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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.

0
votes
1answer
27 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 $\alpha_4=1$...
3
votes
2answers
43 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}\...
3
votes
0answers
41 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: ...
0
votes
4answers
28 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
vote
1answer
40 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 $c_j$?
2
votes
0answers
24 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 ...
0
votes
0answers
12 views

Left ideal of the Weyl algebra $A_1k$ such that gr$I = (x^2, xy, y^2)$ .

Let $R = A_1k$ be the Weyl algebra over a field of characteristic $0$. It is known that gr$R \cong k[x,y]$. There is a Poisson bracket on gr$R$ given by $\{x,y\} = 1$, $\{y,y\} =0= \{x,x\}$. Gabber'...
1
vote
2answers
44 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
votes
1answer
32 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
vote
1answer
25 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
13 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
16 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
12 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
vote
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\...
13
votes
2answers
92 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
vote
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
67 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 $\mathbb{F}_{25}$, ...
2
votes
1answer
51 views

Let : $P(x)=x^{3}+ax^{2}+bx+c$ where $(a,b,c)\in Z^3$

Question : If : $P(x)=x^3+ax^2+bx+c$ where $(a,b,c)\in Z^3$ And $m,n,k$ root of $P(x)$ such that : $m.n=k$ Then show that : $2P(-1)$ multiple of $P(1)+P(-1)-2[1+P(0)]$ My try : We known ...
0
votes
0answers
49 views

Equation : $3x^4+x^3-10x^2-x+3=0$ [duplicate]

Solve in $R$ the following equation : $3x^4+x^3-10x^2-x+3=0$ Im try use sub $y=x+\frac{1}{x}$ But I don't understand whene and where I use this sub Please give me ideas or hint to approach it
1
vote
1answer
16 views

Comparing two rational approximation of the same minimum

Let $P(x)=x^3-6x$. It is easy to see that on $[1,2]$, $P$ reaches its minimum at $x=\sqrt{2}$. Among all the fractions with denominator dividing $n$, the two closest ones are $a_n=\frac{\lfloor n\sqrt{...
0
votes
0answers
22 views

Polynomial function degree [on hold]

what is the lowest degree of a polynomial function which must go through two distinct points (x1, y1) and (x2, y2) of the plane with predefined second derivative d3 at a 3rd abscissa x3? why?
0
votes
1answer
29 views

Proving the accuracy for numerical integration

Given a smooth function $f$, we denote $L_{f}$ the Lagrange polynomial of degree less than or equal to $1$ which is equal to $f$ at the points $x_1$ and $x_2$. Define $I_{f} = \int_{-1}^{1} L_{f}(...
2
votes
1answer
70 views

Name for this method of factoring quadratic and are there any textbooks that describe it?

I remember learning this method of factoring quadratics in middle school or high school, but looking for a name or more information on it leads me to dead ends. Given: $ax^2+bx+c=0$ $d*e=a*c$ $d+...
0
votes
1answer
26 views

Division algorithm for polynomials discrete maths

Problem: State the division algorithm for polynomials. Using this result, show that, if the polynomial $f(x)$ has a root $a$, then the linear polynomial $x-a$ divides $f(x)$. I’m incredibly stuck on ...
0
votes
2answers
38 views

Find for which lambda values this equation gives positive solutions [on hold]

The equation is $x^3 +x^2 +3 = \lambda x$ I thought I could move lambda to the first sector, getting $x^3 +x^2 -\lambda x+3 = 0$ But then? Can you show me the solution?
2
votes
6answers
71 views

Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$

Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$ By factor theorem we know that $(x-a)$ is a factor of $f(x)$ if $f(a)=0$. In this case, $f(x)=x^n -nx +n-1 \implies f(1)=0$ Hence we conclude that ...
0
votes
1answer
17 views

a sum of determinant with polynomial functions

I have $P, Q, R: \ C \rightarrow \ C$ polynomials functions with maximum degree 2 and $a,b,c\in \ C$ such that $\begin{vmatrix} P(a) & Q(a) & R(a) \\ P(b) & Q(b) & R(b) \\ P(c) & Q(...
0
votes
0answers
37 views

The maximum of roots

There are 10 such square trinomials $ P_1, P_2, ..., P_ {10} $, such that their graphs touch each other in pairs. How many maximum different roots can a polynomial have $(P_1-P_2) (P_2-P_3)...(P_9-P_ {...
0
votes
1answer
30 views

Dimension of a kernel of a linear map [on hold]

Here I only need a solution of part (b). Trying to use those given point as a root of the polynomials in kernel but facing problem with the degree of the polynomial.
0
votes
0answers
35 views

When are polynomials factorizable? [on hold]

To have uniquely factorizable polynomials from a set $\mathcal{S}$ with scalars from a set $\mathcal{K}$, what must be the necessary and sufficient conditions on the elements of and operations on ...
2
votes
2answers
77 views

Showing that $f(x)=x^3-3x+1$ has at least two zeros in the interval $[0,2]$

I was given this task by my professor: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be defined by $f(x)=x^3-3x+1$ Show that $f$ has at least two zeros in the interval $I := [0,2]$. My answer is: Since ...
0
votes
0answers
38 views

Identifying polynomials with other mathematical objects

What are the necessary and sufficient conditions to allow for the same manipulation of non-polynomial mathematical entities like matrices, linear transformations, differential operators, etc. as for ...
0
votes
1answer
26 views

Elements of the spectrum of complex numbers

I recently learned that the elements in the spectrum of $\mathbb{C}[x]$ are in the form $x-a$. I understand that a spectrum consists of all prime ideals of a ring, but I'm a little confused as to why ...
1
vote
1answer
31 views

Is a finitely generated subring of a Noetherian ring also Noetherian?

Is a finitely generated subring of a Noetherian ring $R$ also Noetherian? Remark: In fact I'm interested in the case $R=\mathbb C[x_1,...,x_n]$.
1
vote
1answer
34 views

How to construct a polynomial which is strictly negative everywhere, except for finitely many roots?

Given a finite sequence of real vectors $x_1, x_2, \dots, x_m \in \mathbb{R}^d$, how do I construct a polynomial $p \colon \mathbb{R}^d \to \mathbb{R}$, such that $p(x) = 0$ if $x \in \{ x_1, \dots, ...
8
votes
0answers
80 views

What can be P(0), when $P(x^2+1)=(P(x))^2+1$ and P(x) is polynomial? [duplicate]

What can be $P(0)$, when $P(x^2+1)=(P(x))^2+1$ and $P(x)$ is polynomial? Let $P(0)=0$, then $P(1)=1$, $P(2)=2$, $P(5)=5$, $P(26)=26$, $P(677)=677$ ... and so on. Then $P(x)=x$, because all the points ...
3
votes
5answers
109 views

How to factorized this 4th degree polynomial?

I need your help to this polynomial's factorization. Factorize this polynomials which doesn't have roots in Q. $ \ f(x) = x^4 +2x^3-8x^2-6x-1 $ P.S.) Are there any generalized method finidng 4th ...
0
votes
2answers
25 views

Decide whether the following subsets of a vector space are sub-vector spaces.

a) $\{f\in\mathbb{R}[t]:f(1)=0\}=:U_1$ b) $\{f\in\mathbb{R}[t]:\exists a\in\mathbb{R}\text{ with }f(a)=0\}=:U_2$ where $\mathbb{R}[t]$ is the set of all polynomials above K. Does a) mean, that ...
2
votes
1answer
43 views

Irreducibily of polynomials in two variables

Let $\mathbb{F}_q$ be a finite field where $q$ is odd. Let $f \in \mathbb{F}_q[x,y]$ be the following polynomial $$f:=(x^2y^2 - 2x^2y - 2xy^2 - x^2 + xy - y^2 - 2x - 2y + 1).$$ How to prove that $f$ ...
1
vote
1answer
19 views

Every non-negative multivariate polynomial has even degree and the highest degree term has positive coefficient?

Part of my question has been asked before (Every non-negative multivariate polynomial has degree even?) but the proof there is not very satisfactory. The other part of my question involves proving (or ...
0
votes
0answers
40 views

Prove: Permutation of a root is another root of a polynomial

I read that Galois group is a permutation of the zeros or roots, this is new to me, so, I have a question. How can I prove, all roots of a polynomial are permutation of one another? in other words, ...
3
votes
5answers
126 views

Show that $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$.

Show that $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$. Once you have found the inverse like here, the verification is trivial. But how do you come up with such an inverse. Do I just try with general ...
1
vote
0answers
63 views

Polynomial that has no rational roots yet has roots modulo every integer

I want to characterize when the polynomial $p(x) = (x^{2} - a) (x^{2} - b) (x^{2} -ab)$ has a root modulo every integer, yet doesn't have an integer root. I worked out the condition when '$a$' and '$...
46
votes
7answers
7k views

Polynomial division: Is this trick obvious?

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+...
1
vote
3answers
86 views

Proving $f(x)=x^4+4x^3+3x^2+7x-4$ is irreducible

Here's my attempt, I'm almost there but I'm stuck: Using a hint, I wrote the modular reduction: Reducing the coefficients modulo $2$ gives: $\left [ f \right ]_2=x^4+x^2+x=x(x^3+x+1)$. Reducing ...
0
votes
1answer
87 views

How to solve an equation with 6 degree polynomial?

Does anyone have an idea to solve the following equation if it is possible? It's the best to get analytic solution, but if you can help me to show when equation has all real root with certain ...
6
votes
4answers
633 views

Is it possible to determine $P(21)$, if $P(x)$ is a 2nd degree polynomial, $P(11)=151$, and for all $x\in\Bbb R$, $x^2-2x+2\le P(x)\le2x^2-4x+3$?

I was given the following: Given that $P(x)$ is a second degree polynomial, and that $P(11)=151$, and that $$\forall x \in \mathbb{R}, \,\, x^2-2x+2 \le P(x) \le 2x^2-4x+3$$ determine $P(21)$. ...
1
vote
0answers
40 views

Factoring Polynomials into Galois Conjugate Linear Factors

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial of degree $n$. Question: Is there a nice way to describe the set of polynomials $f(x)$ (not necessarily monic) that can be factored as $f(x)= \prod_{i = ...