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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2answers
59 views

Is there a link between the Bunyakovsky conjecture and the Twin Prime conjecture?

Can the proof of one conjecture be considered a proof of the other conjecture? The general method of building an infinite number of prime producing quadratic polynomials was given in the link ...
-1
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0answers
23 views

How to evaluate the coefficient of power series?

For some reason(or trick), I need to calculate something like the coefficient of $X^6$ in $f(X)=\frac1{(1-X)(1-X^2)}\times\frac1{(1-X)(1-X^2)(1-X^3)(1-X^4)}$ evaluated as power series. How should I ...
3
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1answer
67 views

Irreducibility in $k((t))[y]$

Let $k$ be an algebraically closed field of char $0$ and suppose $f(y) \in k[y]$ (need not be monic). Let $t$ be an indeterminate and consider the fraction field $k((t))$ of power series ring ...
0
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0answers
13 views

Legendre polynomial related simple proof question

Given the set of orthogonal polynomials {Qi(x)}i=0 to n , a polynomial Pn(x) of degree ≤ n, can be written as: Pn(x) = a0*Q0(x) + a1*Q1(x) + · · · + an*Qn(x) for some a0, a1, . . . , an. Please help ...
2
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2answers
56 views

Prove that $ (a+b\sqrt{2})^n $ is of the form $k+l\sqrt{2}$.(a,b,k,l,n are integers; n>1)

I have previously proved it for n=1. Using induction, assume $(a+b\sqrt{2})^{x-1}$ is true; it is of the form $k+l\sqrt{2}$. for $(a+b\sqrt{2})^x$; how do i proceed from here? Binomial theorem for ...
2
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2answers
50 views

Help me to prove this statement about quadratic equations? (from Gelfand's Algebra).

$ x^2+px+q=0 ${p,q are integers; a,b are roots}. Prove $a^n+b^n$(n is any natural number) is an integer. This is the third part of the problem.I have previously proved that $a^2+b^2$ and $a^3+b^3$ ...
1
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1answer
41 views

Find all Real polynomials.

"Find all real polynomials for which $f(2) = 3, f(3) = 5$ and $f(5) = 2$." Well my first thought was, since we have three points i can determine a polynomial of second degree such that it satisfies ...
1
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1answer
46 views

Find all polynomials $P(x) \in \mathbb{Z}[X]$ such that $P(n) \mid 2^n-1 $ [closed]

Find all polynomials $P(x) \in \mathbb{Z}[X]$ such that $P(n)\mid 2^n-1 $ where $n \in \mathbb{Z}^+$ I don't have any ideals, may be it relates to number theory ?
2
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2answers
47 views

Find $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$ has $n$ roots $x_1,x_2,\ldots,x_n \le -1$ and such that $a_0^2+a_1a_n=a_n^2+a_0a_{n-1}.$

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$ have $n$ roots $x_1,x_2,\ldots,x_n \le -1$ and $a_0^2+a_1a_n=a_n^2+a_0a_{n-1}$. Find all such $P(x)$ (Poland 1990). I used Viete Theorem ...
2
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1answer
34 views

three cubic homogeneous polynomials satisfy a cubic polynomial

Question: How can we show algebraically that three cubic homogeneous polynomials in two variables satisfy a cubic polynomial of three variables? More specifically, let ...
1
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1answer
44 views

how to show the derivative of the polynomial is bounded by itself in certain space.

How to prove that for every positive integer $d$, there exists $C(d)>0$, such that: For every polynomial with degree $\leq d$, we have $\max\limits_{x\in [0,1]}|p'(x)|\leq C(d)\max\limits_{x\in ...
1
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2answers
40 views

Greatest common divisor of polynomial in Finite Field(256), AES

Have assigment and will use it as example, found solution computationaly, want to understand idea. It is about SubBytes procedure in AES, particulary about finding inverse of polynomial. Suppose we ...
1
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1answer
73 views

Non trivial solutions of a polynomial equation

In a question a user asked for a polynomial which solves $$2P(2x^2-1)=(P(x))^2-1.$$ There are two solutions I could provide, namely the two constant ones. However in the comments to my answer it has ...
3
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1answer
136 views

Find a polynomial P(X)

Find a polynomial $P(x)$ such that it satisfies $$2P(2x^2-1)=(P(x))^2-1$$ How to find all of them?
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1answer
39 views

The number of zeros of a polynomial that almost changes signs

Let $p$ be a polynomial, and let $x_0, x_1, \dots, x_n$ be distinct numbers in the interval $[-1, 1]$, listed in increasing order, for which the following holds: $$ (-1)^ip(x_i) \geq 0,\hspace{1cm}i ...
1
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1answer
23 views

Is Gershgorin bound of roots sharp?

Gershgorin circle theorem tells that the eigenvalues of a matrix $A$ lie in the union of the associated Gershgorin circles. $A=\begin{pmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 ...
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1answer
49 views

What does the notation $\mathbb R[x]$ mean?

What does the notation $\mathbb R[x]$ mean? I thought it was just the set $\mathbb R^n$ but then I read somewhere that my lecturer wrote $\mathbb R[x] = ${$\alpha_0 + \alpha_1x + \alpha_2x^2 + ... + ...
1
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1answer
23 views

Derivation of the discriminant of a cubic polynomial by algebraic manipulation.

The problem was asked before: Using Vieta's theorem for cubic equations to derive the cubic discriminant . I tried to solve it by purely algebraic manipulation but was faced with an explosion of ...
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0answers
30 views

Monic polynomial

Recently I've learned that when a given polynomial is a monic polynomial, then this polynomial root has to be a rational root. As far as I know, to figure out if a given polynomial is a monic ...
1
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1answer
54 views

What can we learn from prime generating polynomials?

Here's a simple polynomial that generates quite a few primes (not necessarily consecutive). $p(n) = n^2 + 23n + 23$ with $n=0,1,2... $ What can such polynomials tell us about primes? Thanks. ...
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0answers
17 views

Question about Karp reduction

friends. I have a curiosity about Karp reduction. What we need to do for reduction from problem X to problem Y is that 1) Transformation from Instance of problem X to Instance of problem Y can be ...
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1answer
26 views

Relation between roots an coefficients in a generic equation: $a_0+a_1\cdot x+\cdots+a_n\cdot x^n$

In a generic equation $$a_0\cdot x+a_1\cdot x^2+ a_3\cdot x^3+\cdots+a_n\cdot x^n$$ there are some relations between roots ($x_1, x_2,\ldots,x_n$) and coefficients ($a_0, a_1,\ldots,a_n$). How can i ...
0
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0answers
23 views

Condition Butterworth polynomial

My course states that a polynomial is a Butterworth polynomial when it satisfies the following condition: $|B(j\Omega)|=\sqrt {1+{\Omega}^{2\,n}}=\sqrt {1+{(\omega/\omega_p)}^{2\,n}}$ I'm really ...
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1answer
31 views

Is there closed form solution for this infinite polynomial or high-order polymonial?

The equation is as follows \begin{align} \sum_{N=1}^{\infty}P(N)x^N=Z, \end{align} where $P(N)$'s are real number satisfying $0\leq P(N)\leq 1$. Another equation is \begin{align} \sum_{N=1}^{\bar ...
1
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1answer
47 views

why there are not polynomials $p,q$ such that $\sqrt{x^2-4}=\frac{p(x)}{q(x)}$

show that there are not polynomials $p,q$ such that $$\sqrt{x^2-4}=\dfrac{p(x)}{q(x)}$$ there a book say it is clear,because if such polynomials existed,then each zero of$x^2-4$ should have even ...
1
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0answers
38 views

An ideal in a ring of polynomials and a field extension.

Let $K\subseteq L$ be fields and $I$ an ideal of $K[x_1,...,x_n]$. I want to show that $IL[x_1,...,x_n]\cap K[x_1,...,x_n] =I$. The inclusion $I \subseteq IL[x_1,...,x_n]\cap K[x_1,...,x_n]$ is ...
1
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1answer
39 views

Inverse pairing function with polynomial constituents

Many bijective pairing functions $f:\mathbb N \times \mathbb N \rightarrow \mathbb N$ exists, including polynomial ones such as the Cantor pairing function $$f(n,m) = \frac{1}{2}(n + m)(n + m + ...
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2answers
58 views

Let a polyminal : $P(x)$ is a irreducible in $\mathbb{Q}[X]$. If $x_0 \in \mathbb{R} :P(x_0)=0$ prove that $P'(x_0) \not=0$ [closed]

Let a polyminal : $P(x)$ is a irreducible in $\mathbb{Q}[X]$. If $x_0 \in \mathbb{R} :P(x_0)=0$ prove that $P'(x_0) \not=0$ Vietnam 2014 (College)
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2answers
45 views

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \tfrac{2\pi x_1}{3}+\sin \tfrac{2\pi x_3}{3}=2\sin \tfrac{2\pi x_2}{3}$. Find $a$.

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \dfrac{2\pi x_1}{3}+\sin \dfrac{2\pi x_3}{3}=2\sin \dfrac{2\pi x_2}{3}$. Find $a$ (Bulgari 1998)
6
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4answers
214 views

How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials?

How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials? The link I've given to the theorem uses elaborate wordings using 'rings', ...
0
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1answer
36 views

Proving a linear transform defined by an integral is injective

Let the fact that $I(p)(x)=\int_0^x p(s) ds$ is a linear transform from $P_4\rightarrow P_5$ be given. Prove that $I$ is injective. Would it be sufficient to just state that for any 2 ...
5
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0answers
83 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
2
votes
2answers
31 views

Existence of polynomials $g$, $h$, with non-negative coefficients, such that $f(x)= \frac{g(x)}{h(x)}$ [closed]

Suppose $a$ and $b$ are real numbers such that the quadratic polynomial $f(x) = x^2 + ax + b$has no non-negative real roots. Prove that ther exist two polynomials g,h, whose coefficients are ...
3
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0answers
30 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
4
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1answer
48 views

Prove the extension to be a Galois Extension

Let $p$ be a prime number. $K$=$\mathbb C(x,y)$ and $F=\mathbb C(x^p,y^p)$.Then, Prove that $K/F$ is a Galois Extension. Trial: Since this $\mathbb C$ is a field of charactersitic $0$,it would be ...
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2answers
71 views

Consider $n$ numbers $a_1,…, a_n$ and $x_1,…, x_n$. Can one find a polynomial, $f(x)\in R[x]$ st $f$ path through $(x_i,a_i) $

Consider $n$ arbitrary integer numbers $a_1,\ldots, a_n$ and real numbers $x_1,\ldots, x_n$. Can one find a polynomial, $f(x)\in \mathbb{R}[x]$ such that the graph of $f$ path through $(x_1,a_1), ...
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0answers
15 views

Finite Inseparable Extension

Preparing for my Galois theory exam in may and i am faced with the following question. Give an example of a finite inseparable extension with a sketched proof of its inseparability I have the ...
3
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2answers
73 views

$x\in \mathbb{R} : P(x) \in \mathbb{Z} \Leftrightarrow Q(x) \in \mathbb{Z}$. Prove that $P(x)-Q(x)=c \in \mathbb{Z}$ or… [closed]

$P(x),Q(x)$ are two polynomials such that $x\in \mathbb{R} : P(x) \in \mathbb{Z} \Leftrightarrow Q(x) \in \mathbb{Z}$. Prove that $P(x)-Q(x)=c$ or $P(x)+Q(x)=d, $ where $c,d \in \mathbb{Z}$.
3
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5answers
155 views

Prove that equation $x^6+x^5-x^4-x^3+x^2+x-1=0$ has two real roots

Prove that equation $$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$ has two real roots I think that: ...
3
votes
2answers
35 views

Let $ (x-1)^n\mid P(x)$ Prove that $P(x)$ has $n+1$ coefficients not zero

Let $ (x-1)^n\mid P(x)$ Prove that $P(x)$ has $n+1$ coefficients not zero It's is 1977 Bulgaria contest, i tried but not succeed
1
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2answers
58 views

Given some zeroes of a real polynomial of a given degree, how can one find the remaining zeroes?

Here is what the problem says: If $2$, $-\sqrt{5}$, and $3+i$ are three zeroes of a $5$th degree polynomial function with real coefficients, find the other zeroes of multiplicity $1$. I don't ...
2
votes
2answers
38 views

$f(x)$ is a polynomial with complex co-effcients taking integer values for all sufficiently large integers $x$ , then $f$ integer for all integers?

Let $f(x)$ be a polynomial with complex co-effcients such that $\exists n_0 \in \mathbb Z^+$ such that $f(n) \in \mathbb Z , \forall n \ge n_0$ , then is it true that $f(n) \in \mathbb Z , \forall n ...
0
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1answer
21 views

On the leading co-efficient of polynomial which takes integer values at every integer argument

If $f(x)$ is a polynomial with complex co-efficients of degree $k$ with leading co-efficient $a_k$ such that $f(n) \in \mathbb Z , \forall n \in \mathbb Z$ , then is it true that $|a_k| \ge \dfrac ...
1
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1answer
36 views

relations between a set of polynomials

I have a set of polynomials. Is there a computer algebra program that gives all the algebraic relations between them ? I will prefer singular if it has this component.
0
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1answer
27 views

Quadratic graph / standard form

If I draw a graph of the quadratic $x^2-9=0$, I have a parabola with roots $x=3$ and $x=-3$ and a vertex of $(0,-9)$ with the parabola opening upwards as $a$ is positive in the standard quadratic ...
0
votes
3answers
78 views

Show that $\mathbb{F}[x^2,y^2,xy]$ is not polynomial

$\mathbb{F}[x^2,y^2,xy]$ is the polynomials in two variables whose terms all have even degrees. Of course, this generating set $x^2,y^2,xy$ is not algebraically independent, but I need to show that no ...
-1
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0answers
70 views

zariski closure of first quadrant

Consider the boundary of the first quadrant in $\mathbb{R}^{2}$. Show that this is not a variety, and then find its Zariski closure. So as we are looking at the first quadrant we can write $$S= ...
-1
votes
1answer
49 views

Show any straight line is irreducible

Show that any straight line in $\mathbb{F}^{n}$ is irreducible, where F is an infinite field. I know V($ax+b$) would be a variety that represents any straight line and then V is irreducible if I(V) ...
3
votes
3answers
79 views

$P,Q$ are polynomials with real coefficients and for every real $x$ satisfy $P(P(P(x)))=Q(Q(Q(x)))$. Prove that $P=Q$

$P,Q$ are polynomials with real coefficients and for every real $x$ satisfy $P(P(P(x)))=Q(Q(Q(x)))$. Prove that $P=Q$. I see only that these polynomials are same degree
3
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2answers
30 views

Elementary bound theorem of a monic real polynomial

An elementary bound theorem on the roots of a real monic polynomial states that $$M := \operatorname{max} (1, |a_0| + \cdots + |a_{n-1}|) := \operatorname{max} (1, B)$$ is an upper and lower ($-M$) ...