# Tagged Questions

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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### Finding generators for a polynomial ideal given some polynomials belonging to it

Let $k$ be a finite field, $n$ a positive integer and $R := k[x_1,\ldots,x_n]$ the polynomial ring in $n$ variables. Let $f_1,\ldots,f_n\in R$ be polynomials with the following property: $f_i$ has ...
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### Prove that $x^4+2x^2-6x+2=0$ when $x\in\mathbb{R}$ has exactly two solutions

Show that $x^4+2x^2-6x+2=0$ when $x\in\mathbb{R}$ has exactly two solutions. I first showed that the IVT guarantees that there exists at least one zero in $(0,1)$ and at least one zero in $(1,2)$. I ...
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### 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 ...
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### Why can a quartic polynomial never have three real and one complex root?

It seems that a quartic polynomial (degree $4$) either can have $0$ real, $1$ real, $2$ real, or $4$ real roots, and the rest is complex roots. Why can't it have $3$ real roots and $1$ complex?
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### What is trailing zero?

I am a newcomer.Would you like to give me a hand.Thank you so much. I have studing a paper related with GCD algorithm these days. In this paper,there is conclusion related with "trailing zero".I ...
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### $f(x)$ 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 $f(x)$ ...
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### Irreducible polynomial over $\mathbb{Q}$ implies polynomial is irreducible over $\mathbb{Z}$

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial of degree $\geq 2$. Then choose correct a) if $f(x)$ is irreducible in $\mathbb{Z}[x]$ then it is irreducible in $\mathbb{Q}[x]$. b) if $f(x)$ is ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Is any matrix representation of a monomial ordering invertible?

We know that any monomial ordering has a matrix representation. Let $\prec$ be a monomial ordering and $M$ be its matrix representation. Is $M$ invertible?
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### 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 ...
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### 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 ...
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### Primitive polynomials

I am revising for a discrete mathematics exam and as quite stuck on this question. Show that the polynomial $f = x^2 + 2 x + 3 \in \mathbb{Z}_5[x]$ is primitive. How many monic primitive quadratic ...
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### 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. ...
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### polynomial ring with isomorphic quotients

If $R$ is a commutative ring and $f(x), g(x) \in R[x]$ two polynomials such that $R[x]/f(x)\cong R[x]/g(x)$ as $R$-algebras, what can we say about $f$ and $g$? Or given $f(x)\in R[x]$, what can we ...
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### Denominator in rational gcd of integer prolynomials

A recent question tells us that even if two polynomials $f,g\in \mathbb Z[X]$ have no common factor as polynomials, their values at integer points may have common factors. That question gives this ...
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### Multiplication of polynomials of the same degree

Consider polynomials of the form $$p(x)=x^{n-2r}\sum_{i=0}^ra_ix^{2i},$$ where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{...
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### 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?
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### 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 ...
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### 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 ...
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### 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. ...
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### 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)...
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### 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 ...
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### 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, ...
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### 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
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 $\text{Re}(z)\in \mathbb{Q}$ ?
### 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 ($x_1,x_2 \in \mathbb{Z}$) ...