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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3answers
73 views

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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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 ...
2
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1answer
57 views

Convert $x \not\equiv 0$ mod $pq$ to a modulo polynomial [on hold]

$p,q \in \mathbb{P}$, primes For $x \not\equiv 0 \bmod p$ you can write $(x-1)(x-2) \dots (x - (p-1)) \equiv 0$ mod $p$ Is there a way to do the same for a a composite modulus $pq$? Note: $(x-1)(x-...
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0answers
17 views

How to shorten dot product

I would like to shorten a dot/scalar product: $$f(s)=sP_1+s^2P_2+\big((P_2-P_1)^TsN_1\big)N_1$$ Here $s$ are scalars, $P$ are points and $N$ are unit normal vectors in $R^3$. The function $f(s)$ ...
0
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2answers
37 views

What is the difference between a polynomial and a function or can they be used interchangebly?

I have been wondering over this basic question (seems rather trivial at first sight) for a long time- What is the difference between a polynomial and function? My confusion arises form the ...
0
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1answer
33 views

What does it mean for $x$ to be not upper bounded by any polynomial in $y$?

I am reading a statement saying the value $2^n-3$ is not upper bounded by any polynomial in $|S|$ where $|S|=n \cdot \binom{n}{2}$. I am just trying to understand the intuition behind this. My ...
2
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3answers
105 views

Are polynomials with integer coefficients uniquely determined by their coefficients?

Assume f and g are polynomials, where f = g. More explicitly; $$ f = a_{n}z^{n} + \ldots + a_{1}z + a_{0} = b_{n}z^{n} + \ldots + b_{1}z + b_{0} = g $$ where $ \ \ a_{0} \ , \ldots, \ a_{n} \in \...
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4answers
1k views

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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0answers
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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 ...
3
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1answer
58 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 + ...
-1
votes
0answers
26 views

Finding the area under a 5th degree polynomial curve given the polynomial fitting?

Using an outdated graphing program, I have a 5th degree polynomial fitted curve for my data (protein level in a treatment). I would like to find the integral of the curve given I only have the 'p0-p5' ...
13
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8answers
433 views

The proofs of the fundamental Theorem of Algebra [closed]

There are many proofs of the fundamental theorem of algebra. Which are the most beautiful proofs?
5
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0answers
71 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 ...
3
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1answer
59 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 ...
2
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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.
2
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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 ...
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4answers
3k views

Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=...
3
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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}_{\...
2
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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 $...
0
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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
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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 ...
17
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1answer
267 views

Define $f(x),g(x)\in \mathbb{R}$. Prove $f(x)=g(x)$.

Problem: Define $f(x),g(x)\in \mathbb{R}$ are polynomials and both of them have at least one real root and satisfy: $$f(1+x+g(x)^{2})=g(1+x+f(x)^{2}),\forall x\in\Bbb{R}$$ Prove $f(x)\equiv g(x)$. ...
1
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1answer
29 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 $2$ ...
-1
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1answer
49 views

Why does $a-b$ divide $f(a)-f(b)$ if $f$ is an integer polynomial? [closed]

In the question Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $n\ne 4$, is irreducible over $\mathbb Z$, Calvin Lin explain that $(a-b) | (f(a)-f(b))$ for all integer polynomiasl. Is anyone ...
4
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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 ...
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0answers
50 views

Finiteness of solutions to system of polynomial equations $P(x)P(y)=1$ & $Q(x)Q(y)=1$

Can that finiteness be proved for polynomials $P^n\neq\pm Q^m,\quad n,m>0\;$ by known methods? For univariate polynomials $A,B,X,Y$ $$AX+BY\equiv0$$ iff $$A\equiv\frac{HY}{(X,Y)},B\equiv-\...
4
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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
2answers
137 views

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 ...
0
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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 ...
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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 ...
0
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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 ...
0
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0answers
53 views

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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3answers
39 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 ...
3
votes
4answers
456 views

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 ...
4
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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. ...
6
votes
4answers
178 views

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 ...
4
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1answer
77 views

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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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{...
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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?
2
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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 ...
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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 ...
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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. ...
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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)...
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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
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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
3
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2answers
52 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 $\text{Re}(z)\in \mathbb{Q}$ ?
1
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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 ($x_1,x_2 \in \mathbb{Z}$) ...