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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5
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0answers
72 views

How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and ...
2
votes
2answers
71 views

Number of irreducible polynomial over a field. [closed]

Find the number of irreducible monic polynomials of degree $2$ over a field with five elements. Please anyone help me.
3
votes
1answer
62 views

How does Gauss's lemma follow from Nagata's lemma?

In section 4 of Samuel's Unique Factorization it's said Gauss' lemma is an easy consequence of Nagata's lemma. How does this work, i.e., how to deduce Gauss' lemma from Nagata's lemma? I'm asking ...
3
votes
1answer
59 views

Is the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + t^2)$ irreducible over $k(t)$?

Let $k$ be an algebraically closed field of characteristic 2 and let $k(t)$ be rational function field of one variable. Consider the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + ...
2
votes
0answers
20 views

How to identify properties of the zeroes of this polynomial? [on hold]

If $f_0(x)=1$, and $f_{n+1}=\frac{d}{dx}((x^2-1)f_n(x))$, prove that every $f_n$ has exactly $n$ distinct zeroes, all located in the interval $(-1,1)$. It's got me stumped, so any help/pointers would ...
0
votes
1answer
17 views

Field structure of non solvable field extensions

I was considering the base field $D$ which is some solvable extenstion of $ \mathbb{Q}$, and a polynomial that isn't solvable in radicals such as $ x^5 - x + 1$. If we let $\zeta$ be a root of this ...
3
votes
5answers
236 views

Find a positive integer solution to $xyzw=504(x^2+y^2+z^2+w^2)$

Find positive integer values of $x,y,z,w$, such that $$xyzw=504(x^2+y^2+z^2+w^2)$$ I found it at some point and now I am unable to find the solution anymore, maybe this equation isn't satisfiable? ...
0
votes
1answer
27 views

Two variable curve fitting

I need to fit an expression of the form $f(x,y)$ for which the data comes from an experiment. From the experiment data I found the following equations. $$f(x,800)=0.1079x^2−0.1699x+0.4216$$ ...
0
votes
1answer
27 views

Factoring Polynomials: How do I express the area and perimeter in factored form?

Our topic is factoring polynomials, and I can't seem to solve this question: Express the area and perimeter of the shaded region in factored form. We've discussed how to solve for the ...
1
vote
1answer
31 views

The greatest common divisor of $(O_n, T_n+2)$ where $O_n$ and $T_n$ are the oblong and triangular numbers respectively.

Suppose that $T_n$ is odd. Can we find infinitely many $n$ such that $(O_n, T_n+2)=1$? Is it trivial and obvious? My hunch based on some hand calculations is to look at $n$ congruent to $0$ or $2$ ...
4
votes
1answer
35 views

Problems reducible to polynomial root finding

In the past, I have noticed several problems for which the solution goes something like this: Reduce the problem to a polynomial equation Find the roots of the polynomial Interpret appropriately in ...
0
votes
0answers
20 views

what are some applications of Horner's rule?

As seen in the book Theory of equations (Uspensky) theres a very fun way to calculate a polynomial $f(x)$ in the powers of $(x-c)$. (Horner's rule) What is a possible application of this new ...
2
votes
3answers
63 views

Find the first $4$ Hermite polynomials using a recursion relation

Given the Probabilists' Hermite differential equation: $$U''-xU'+\lambda U=0\tag{1}$$ A book question asks me to: Find the first $4$ polynomial solutions (for $...
1
vote
0answers
49 views

Solve $ \int{\sqrt{1 + (3x^2 + 2x - \frac{29}{2})^2}} dx $

I have to solve this indefine integral: $$ \int{\sqrt{1 + (3x^2 + 2x - \frac{29}{2})^2}} dx $$ I tried to make the square: $$ \int{\sqrt{9x^4 +12x^3-29*3x^2 -58x + \frac{29^2 +4}{4}}} dx $$ but ...
3
votes
1answer
70 views

Algebraic or Analytic Proof of a Polynomial Identity

Let $m$, $n$, and $r$ be integers with $0\leq r \leq \min\{m,n\}$. Define $$f_{m,n,r}(q):=\left(\prod_{j=1}^r\,\left(q^m-q^{j-1}\right)\right)\,\left(\sum_{\substack{{j_1,\ldots,j_r\in\mathbb{Z}_{\...
0
votes
3answers
40 views

How do you figure out the signs in factorization of this high-degree polynomial?

Rather than carrying out the whole factorization, in this case you only need to be able to identify patterns (this question is from a study guide, not an actual test or problem set). You are given ...
4
votes
1answer
110 views

Proving roots of a polynomial are real and distinct.

Let $p(x)$ be a polynomial with all roots real and distinct such none of its roots is equal to zero. Prove that the polynomial $x^2p''(x)+3xp'(x)+p(x)$ also has all roots real and distinct. Unable ...
4
votes
4answers
119 views

$f(x)$ is a quadratic polynomial with $f(0)\neq 0$ and $f(f(x)+x)=f(x)(x^2+4x-7)$

$f(x)$ is a quadratic polynomial with $f(0) \neq 0$ and $$f(f(x)+x)=f(x)(x^2+4x-7)$$ It is given that the remainder when $f(x)$ is divided by $(x-1)$ is $3$. Find the remainder when $f(x)$ ...
4
votes
1answer
50 views

Is $x^{2\cdot 3^n}+x^{3^n}+1$ irreducible (mod 2)?

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, it seems that the polynomials of the form $$x^{2\cdot3^n}+x^{3^n}+1 \pmod 2$$ are irreducible. ...
0
votes
1answer
27 views

Unique integer solutions to $\sum\limits_{i=1}^n a_i = A$ when $l \leq a \leq u$ and $a,A,l,u \in \mathbb{N}$

I'm trying to find a analytical way for finding the total amount of unique solutions to equation: $$\sum\limits_{i=1}^n a_i = A, \text{when } l \leq a \leq u,$$ where $a,A,l,u \in \mathbb{N}$. For ...
2
votes
0answers
41 views

Multiplication of polynomials of the same degree

Consider polynomials of the form \begin{equation} p(x)=x^{n-2r}\sum_{i=0}^ra_ix^{2i}, \end{equation} where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{...
0
votes
1answer
18 views

Connection between saturated ideals an CM algebras.

Let $I$ be an homogenous ideal of the polynomial ring $K[x_1,\dots,x_n]$. Is there any relations between $I$ being saturated and $R/I$ being a Cohen-Macaulay?
7
votes
1answer
56 views

If $B$ is an ideal of $A$ then $B[x]$ is an ideal of $A[x]$ - what's wrong with my proof?

This is exercise E.2 from chapter 24 of Pinter's A Book of Abstract Algebra: If $B$ is an ideal of $A$, $B[x]$ is not necessarily an ideal of $A[x]$. Give an example to prove this contention. It ...
1
vote
0answers
35 views

Surjectivity on the image of a annulus

I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself. ...
0
votes
0answers
24 views

factorising polynomials related proofs

$P(x)= a{x}^3+b{x}^2+c{x}+d$ where $a,d$ are not equal to zero. (All the coefficients are integer) Now $P(x)$ is divided by $x-r$. Here why r needs to be an integer to be a factor of d(constant term)...
0
votes
0answers
24 views

Plaid in generic position. Counting faces.

I write $\pi_n$ to denote a group of $n$ parallel lines. Consider a family of $(\pi_1,\pi_2,\ldots,\pi_s)$ parallel groups each with $(n_1,n_2,\ldots,n_s)$ parallel lines. Arrange the family of ...
2
votes
1answer
57 views

How do I get a sequence from a generating function?

For example if I have the generating function $\frac{1}{1-2x}$ then it corresponds to the sequence $1 + 2x + 4x^2 + 8x^3 +~...$. I know how to start from the sequence and get the generating function, ...
0
votes
0answers
16 views

How to use remainder theorem if divisor is constant?

How to prove remainder theorem if divisor is constant? Like F(x) = 9x-3, polynomial d(x) =3, divisor
1
vote
1answer
29 views

Find $ m \in \mathbb{Z} $ for which $ x_1 $ and $ x_2 $ are integers

$$ (m+1)x^2 - (2m+1)x - 2m = 0 $$ $$ m \in \mathbb{R}-\{-1\} $$ Find $ m \in \mathbb{Z} $ for which $ x_1 $ and $ x_2 $ (the solutions of equation, the roots) are integers ($x_1,x_2 \in \mathbb{Z}$) ...
0
votes
1answer
37 views

Why has the equation positive root?

Let $x \in \mathbb{R}$ and $\lambda ,{\lambda _0} \in \mathbb{C}$ and $r\in(0,1)$. $w(x) = {\alpha _m}{x^m} + \cdots + {\alpha _1}{x^1} + {\alpha _0}$. $f(\lambda)$ is function such that $f(\...
0
votes
3answers
99 views

Is the expression $\frac{x}{x+5}$ a polynomial

I recently took an online test for my math class and was asked the question "What type of polynomial is $\frac{x}{x+5}$." The supposed correct answer was "not a polynomial" because there is a variable ...
0
votes
4answers
65 views

Question about rational roots of polynomial

Let $p(x) = c_0 + c_1x + \ldots + c_d x^d$ be a polynomial with integer coefficients $c_0, \ldots, c_d \in \mathbb{Z}$ and $c_d \neq 0$, so $p(x)$ has degree $d$. Let $a/b$ be a rational root of $p(x)$...
3
votes
0answers
80 views

Is this equation $(n+1)~x^{2n+1}-n~x^{2n}-n=0$ solvable in radicals for some $n \geq 2$?

Consider this polynomial equation: $$(n+1)~x^{2n+1}-n~x^{2n}-n=0,~~~~n \geq 2,~~~n \in \mathbb{N}$$ It's related to another question of mine, but I don't think the context matters here. I'm ...
3
votes
2answers
53 views

Roots of Unity with Rational Real Parts

All of the $4^{\text{th}}$ and $6^{\text{th}}$ roots of unity have real parts that are rational numbers. Are these the only roots of unity $z$ such that $\text{Re}(z)\in \mathbb{Q}$ ?
0
votes
1answer
36 views

Notation R/qR meaning

Suppose $R$ is defined as $\mathbb{Z}[x]/f(x)$ where $f(x)=x^3+1$. From what I understand the result is the class of elements from the Euclidean division by $f(x)$. But what does $R_q=R/qR$ mean? Is ...
0
votes
0answers
24 views

A question relating to cyclic polynomials.

A polynomial is cyclic if f(x1,…,xn)=f(x2,…,xn,x1) This implies that f(x1,…,xn)=f(xk+1,…,xn,x1,…,xk)for any k. Whyy?? How?? (This excerpt has been taken from an answer on the definitions of the ...
1
vote
2answers
66 views

Can a polynomial of $n$ degree have $n+1$ distinct real roots?

Question : Let $f(x) = \sum^n_{k=0}c_kx^k$ be a polynomial function then prove that if $f(x) = 0$ for $n+1$ distinct real values, then every coefficient $c_k$ in $f(x)$ is $0$ , thus $f(x) = 0$ for ...
2
votes
1answer
28 views

Infinite variable polynomial

I'm curious as to how to construct an infinite variable polynomial. Is there an nice formulation of such a thing? I've attempted using functions and functionals to construct one, but that didn't lead ...
2
votes
1answer
64 views

Use of substitutions in solving equations

I am currently working on this problem, I am asked to solve the following $x^2 - 4 - x\sqrt{x^3 + 3x} = 7$. I am able to manipulate the above to obtain $x^5 - x^4 +3x^3 + 22x^2 - 121 = 0$. The ...
3
votes
1answer
461 views

Proof that the following function is a polynomial

I've been trying to get my head around this problem for a long time, yet I have not been able to make much progress. Let $\ell_0(j) = \left\lfloor \frac{1}{2}\left( \sqrt{8j^2 - 8j + 1} + 2j - 1 \...
1
vote
1answer
28 views

Manipulation of Polynomials and Goodness-of-Fit

This question is directly connected with my problem in Python section HERE. Basically I've programmed a method with mathematical and physical background that forms a polynomial of selected degree ...
3
votes
2answers
253 views

Expected value of the zeros of random polynomials of degree two

Let $a_1,a_0$ be i.i.d. real random variables with uniform distribution in $[-1,1]$. I'm interested in the random zeros of the polynomial $$p(x) = x^2 + a_1x + a_0. $$ One thing (between many) thing ...
0
votes
0answers
9 views

Is there a simple algorithm to compute polynomial inverses over cyclotomic polynomials?

I'm working with polynomial inversions in a ring built over the nth-cyclotomic polynomial, with $n = 2^i$. As usual, I'm applying Extended Euclidean algorithm on this, an approach that does not scales ...
3
votes
2answers
31 views

Why is the CRC, essentially polynomial division over GF(2), linear?

On the Wikipedia page for Cyclic Redundancy Check, it says that: CRC is a linear function with a property that ${\displaystyle \operatorname {crc} (x\oplus y)=\operatorname {crc} (x)\oplus \...
1
vote
1answer
23 views

Factor polynomial over $C, Q, R$, if one comlex root is given

The polynomial is: $P(x)=x^6+x^4-x^3+x^2+1$. I need to factor it over $C, Q, R$ if one complex root is $\sqrt[3]{1}$. Also find all fields in which $P$ is reducible. Now, I know how to find one ...
1
vote
2answers
45 views

find a and b F(x) , g(x) [closed]

Functional $F(x)=(x^2-x+1)Q(x)+x-1$ $G(x)=(x^2-x+1)T(x)+x+1$ $F(x).G(x)=(x^2-x+1)H(x)+ax+b$ find a and b
1
vote
5answers
385 views

Remainders of polynomials of higher degrees. [on hold]

Let $R(x)$ be a remainder upon dividing $x^{44}+x^{33}+x^{22}+x^{11} +1$ by the polynomial $x^4 +x^3 +x^2 +x +1$. Find: $R(1)+2R(2)+3R(3)$. Answer provided is $0$
2
votes
3answers
49 views

Quadratic Functional equations.

Suppose $f$ is a quadratic ploynomial, with leading cofficient $1$, such that $$f(f(x) +x) = f(x)(x^2+786x+439)$$ For all real number $x$. What is the value of $f(3)$?
0
votes
1answer
35 views

Polynomial sum of the coefficients

There's a problem I saw in the exam: Let $p(x)$ be a polynomial such that: $p(x) = (x–2)^{2012}(x + 2012) + (x–2)^{2011}(x + 2011) + … + (x–2)(x + 1)$ Find the sum of the coefficients of ...
0
votes
1answer
35 views

Polynomials $q$ satisfying $ q(z, \bar{w}) + q(w, \bar{z}) \le q(z, \bar{z}) + q(w, \bar{w})$ for all complex z and w

Does there exist a real number $b$ such that the polynomial $q(x,y) := - x y^3 + b x^2 y^2 - x^3 y $ satisfies $$ q(z, \bar{w}) + q(w, \bar{z}) \le q(z, \bar{z}) + q(w, \bar{w}) \quad \text{for all } ...