing textbook "Engineering Mathematics", and am on the "Partial Fractions" chapter. As part of an exercise I need to do long division of two polynomial ...
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0answers
10 views

How to analytically calculate a specific pseudoinverse for multinomial root finding

Consider the following problem: $$d_1=a_{1,1}x+a_{1,2}y+a_{1,3}x^2+a_{1,4}2xy+a_{1,5}y^2$$ $$d_2=a_{2,1}x+a_{2,2}y+a_{2,3}x^2+a_{2,4}2xy+a_{2,5}y^2$$ and suppose we can represent it as follows: $$\...
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1answer
48 views

Show that $\frac{x(x-1) \dots (x-n+1)}{n!} \in \mathbb{Z}$ with $x \in \mathbb{Z}$ [duplicate]

Problem: Let polynomial $Q_n (x) = \frac{x(x-1) \dots (x-n+1)}{n!} \in R[x]$ for some ring $R$. Show that $\forall t \in \mathbb{Z}, Q_n (t) \in \mathbb{Z}$. My solution: For each $t \in \mathbb{Z}$, ...
2
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2answers
50 views

If $F\subseteq K$ are fields, $\alpha \in K$ Prove $\alpha$ is algebraic over $F$

If $F\subseteq K$ are fields, $\alpha \in K$, and $K$ is an extension field of $F$. Prove the following are equivalent: $\alpha$ is algebraic over $F$ $F(\alpha)=F[\alpha]$ $|F(\alpha):F|$ is ...
2
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1answer
29 views

Show that any complex polynomial of degree $n \ge 2$ has finitely many neutral or attracting orbits

Let $f: \mathbb{C} \to \mathbb{C}$ be a complex polynomial of degree $n \ge 2$. A point $w \in \mathbb{C}$ is a periodic point of $f$ of minimal period $p$ if $f^{\circ p}(w) =w$ and $p$ is the ...
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1answer
29 views

For complex polynomial $f(z)$ show there are finitely many $w$ such that $f(v) = w \neq v$ implies $f'(v) =0$.

Let $f(z) : \mathbb{C} \to \mathbb{C}$ be a complex polynomial of degree $n \ge 2$. Consider the set $$A = \{w \in \mathbb{C} : \text{there exists } v \neq w \text{ with } f(v) = w \text{ and } f'(v)...
0
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1answer
29 views

Non-isomorphic graphs with same Tutte polynomial

I've been looking for some non-isomorphic graphs with the same Tutte polynomial. I'm aware of this thread and this thread, however my understanding of matroids is non-existent, and they are a bit ...
1
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6answers
92 views

real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is

Number of real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is Plan Put $x^2+6x+7=f(x)$. Then i have $f(f(x))=x$ For $f(x)=x$ $x^2+5x+7=0$ no real value of $x$ For $f(x)=-x$ $x^2+8x+7=0$...
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2answers
69 views

Find the value of $\alpha^{\frac13}+\beta^{\frac13}$

If $f(x)=x^2-5x+8, f(\alpha)=0$ and $f(\beta)=0$ then find the value of $\alpha^{\frac13}+\beta^{\frac13}$ $$\alpha+\beta=5$$ $$\alpha \beta=8$$ $$\alpha^{\frac 1 3}=\frac 2 {\beta^{\frac 1 3}}$$ ...
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1answer
21 views

$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in $ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$

This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...
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3answers
30 views

Prove The Derivative Rules in the Ring of Polynomials

Let R be a commutative ring with unity element 1. Let $f(x)\in R[x]$ and define its derivative as $f'(x)=r_1 +2(r_2)x+...+n(r_n)x^{n-1}$. Prove that $(f+g)'(x)=f'(x)+g'(x)$ and that $(fg)'(x)=f'(x)g(x)...
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0answers
34 views

When does $\sum_{k=0}^M a_k x^{2k}$, $a_k \in \mathbb{R}$ have no roots in $[0,1]$?

When does $\sum_{k=0}^M a_k x^{2k}$, $a_k \in \mathbb{R}$, $M \in \mathbb{N}$ have no roots in $[0,1]$? There is nothing special about the $1$, the question can be generalized to $[0,c]$ but that ...
2
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1answer
36 views

A polynomial whose roots form an arithmetic progression

Let $f$ be a fourth degree polynomial whose roots form an arithmetic progression. Prove that $f'$'s roots also form an arithmetic progression. I didn' t make much progress, I just wrote $f(x) =a(x-b-r)...
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0answers
16 views

Coloring of hypergraph with polynomial implication

Sorry to bother again with my misunderstandings, but I encountered yet again an issue with the topic of coloring in graphs and again I require some help to deal with this exercise: Let $\mathbb{F}$ ...
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0answers
32 views

Dealing with a polynomial sum

I am trying to approximate the following function, $$f(x)=\frac{\sum^{N}_{c=1}\gamma_c x^{c+k-2}-\sum^{N}_{c=1}\beta_c x^{c-1}}{\sum^{N}_{c=1}\alpha_c x^{c}}$$ where $\alpha_c$'s, $\beta_c$'s and $\...
1
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0answers
31 views

Polynomial question related to distinct roots [on hold]

If $a$, $b$ and $c$ are distinct real numbers and $$\frac{a^3 - b^3} {a+b} + \frac{b^3-c^3}{b+c} + \frac{c^3-a^3}{c+a} = m\left[ \frac{1}{c} \left(\frac{a-b}{a+b}\right) + \frac{1}{a} \left( \frac{b-c}...
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5answers
24 views

Which of these collections of elements span $P_2(\mathbb{R})$

Consider the $\mathbb{R}$-vector space $P_3(\mathbb{R})$ of real polynomials $$a_0+a_1X+a_2X^2,$$ of degree $\leq 2$. Which of the following collections of elements span $P_3(\mathbb{R})$? $1,X,X^2$ $...
0
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1answer
40 views

How do I produce a Bézier curve with 4 points and without just using a website?

I have these for points, (1,-2), (4,3), (12,3) and (15,-3), and I was wondering how do I make a model out of this, and how would this relate to polynomials and Pascal's triangle? I have this formula: ...
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5answers
43 views

Proof strategy polynomial division

I'm given the following polynomial $$ p(x) = x^4 -2x^3 +6x^2-10x+5 \in \mathbb{Q}[x] $$ I need to prove that $(x-1)^2|p(x)$. Which are the possible ways to prove that? One way is to do the ...
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0answers
24 views

Number of distinct roots between complex roots among three polynomials

I want to prove that any three relatively prime polynomials $A, B, C \in \mathbb C[X]$ verifying $A+B+C=0$ have at least $1+\max(\deg A, \deg B, \deg C)$ distinct roots in total among each other. I ...
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0answers
28 views

Where can I study polynomials online?

I have one good book on polynomials. It is called (Level of Knowledge of Polynomials) by its author S. L. Tabachnikov. In principle, I could study it, but I am still mastering the Agebra I. Not long ...
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0answers
10 views

Set of roots Hermite polynomials(probabilistic type)

What is the nature of the set of all roots of the Hermite polynomials? They’re known to be all real. Is it a dense set? If not, what are the limit points?Are the limit points also roots? Are the limit ...
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2answers
45 views

Count number of roots of polynomial modulo prime power

I found this problem in a number theory course, I am assuming (but not sure) it is supposed to be an application of Hensel's lemma. For every $n \in \mathbb{N}_0$, determine the number of solutions ...
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2answers
44 views

Show that $e^{x}=\left | P(x) \right |$ has at least one real solution

We have $P$ a polynomial that is not identically $0$. Show that $e^{x}=\left | P(x) \right |$ has at least one real solution. I got $f(x) = e^{x}- \left | P(x) \right |$, which is continuous. I need ...
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0answers
47 views

$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$

$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$ I think $p$ is supposed to be a prime for the only ...
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0answers
25 views

Number of solutions of a polynomial over $\mathbb{Z}/5^k\mathbb{Z}$

I have to determine the number of solutions of the polynomial $y^2 = x^3 + x^2 + 5$ over $\mathbb{Z}/5^k\mathbb{Z}$ for every $k \in \mathbb{Z}$, with $k \geq 1$. For $k=1$, the solutions are given ...
0
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1answer
31 views

Show that $\{x^4+x^3-1,x^3-x,x^2+1\}⊂E$ is a generating set of span($E$).

I'm given the following set of polynomials: $$E:=\{x^4+x^2+x,x^4+x^3-1,x^3-x,x^2+1\}$$ I know that $E$ is linearly dependent because when $\alpha_1 = -1$, $\alpha_2=1$, $\alpha_3=-1$ and $\alpha_4=1$...
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2answers
49 views

How to determine the base of $\ker\phi$ for polynomial function?

Given is a base defined as $$B:=(x\mapsto1,x\mapsto x,x\mapsto x^2,x\mapsto x^3 ,x\mapsto x^4)$$ A set V defined as $$V:= \{ f: \mathbb{R} \mapsto \mathbb{R}\ |\ \exists\ {a_0},...{a_4} \in \mathbb{R}\...
4
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0answers
65 views

$a$ and $b$ are solutions of $ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $, $a+b=?$

$a$ and $b$ are solutions of $$ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $$ What is $a+b=?$ $$ $$ Are there better approaches than the one below? Solution: ...
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4answers
36 views

Polynomial factorization in $\mathbb{R}$ and $\mathbb{Z}_{[n]}$

I've the following polynomial: $$ a(x) = x^6 + x^5 + 2x^3 - 3x^2 +x -2 \in \mathbb{K}[x] $$ Set $\mathbb{K} = \mathbb{R}$. A factorization of $a(x)$ is: $$ a(x) = (x^2 + 1)^2(x-2)(x+1) $$ Now set ...
1
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1answer
44 views

A polynomial coefficient

Let $P(x):=1+a_1x+a_2x^2+\cdots+a_nx^n$, then $$(P(x))^m=1+c_1x+c_2x^2+\cdots+c_{mn}x^{mn},$$ how to find the coefficient $c_j$?
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0answers
27 views

Finding the real roots of a univariate polynomial on the interval [0,1]

I have numerous, univariate polynomials with degree in excess of 100 and with very, very large coefficients (Here's an example coefficient ...
1
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2answers
49 views

Zero of a polynomial and divisbility

I have a polynomial $p(x)$, $deg(p) = n$. I know that $\alpha$ is a zero of $p(x)$. Then $(x-\alpha)|p(x)$. Is it wrong to say that $(x-\alpha)^m|p(x)$, $m \in \mathbb{N}, m>1 $?
4
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1answer
38 views

Degree 2 Recurring monic polynomials

Consider a monic polynomial $x^2+ax+b=0$, with real coefficients. If it has real roots $p$ and $q$, such that $p\leq q$, then you construct a new monic polynomial as $x^2+px+q=0$. If this polynomial ...
1
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1answer
27 views

Number of ring homomorphism from $\Bbb Z[x,y]$ to $\Bbb F_2[x]/(x^3+x^2+x+1)$

Find the number of ring homomorphism from $\Bbb Z[x,y]$ to $\Bbb F_2[x]/(x^3+x^2+x+1)$. Now I have observed that $(x^3+x^2+x+1)=(x+1)^3$ in $\Bbb F_2[x]$. Then $\Bbb F_2[x]/(x^3+x^2+x+1)=\Bbb F_2[x]/(...
1
vote
0answers
15 views

Division of the leading terms of numerator and denominator in “Polynomial long division” algorithm [duplicate]

I am trying to understand why polynomial long division algorithm works. I have found an answer on Quora, please follow the link: Why does polynomial long division work? As you can see the explanation ...
0
votes
0answers
17 views

Boundedness of a general rational function

Is a rational function of polynomials (with real coefficients) $f:\mathbb{R} \to \mathbb{R}$, where $f(x)=\frac{q(x)}{p(x)}$, bounded above for all $x \in \mathbb{R}$ if the polynomial $p(x)$ has no ...
0
votes
1answer
13 views

Classification of image (in interval) of polynomial (non constant)

There is something I am trying to prove: Let $f:\mathbb R\to\mathbb R$ be a nonconstant polynomial. Show that the image of the function is either the real line, $[a, \infty)$, or $(-\infty,a]$ for ...
1
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3answers
30 views

How to find basis of linear subspace $V$ when $V$ contains all polynomials with the degree up to 4?

I have been given the following definition $V := \{ f : \mathbb{R} \to \mathbb{R} \mid \textrm{ there are } a_0,\ldots,a_4 \in \mathbb{R}$ and $f(x) = \sum_{i=0}^4 a_i x^i$ for all $x \in \mathbb{R}\}$...
0
votes
1answer
46 views

What meaning has the derivative of the Discriminant of polynomials

Let me ask it differently: 1st example: I have the polynomial $$p(x)=x^4 - \sqrt{\frac{\epsilon}{16} + 8} \, x^3 + r\,x^2 + r\sqrt{\frac{\epsilon}{16}+8} \, x - 2r^3$$ which has 4 roots, three of ...
1
vote
1answer
35 views

Chebyshev polynomial property

I want to prove inequality (5.13) but I have a problem with (5.16). I have: $$ \sin(n\theta) = \sin\theta \cos(n-1)\theta + \sin(n-1)\theta \cos\theta = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\...
14
votes
2answers
154 views

If $x = \frac{\sqrt{111}-1}{2}$, calculate $(2x^{5} + 2x^{4} - 53x^{3} - 57x + 54)^{2004}$.

I already have two solutions for this problem, it is for high school students with an advanced level. I would like to know if there are better or more creative approaches on the problem. Here are my ...
1
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0answers
26 views

Find an integer coefficient polynomial for this number [duplicate]

At my latest exam was the following problem: Prove that $\sqrt{2}+\sqrt[3]{2}$ is irrational. The solution is to find a polynomial with integer coefficients, a root of which is $\sqrt{2}+\sqrt[3]{...
2
votes
1answer
77 views

Irreducibility of $t ^ { 4 } + t - 1$ over $\mathbb{F}_{25}$

Problem: Let $f ( t ) = t ^ { 4 } + t - 1 \in \mathbb { F } _ { 5 } [ t ]$, show that $f(t)$ has no roots in $\mathbb{F}_{25}$. I tried the following: suppose $f(t)$ has a root in $\mathbb{F}_{25}$, ...