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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2
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3answers
245 views

Finding integral solutions to the equation $x^4-ax^3-bx^2-cx-d=0$

How many integral solutions exist for the equation: $$x^4-ax^3-bx^2-cx-d=0\qquad a\ge b\ge c\ge d\qquad a,b,c,d\in\Bbb{N}$$ I have no idea where to begin even.Please help.
2
votes
2answers
116 views

Show that the polynomial $x^2 + x + 1$ is irreducible in $\mathbb Z/2\mathbb Z[x]$

I've looked up Eisensteins criterion, but I don't understand how to apply it to show that $ x^2 + x + 1$ is irreducible. Edit: Ok, I see now that Eisensteins criterion does not apply here.
0
votes
1answer
54 views

find minimal polynomial of $T(p)=p'+p$

I'm trying to solve the following question: let $T: \mathbb C_n[x] \to \mathbb C_n[x]$, $T(p)=p'+p$ find the characteristic and minimal polynomial of $T$. What I'm trying to do is the following: I ...
-1
votes
1answer
85 views

Irreducibility of $1+X+\ldots+X^{p-1}$ over $\mathbb{Q}$ [duplicate]

Show that the polynomial: $$g(X)=1+X+X^2+\ldots+X^{p-1}$$ is irreducible over $\mathbb{Q}$, where $p$ is prime. I am not sure how to approach this.
0
votes
1answer
20 views

Solving for $x$ [not homework]

How do I bring the remaining $x$ to the LHS? $\pm x=\frac{(2-x)\sqrt{|q_2|} } {\sqrt{|q_1|}}$ to get $x=\frac{2 \sqrt{|q_2|}}{\sqrt{|q_2|} \pm \sqrt{|q_1|}}$ I'm just not sure about the ...
0
votes
2answers
33 views

Function tranlsation $g(x) = f(x) + 15$

I can't seem to work this answer out when practicing for exams. Here's the question: You are given that $f(x) = (2x - 3)(x + 2)(x + 4) \cdots$ From this I know $f(x)$'s roots: $\frac{3}{2}$, ...
1
vote
2answers
40 views

Let $p(x)$ have a zero $a\in \mathbb{Q}$ …

"Let $p(x)$ have a zero $a\in \mathbb{Q}$..." Where $p(x)$ is a polynomial. I came about this part of a statement and I was not entirely sure what it meant. Although, I assumed that it meant that ...
6
votes
1answer
94 views

Galois group of $X^5-n$

In the following situation I want to find the Galois group of a specific polynomial: Let $n > 1$ be a square-free integer, $f =X^5 - n \in \mathbb{Q}[X]$, $x:=\sqrt[5]{n}$ and $\zeta=\text{e}^{2 ...
3
votes
1answer
83 views

Showing a polynomial is irreducible

We define $K:=\mathbb{Q}[\sqrt{-3}]$, in particular $e^{2\pi i/3} \in K$. If $f \in \mathbb{Q}[X]$ is a monic, irreducible polynomial with $\text{deg}(f)=3$, why is $f$ also irreducible over $K$?
3
votes
1answer
66 views

Minimal polynomial in Galois extension

Let $K \subset L$ be a Galois extension and $x \in L$ such that $L=K[x]$. $H \leq \text{Aut}(L|K)$ is a subgroup of the galois-group. I want to show that the minimal polynomial of $x$ over ...
2
votes
1answer
41 views

does it have a name : $\prod\left(1-x_i\right)$

I want to know whether there is a formula or theorem on the expansion of this expression: $$ \prod_{i=1}^n \left(1-x_i\right). $$ I only know the bionomial theorem and multinomiol theorem, but this ...
2
votes
1answer
67 views

Polynomials in $Z_p[x]/f(x)$

For shorthand, suppose $K=\mathbb{Z}_p[x]/f(x)$, $p$ a prime, and $\deg(f)=n$ where $f\in \mathbb{Z}_p[x]$. Then, how do we show that (1) $K$ can be written as $\mathbb{Z}_p[\theta]$, where $\theta$ ...
0
votes
0answers
257 views

Integrating quotients with polynomials in numerator and denominator that are raised to fractional powers

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
0
votes
1answer
43 views

Let $K = $ algebraic numbers. Then is $\operatorname{Span}_K(\pi, \pi^2, \dots)$ a vector space of transcendentals?

$V = {\rm Span}_K(\pi, \pi^2, \dots)$ is clearly a $K$-vector space. If we let $K = \Bbb{Q}$ temporarily, then every element of $V$ is transcendental as it's a finite linear combination $Q(X), \ X = ...
2
votes
3answers
201 views

Using telescoping property to prove difference of powers

Ok so I have started working through Apostol calculus and as you can see I am stuck. The problem is that I can not see the telescoping pattern anywhere for following problem. Prove that $$a^n - b^n ...
0
votes
2answers
61 views

$x^3+3x^2+4x+5=0$ and $x^3+2x^2+7x+3=0$, how many common roots they have?

My attempt, Equate both, at the end you will get $x^2-3x-2=0$ That means $x=-1$ and $x=2$. But what after that. Please provide solutions as well.
1
vote
3answers
68 views

Inequality for quartic polynomial depending on a parameter

Let $f(x) = \frac 14 x^4 - \frac \alpha2 x^2 - (\alpha-1) x - \frac \alpha 2 + \frac 3 4 $. I want to show that there exists an $\alpha>1$ such that $f(x)\geq 0 $ for $x\leq 0$. Even more, it ...
1
vote
1answer
50 views

What is the remainder useful for when dividing a polynomial?

I'm studying AS maths, and am trying to connect my thoughts around polynomials and the factor and remainder theorems. I understand the factor theorem and its application: it helps me find roots of a ...
0
votes
0answers
133 views

Prove a polynomial is irreducible over $\mathbb Q$

Show that the polynomial $x^{n-1}+x^{n-2}+\cdots+x+1$ is irreducible over $\Bbb Q$ if and only if $n=p$ is a prime. (For the direction when $n=p$, make a change of variable $x\to x+1$ and use ...
0
votes
3answers
291 views

Using Rouche's Theorem to find the number of zeros of $p(z)=z^8 +10z^3 −50z+1$ in the right halfplane

I'm studying for a complex analysis qualifying exam and was wondering if someone could help me out with this. I am not sure how to apply Rouche's Theorem to this. How many zeros does the polynomial ...
6
votes
2answers
67 views

Disk with root in center with no other roots in polynomial

Say we have a polynomial $p$ with roots $r_1,r_2...r_n$, I'm looking for a way to find a disk which, if placed on the center of any root, does not contain any other root (multiple roots considered as ...
0
votes
1answer
24 views

Multivariate polynomials over finite fields representation with prime field coefficient.

Let $p(x_1, \ldots, x_r)\in\mathbb{F}_{p^m}$ be a multivariate polynomial over finite field of order $p^m$. I would like to represent $p$ as a sum: $p(x_1, \ldots, x_n)=\sum_{i=0}^{k} \xi^i q_i(x_1, ...
8
votes
2answers
186 views

Minimising an expression - involving polynomial

I found this one on a forum but it has been unanswered from long there. I am curious to know if there is a solution to this problem. Here it is: Let n be a positive integer. Determine the smallest ...
1
vote
0answers
84 views

Matlab: Calculate Lagrange interpolaton polynomial

I'm new to Matlab so I don't even know where to start. I need to calculate Lagrange's interpolation polynomial of some function f in the case when the nodes are ...
0
votes
1answer
29 views

$(x+b)^3\mid P(x)+a$ and $(x-a)^3\mid P(x)-a$

$a,b\in\mathbb{C}$, $b!=0$ I need to find all the polynomials $P$ of degree $5$ verifying: $ \begin{cases}(x+b)^3\mid P(x)+a\\ (x-b)^3\mid P(x)-a\end{cases} $ PS : there was en error, i fixed it ...
3
votes
1answer
83 views

Finding a disk containing all roots of a complex polynomial

I'm trying to list all roots of a polynomial so I found this paper, in Part 9 on page 29 it gives a simple recipe to find all the roots. But there is this remark: We have assumed throughout the ...
1
vote
4answers
73 views

Constructing a polynomial with certain zeroes.

I want to construct a polynomial $f(x)$ that has zeroes at $-9,\,-5,\,0,\, 5,\, 9$. Can somebody provide a method (or perhaps some hints) for solving this?
0
votes
1answer
60 views

Finding some homogeneous generators of an ideal.

Suppose that $\mathfrak a$ is an homogeneous ideal of $K[T_1,\ldots, T_n]$ where $K$ is a field of characteristic $0$ and $T_1,\ldots,T_n$ are indeterminates. Moreover suppose that $\mathfrak a$ has a ...
1
vote
0answers
54 views

Non-symmetric polynomials, game

This is a game I thought was easy but appears to be too hard for me... I'm trying to find a polynomial in x,y,z (they commute) such that permutations of the variables only give rise to 2 different ...
1
vote
1answer
66 views

A theorem about ideals of $K[T_1,\ldots,T_n]$ and their generators

Suppose that $L\subseteq K$ is a field extension ( we are in characteristic $0$) and moreover that $\mathfrak a\subseteq K[T_1,\ldots,T_n]$ is an ideal ($T_1,\ldots,T_n$ are indeterminates). I have ...
0
votes
1answer
103 views

Given a polynomial of degree 5, get minimum and maximum without using derivatives

Given a quintic polynomial (in my case, $x^5+2x^4+16x-32$), I am supposed to get its maximum and minimum value for the interval $I=[-2;2]$ without using the derivative of the corresponing polynomial ...
1
vote
1answer
49 views

Bounding the Number of Roots of Integer Polynomial

Let $P(x)$ be a non constant polynomial in $\mathbb{Z[x]}$. Let $n$ be the number of roots of $P(x)^2-1 = 0$. Show $n \le \deg P+2$.
1
vote
2answers
54 views

Integer-valued polynomial

Let $f(x) \in \mathbb{Q}[x]$, and suppose $f(n)$ is an integer for all large integer $n$. Prove that $f(n)$ is an integer for small positive integers $n$. I read the answer from here is the hilbert ...
0
votes
2answers
60 views

Greatest common divisor of polynomials over $\mathbb{Q}$

I have two polynomials: $f: x^3 + 2x^2 - 2x -1$ and $g: x^3 - 4x^2 + x + 2$. I have to do two things: find $gcd(f,g)$ and find polynomials $a,b$ such as: $gcd(f,g) = a \cdot f + b \cdot v$. I have ...
2
votes
1answer
120 views

Help to understand this polynomial trick

I'm trying to understand this answer which I copy here (I didn't ask to the user because he left MSE). Could someone verify if my answer is correct and help me to understand the highlighted ...
0
votes
3answers
99 views

Find all values of $a$ for which there are two real solutions of $x^3-2ax^2+a^2x-3=0$

Find all values of $a$ for which there are two real solutions of the equation. $$x^3-2ax^2+a^2x-3=0$$ Ans = $1.5\sqrt[3]{6}$ I tried to research the function by dint of derivative, but it didn't ...
1
vote
0answers
42 views

When is the dot product of roots of certain multivariate polynomials also a root?

Problem. Let $n\in\mathbb N$ be fixed, and suppose that we are given three collections $$z_1,\ldots,z_n\in\mathbb Z,~a_1,\ldots,a_n\in\mathbb R,\text{ and }b_1,\ldots,b_n\in\mathbb R.$$ Suppose we ...
4
votes
0answers
81 views

A polynomial $\ f(x)$ has integer coefficients such that $\ f(0)$ and $\ f(1)$ are both odd numbers. Prove that $\ f(x) = 0$ has no integer solutions. [duplicate]

Let there be a polynomial $\ f(x)$. It has integer coefficients such that $\ f(0)$ and $\ f(1)$ are both odd numbers. Prove that $\ f(x) = 0$ has no integer solutions.
0
votes
4answers
40 views

What is the matrix corresponding it a linear transformation of a polynomial?

Given the linear map $T(f(x)) = f(2x+1)$ where $f(x)$ is a polynomial of degree $3$, what is the matrix corresponding to $T$?
0
votes
1answer
57 views

Discrete Logarithm Problem in $GF(p^m)$

I have question regarding DLP in $GF(p^m)$ I know the algorithms for solving the DLP in $GF(p)$ like Baby Step-Giant Step, Pohlig-Hellman etc... But what if we move into the $GF(p^m)$ and are ...
1
vote
0answers
73 views

Polynomials and NSA

I'm looking for some applications of criteria of irreducibility of integer polynomials inside and outside mathematics. I was reading the CV of Filaseta, a great researcher in this area and he has ...
2
votes
1answer
78 views

Real solutions of the polynomial

Let $a, b, c$ be distinct real numbers. Then find the number of real solutions of $(x − a)^5 + (x − b)^3 + (x − c)$ I can't understand how there will be any solution. The polynomial is not equated ...
2
votes
1answer
76 views

Why are two statements about a polynomial equivalent?

I am reading a claim that the following two statements are equivalent. One of the roots of a polynomial $v(t)$ is a $2^j$-th root of unity, for some $j$. The polynomial $v(t)$ is divisible either by ...
2
votes
1answer
114 views

Find the number of irreducible polynomials in any given degree

For any prime $p$ find the number of monic irreducible polynomials of degree $2$ over $\mathbb Z_p$. Do the same problem for degree $3$. Generalize the above statement to higher degree ...
1
vote
1answer
84 views

Finite Subgroup of multiplicative group of Field is Cyclic proof

I know this question has been asked before, but I would like to specifically address one part of the proof I'm reading that confuses me. I am following Fraleigh 7th edition, page 213. The theorem is ...
0
votes
0answers
16 views

Is there anything interesting we can do with this fact on iterates of polynomials over a finite ring?

Let $R$ be a finite ring and $r \in R$ such that $f \in R[X]$ is a min. poly of $r$ over $R$. Ie. $f(r) = 0$ and $f$ has minimal degree. In particular if $g(r) = 0$ and $g \in R[X]$, we have that $f ...
4
votes
3answers
335 views

Positive integer solutions of $a^3 + b^3 = c$

Is there any fast way to solve for positive integer solutions of $$a^3 + b^3 = c$$ knowing $c$? My current method is checking if $c - a^3$ is a perfect cube for a range of numbers for $a$, but this ...
2
votes
2answers
64 views

Prove that $p(z) = 2z^5 + 6z - 1 $ have roots (in two sets)

Prove that $p(z) = 2z^5 + 6z - 1 $ have one real root in $(0,1)$ and four root in $\left\{ z \in \mathbb{C} : 1<|z|<2 \right\}$. I suppose that we should use Rouché's theorem but I have no ...
3
votes
3answers
507 views

How and in what context are polynomials considered equal? [duplicate]

There's two notions of equivalent polynomials floating around, one saying that $f = g$ iff they're equivalent as maps, and the other saying $f = g$ iff they're equal on each coefficient when written ...
1
vote
2answers
117 views

Roots of polynomial $x^3-3\sqrt 5x^2+13x-3\sqrt 5$ given the factor $x-\sqrt 5$

Given that $x-\sqrt 5 $ is a factor of the cubic polynomial $x^3-3\sqrt 5x^2+13x-3\sqrt 5$, find all the values of the polynomial After the long division method I get $x^2-2\sqrt 5x+3$. Now ...