Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=|a_{1}|+\cdots ...
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1answer
8 views

quadratic form polynomial divisibility vs. matrix pointwise multiplication.

Given matrix $V',W',Y'$ is of $d\times m (d\le m)$ ; column vector $c$ is of size $m$; $r_i, i=1,...,d$ are distinct; and each row of the matrix A is $A_i=(r_i^0 ... r_i^{d-1})$. So, A is of $d\times ...
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2answers
27 views

Checking irreducibility of a polynomial over a finite field

A part of a coding theory course I am doing includes some questions on irreducible polynomials. I have a question with solution but am worried I have interpreted it incorrectly. So for $\mathbb F_5$ ...
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33 views

Find all positive values for j,k,l such that j, k, l are positive integers and (j-k)|l, (k-l)|j, (l-j)|k.

Find all possible values of $j,k,l$ such that $j, k, l$ are positive integers and $(j-k)|l, (k-l)|j, (l-j)|k$. As I understand that using divisibility properties, it is possible to come to some ...
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2answers
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Find a second root of $x^3+px+q$ given the first root

This is a problem from Artin where given one root $a$, you have to find an equation for a second root in terms of $a$, $p$, $q$, and the square root of the discriminant $\delta$. Here's what I have ...
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0answers
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Is the Möbius inversion applicable in the case of number functions with values in $Q(x)$

I am looking for the cause of an erroneous calculation I did the details I cant present here. I guess a "Möbius inversion" I apply might be the cause. Normally the Möbius inversion is valid for ...
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1answer
50 views

Solving polynomial equation using Fermat little theorem

I am a bit confused on notation. I can't find a reference in notation in my textbook as to what this means. Here it goes: Let p = 13. Compute $\phi$$_{11}$$(3x^{233} + 4x^6 + 2x^{37} + 3)$ This ...
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33 views

Every polynomial has a root

Let $A$ be a commutatif ring, and $f\in A[T]$ une polynome. Then in the $A$-algebre $B=A[T]/(f)$ the polynomial $f$ has a root, namely $T \mod (f)$, because $f(T)\mod (f)=f(T)\mod (f)=0$. Do you ...
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Is there a formula which would let me know how many irreducible polynomials there are to the power n, in $z_n$? [duplicate]

I found that $x^2+x+1$ is the only polynomial to the power 2 that is irreducible in $z_2$. Moreover I found that $x^3+x+1$ and $x^3+x^2+1$ are the only polynomials to the power 3 that are ...
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The complex roots of a biquadratic polynom

In my recent post I have a problem with the following function: $x^4-4x^2+16$, and what I need is to find the complex roots. Here is my answer: First step, I make the substitution $x^2=y$ which ...
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0answers
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Faber polynomials in Matlab [on hold]

I want to compute some faber polynomials associated to an ellipse centered at a point \sigma (in the complex plane) in Matlab. Say the ellipse has minor axis a and major axis b. If someone know how to ...
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0answers
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Is the multiplication modulo $p$ for polynomials well-defined?

Is the multiplication modulo $p$ for polynomials well-defined ? I mean let $g,h\in\mathbb Z[x]$ and let $\bar g$ be the polynomial obtained from $g$ by reducing all the coefficients of $g$ modulo ...
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2answers
677 views

Is Vieta the only way out?

Let $a,b,c$ are the three roots of the equation $x^3-x-1=0$. Then find the equation whose roots are $\frac{1+a}{1-a}$,$\frac{1+b}{1-b}$,$\frac{1+c}{1-c}$. The only solution I could think of is by ...
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2answers
665 views

Why is the zero polynomial not assigned a degree?

Yesterday, I read in my textbook, We assign degree to every polynomial and even a non-zero constant is assigned a degree $0$ but $0$ itself is not assigned a degree. Why is that? Why we don't ...
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1answer
50 views

Location of the roots of $f'$ (Laguerre's theorem)

Let $f \in \mathbb{R}[X]$ be a polynomial of degree $n$ having $n$ distinct roots $a_1,...,a_n$. Let $b_1<...<b_{n-1}$ be the roots of its derivative $f'$ (note that $b_i \in ]a_{i}, a_{i+1}[$ ...
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3answers
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If $\alpha$ and $\beta$ are the zeros of the polynomial $p(s)=3s^2-6s+4$, find the value of

If $\alpha$ and $\beta$ are the zeros of the polynomial $p(s)=3s^2-6s+4$, find the value of $\frac{\alpha}{\beta}$+$\frac{\beta}{\alpha}$+2$(\frac{1}{\alpha}$+$\frac{1}{\beta})$+3$\alpha\beta$ By ...
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1answer
32 views

existence of multiplicity of roots [on hold]

Im confuse..I read in an article that in dealing with polynomials, a quadratic equation can have either 2 real roots, 1 equal real root or 2 complex roots...but in dealing with random polynomials only ...
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1answer
31 views

Why is $z^4-1-i=0$ a polynomial equation which does not have real coefficients?(complex-number)

Why is $z^4-1-i=0$ a polynomial equation which does not have real coefficients? Its coefficient is $1$ and $1$ is a real number, isn't it?
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Binary division using polynomial

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
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1answer
16 views

Division binary using polynomials

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
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0answers
38 views

Corollary of Gauss's Lemma (polynomials)

I am trying to prove the following result. I have outlined my attempt at a proof but I get stuck. Any help would be welcome! Theorem: Let $R$ be a UFD and let $K$ be its field of fractions. ...
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Express homogeneous polynom on unit sphere by higher-degree polynom

I have come across the following statement in 10.1007/BF02391776 (Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, p. 159, middle): ...
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Closed-form for one of the solutions of a specific polynomial equation of degree five of higher with integer coefficients [duplicate]

Because of Abel–Ruffini theorem we know that there is no solution in radicals to polynomial equations of degree five or higher with arbitrary coefficients. For example we know that there is no ...
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33 views

Show: Real roots of a polynomial

I´ve got some problems with this task I found in a german script. Be $\ N \in \mathbb{N}$. Define matrix $\ \Pi := (\pi)_{0\leq i,j \leq N}$ with \begin{equation} \pi_{i,j} := \binom{N}{j} p_i^j ...
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5answers
416 views

Factor a polynomial

How do I factor this polynomial: $$x^4-x^2+1$$ The solution is: $$(x^2-x\sqrt{3}+1)(x^2+x\sqrt{3}+1)$$ Can you please explain what method is used there and what methods can I use generally for 4th or ...
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4answers
54 views

Compute the largest root of $x^4-x^3-5x^2+2x+6$

I want to calculate the largest root of $p(x)=x^4-x^3-5x^2+2x+6$. I note that $p(2) = -6$ and $p(3)=21$. So we must have a zero between two and three. Then I can go on calculating $p(\tfrac52)$ and ...
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Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it devide -4 and the constant coefficient 16, don't devide the ...
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Composition of polynomials over finite fields

Consider the set of polynomials of degree at most $n$ over a finite field $k_q$ with $q$ elements where $q$ is prime: $$ P_{n,q} = \left\{ x + c_2 x^2 + \cdots + c_n x^n:\ c_i \in k_q \right\}. $$ It ...
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1answer
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X - axis of a linearized polynomial.

The other day in my Physics class we had some sample data that we wanted to linearize. The graph resembled a root curve. So to linearize it, we took the square root of all the x data and replotted ...
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Uniform convergence of polynomials (including first and second derivatives)

I am searching for a proof of the following statement: Given a twice continuously differentiable (real-valued) function on $\mathbb{R}^n$ and a compact set $K$, one can find a sequence of polynomials ...
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1answer
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Find the number of divisors of $f'(1)$

The question is that: Let $f(x) = x^{25} + 2x^{24} + 3x^{23} + 4x^{22} + \cdots + 25x$. Find the number of positive divisors of $f'(1)$. How to find this number easily? Is there only one way: ...
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2answers
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Extreme point of quadratic equation

For the below question read here: Write a function quadratic that returns the interval of all values $f(t)$ such that $t$ is in the argument interval $x$ and $f(t)$ is a quadratic function: ...
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1answer
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Exist another method to solve the problem?

We have $x_1,\:x_2,\:x_3\:\in \:\mathbb{C},\:\:f=x^3+x^2+mx+m,\:m\in \mathbb{R}$. We need to find $m\in\mathbb{R}$ such that $|x_1|=|x_2|=|x_3|$. Here is what I tried: $f=x^3+x^2+mx+m=(x^2+m)(x+1)$, ...
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2answers
335 views

Proving a polynomial has a solution in the interval (0,1)

I have no idea how to start this problem. I am assuming that the Mean Value Theorem is needed in the proof but I am not exactly sure how to apply it to the given polynomial. Any hints/help would be ...
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1answer
70 views

Inverting a map from a finite 3D grid to 1D

I need help with this mathematics question. I have made a program on the computer that flattens a 3D array into a 1D array. A 3D array needs an x, y and z but by using this formula (max x * max y * ...
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1answer
22 views

Reducible Polynomials in finite fields

I am stuck on the following question. Verify that $x^5 + x + 1$ is reducible in $Z_2[x]$ and find its factors. Help would be much appreciated whether it is the answers with how you did it or just the ...
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2answers
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Working out a polynomial from it's solutions when set equal to zero

If I have a polynomial of degree $n$ with leading coefficient $1$, that when set equal to zero has as its only solution $x=0$, how do I prove that this polynomial can only be $x^n$?
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1answer
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Is the LUB and GLB always at the most 1 unit away from a root? [on hold]

In a polynomial, like $x^4+x^3-18x^2-16x+32$, is the LUB and GLB always at the most 1 unit away from a root? Foe example, is there any case where the greatest root is at (1,0) and the LUB is at 5? ...
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3answers
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Find a nontrivial unit polynomial in $\mathbb Z_4[x]$

Find a unit $p(x)$ in $\mathbb{Z}_4[x]$ such that $\deg p(x)>1$ What I know: A unit has an inverse that when the unit is multiplied by the inverse we get the identity element. But I am confused ...
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1answer
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Find the gcd of polynomials

This is for a modern algebra course. Find the greatest common divisor of each of the following pairs of $p(x)$ an $q(x)$ of polynomials. If $d(x)=gcd(p(x),q(x))$, find two polynomials $a(x)$ and ...
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If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
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1answer
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Find a polynomial that satisfies the following conditions. [on hold]

Let $f(x)$ is a polynomial of the degree $4$. Suppose $f$ verifies the following: $(x-1)$ is a factor of $f(x)$; When $f(x)$ is divided by $(x+1)^2$ the remainder is $-4(2x+1)$; When ...
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Unusual series expansion - how to find coefficients?

I have encountered an equality $$ \sum_{n=0}^\infty p_{3n}(t)\left(\frac t{(1-t)(1-2t)}\right)^n=\frac{1-2t}{1-4t}. $$ I know that $p_0(t)=1$, $p_3(t)=2(1-t)^2$, while for $n>1$, $p_{3n}(t)$ is a ...
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2answers
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Decomposition of continuous function

I've been playing with this a bit and not making much progress. If I have a increasing, continuous function $f$ on $[0,1]$ such that $f(1/2)=0$, is it always possible to find some function $\hat{f}$ ...
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1answer
35 views

How to evaluate real root of a polynomial equation? [on hold]

If $\alpha$ is a real root of the polynomial equation $$300x^{299}+299x^4+343x^3+23x+300=0$$ Then how to find out the value of $[\alpha]\space $ where, '$[ \space]$' denotes greatest integer? I have ...
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0answers
4 views

Regular Subresultant PRS from Euclidean PRS?

Is there any way to compute regular subresultant polynomial remainder sequence If I know the Euclidean polynomial remainder sequence of two univariate polynomials?
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Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
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1answer
12 views

Polynomials and Euclidean algorithm

I have the next problem. Determine the gcd $d(x)$ of two polynomials with real coefficients $a(x) = x^{4}-1$ and $b(x) = x^{3}-x^{2}-x+1$. And then, determine two polynomials with real coefficients ...
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0answers
15 views

Ideal generated by two polynomials

Let a sequence of polynomials $\{f_n\}_{n=0}^\infty$ in $\mathbb{Q}[x,y]$ be given in the following way: $$f_0=1,$$ $$f_1=-x,$$ $$f_2=x^2-y,$$ $$f_{n+2}=-xf_{n+1}-yf_n.$$ For each $n\geq 0$, find ...
2
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1answer
42 views

dimension of quotient by algebraically independent elements

Let $f_1,\dots,f_s$ be algebraically independent polynomials of $A:=k[x_1,\dots,x_n]$, $s \le n$. Recall that algebraically independent means that there is no non-zero polynomial $g \in ...