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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Roots of $x^4-5x+5$

Suppose $z$ is a complex root of $x^4-5x+5$. What is the extension degree of $\mathbb{Q}(z):\mathbb{Q}$? I suspect it is 4 but I don't have any strategy how to prove it.
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Finding root of a polynomial

Let's say I've got $ax^3 + bx^2 + cx + d = 0$ I want to find x for a general-case. Mathematica gave me a very long solution, and even longer for one which has $kx^4$. Can you please help me ...
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123 views

Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$?

Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$? The values are numbers such that each prime factor of the form $4k+3$ occurs ...
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How to show that $GCD(f,f')=(x-a_1)^{r_1-1}\cdots(x-a_l)^{r_l-1}$

Given any polynomial $f\in \mathbb C[x]$ of degree $n>0$, $f$ can be written in the form $f=c(x-a_1)^{r_1}\cdots(x-a_l)^{r_l},$ where $a_1,...,a_l$ are distinct. Also, $f'$ is the product ...
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What is the name for the polynomials of the form : $ P_n(x)=2^{-n} \cdot ((x+\sqrt {x^2-4})^n+ (x-\sqrt {x^2-4})^n)$?

Polynomials of the form : $ T_n(x) =2^{-1} \cdot ((x+\sqrt {x^2-1})^n+ (x-\sqrt {x^2-1})^n)$ are known as Chebyshev polynomials of the first kind . Consider the polynomials of the form : ...
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473 views

Construct ideals in $\mathbb Z[x]$ with a given least number of generators

How do you construct, for each $n\geq 1$, an ideal in $\mathbb Z[x]$ of the form $(a_1,a_2,\dots,a_n)$ with $a_i\in \mathbb Z[x]$ such that it is impossible to have ...
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Necessary and sufficient condition for $f(q^n)$ to be in $\mathbb{Z}[q,q^{-1}]$ when $f\in\mathbb{Q}(q)[x]$?

In this question, user begins shows that, for each $k\in \mathbb{N}$, there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are in $\mathbb{Q}(q)$, the field of rational functions, ...
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Factoring polynomial in two variables.

Given a polynomial $P(x,y)$ I would like to know what the criteria are for factoring out linear factors. For instance, in one variable, if $Q(a) = 0$, then one may say $Q(x) = (x-a)R(x)$. In two ...
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695 views

How do you solve equations of any degree?

I have stuck solving this problem of financial mathematics, in this equation: $$\frac{(1+x)^{8}-1}{x}=11$$ I'm stuck in this eight grade equation: ...
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Neat problems using roots of polynomials

On a recent test in a course I'm TAing, students were asked to prove that sin(x) is not a rational function by using the fact that polynomials only have finitely many zeroes. During my tutorial, I'd ...
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65 views

Factorizing a polynomial

Can we factorize the polynomial $f(a)=1+a$ so that it is a product of 2 polynomials each of which is not a unit in the ring $R[a]$? I don't think it is possible but I am not sure why. The reason I ...
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441 views

Set Theory: Polynomial Relations

I'm having a bit of trouble understanding exactly what this question is asking me in my Sets and Proofs homework: If a polynomial p over $\mathbb{R}$ is an expression of the form $p(x) = a_nx^n + ...
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76 views

Why does $\text{disc} f=\text{res}(f,f')$?

Over the complex numbers, I'm familiar with the fact that the discriminant of a polynomial $f$ and the resultant of $f$ and $f'$ are equal. Now say you have an arbitrary polynomial $$ ...
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425 views

Why solve polynomial equations?

Most people learn in linear algebra that its possible to calculate the eigenvalues of a matrix by finding the roots of its characteristic polynomial. However, this method is actually very slow, and ...
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Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer?

Are there any computer algebra systems with the functionality to allow me to enter in an explicit symmetric polynomial and have it return that polynomial in terms of the elementary symmetric ...
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530 views

An application of Vandermonde determinant

Let $\lambda_1,\ldots,\lambda_n$ be complex numbers such that for each positive integer $k\geq 0$, $$\sum_{i=1}^n \lambda_i^k=0.$$ Here I am supposed to show that $\lambda_i=0$ for each $i\in ...
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649 views

Quotient Rings of Polynomials Over Finite Fields

I have this question which I don't know how to approach: Let ${F}_{2} = {Z}/2Z$, find representatives for the residue classes of ${F}_{2}[X]$ modulo the polynomial $f(x)$ and compute the ...
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About $t$-analogue of the Euler polynomials.

A certain way to define the $t$-analogue of the Euler polynomials $C_n(x)$ is by $$ C_n(x,t)=\sum_{\pi\in S_n}x^{\text{des}(\pi)+1}t^{\text{maj}(\pi)} $$ where $des(\pi)$ is the descents in $\pi$, ...
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645 views

Determine the values of $k$ for which the equation $x^3-12x+k = 0$ will have three different roots.

Determine the values of $k$ for which the equation $x^3-12x+k = 0$ will have three different roots. I can do this with calculus, I was just wondering what could be the pure algebraic approach to ...
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Polynomial in $\mathbb{Q}[x]$ sending integers to integers?

We can view the binomial coefficient $\binom{x}{k}$ has a polynomial in $x$ with degree $k$. So taking some $f\in\mathbb{Q}[x]$, why is $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$, precisely when the ...
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674 views

Roots of complex polynomial

I have a complex polynomial $z^2 + 2z + 2$. From trying out roots repeatedly I can see the correct roots are $z = -1 + i$ and $z = -1 - i$ But what is the standard method for getting roots of a ...
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For a polynomial $P$ for which $P(x+5) - P(x) = 2 ,\forall x$. What is the least possible value of $P(4) - P(2)$?

There are infinite number of polynomials $P$ for which $P(x+5) - P(x) = 2,\forall x \in \mathbb{R}$. How could we determine the least possible value of $P(4) - P(2)$?
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Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\backslash 0$?

Let $k$ be a field. Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\backslash 0$? I can do it for specific polynomials, but I'm struggling to structure a ...
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Multiple perturbations to cubic equation

Suppose $\alpha\in(0,\frac12)$ and $\beta\in(0,\infty)$ are fixed. Initially I have $N\in\mathbb N\backslash\{0\}$ and $n\in\{0,\ldots,N\}$. I'd like to know, as a function $n$, the solution of the ...
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96 views

Location of a root of a cubic polynomial

For $\alpha\in(0,\frac12)$, $\beta\in(0,\infty)$, $N\in\mathbb N\backslash\{0\}$ and $n\in\{0,\ldots,N\}$, how can I prove that exactly one zero of the cubic polynomial $$ ...
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1answer
57 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
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subalgebras of a polynomial ring

If $k$ is a field, I know that any subalgebra $A \subset k[x]$ is finitely generated, but I wonder if there is a good algorithm to find a set of generators for $A$. In particular, if $(f) \subset ...
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Factorizing polynomial $x^5+x+1$

I'm given a problem to factorize $$ P(x)=x^5+x+1 $$ I've done the following: $$ P(x)=(x^5+x^4+x^3)-(x^4+x^3+x^2)+(x^2+x+1)= (x^2+x+1)(x^3-x^2+1)$$ Is it possible to prove that this cannot be ...
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Complexity of finding solutions for a system of polynomial equations

Problem A: Given a set of polynomial equations $ f_1=0,\ldots,f_m=0 $, where the $ f_i $'s are multivariate polynomials with $ n $ variables over a field $\mathbb F$, decide whether there is a ...
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846 views

Product of all irreducibles with degree divisible by $n$ in $\mathbb{F}_{q^n}$?

In the finite field of $q^n$ elements, the product of all monic irreducible polynomials with degree dividing $n$ is known to simply be $X^{q^n}-X$. Why is this? I understand that $q^n=\sum_{d\mid ...
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A nicer recurrence for the Eulerian polynomials.

I was perusing the subject of Eulerian polynomials. I'm assuming the definition that the Eulerian polynomial is defined by $C_n(t)=\sum_{\pi\in S_n}t^{1+d(\pi)}$, where $d(\pi)$ is the number of ...
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Recurrence $C_n(t)=t(1-t)C'_{n-1}(t)+ntC_{n-1}(t)$ for Eulerian Polynomials?

I was reading about Eulerian polynomials on OEIS, and there is a recurrence given for them, namely: $$ C_0(t)=1 $$ and $$ C_n(t)=t(1-t)C'_{n-1}(t)+ntC_{n-1}(t)\qquad (n\geq 1). $$ How can ...
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Characteristic and minimal polynomial - leading coefficient and norming

When calculating the characteristic polynomial as $$\det \; (A−t E_n)$$ I get the same polynomial as when I calculate the characteristic polynomial as $$\det\;(t E_n−A).$$ Only the signs are changed. ...
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77 views

How do we bound the cardinality of this group

Let $g$ be a positive integer. How do I bound the number of elements of the group $Sp(2g,\mathbb{Z}/15)$? Is there a polynomial bound in $g$, or can we not do better than exponential in $g$?
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simplify complex polynomial $p(t) \in \mathbb{C}[t]$

How to simplify the following polynomial? $$ \begin{align} (t - \sqrt{3} \; e^{ \frac{\pi}{3} i }) (t - \sqrt{3} \; e^{ -\frac{\pi}{3} i }) &= t^2 - \sqrt{3} \; e^{ \frac{\pi}{3} i } \; t - ...
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Quick ways to _verify_ determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …

What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them? So if I calculated determinant, minimal ...
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406 views

polynomial long division - Euclidean algorithm

Why is it that I can change the sign of a polynomial during the Euclidean algorithm as in the following example: $$ \begin{align} (t^5 + t^3 -4t) : (t^5 + 2 t^3 - t) &= 1 \quad ...
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Congruence question

Hi I would like a hint with the following congruence question. $$1+x^{1}+x^{2}+\cdots +x^{6}\equiv 0\mod{29}$$ Is there a formula I should be looking for to group the left hand side?
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roots of complex polynomial - tricks

What tricks are there for calculating the roots of complex polynomials like $$p(t) = (t+1)^6 - (t-1)^6$$ $t = 1$ is not a root. Therefore we can divide by $(t-1)^6$. We then get $$\left( ...
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Simplifying $1 - x + x^2 - x^3 + … + x^{98} - x^{99}$ to an equivalent expression.

I am doing an exercise to see the error when solving this polynomial for $x = 1.00001$ using nested multiplication. I believe the correct way to achieve this simplification (based on a lecture) is to ...
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sum of reciprocals of derivative of polynomial at its roots

If $P(x)$ is a polynomial of degree $n > 1$ with only simple roots $a_1,\ldots,a_n$, is it true that $\frac 1{P'(a_1)} + \cdots + \frac 1{P'(a_n)} = 0$, and, if so, what is the proof? I ...
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Every positive polynomial is above a completely Q-factorized positive polynomial?

Let $P$ be a unitary polynomial with rational coefficients in one variable $x$, such that $P(x) \geq 0$ for all $x \in \mathbb R$. Then $P$ is of even degree, say $2d$. Is it true that there always ...
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Polynomial and exponential regression [duplicate]

Possible Duplicate: Determining computational complexity of stochastic processes I have some points $(x_i,y_i)$ generated by a program. These values are not exact, but are random ...
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71 views

What kind of polynomials?

I consider polynomials $p_n(z)$ such that $p_0(z) = 1$, $p_{n+1}(z) = ( p_{n}'(z)-p_{n}(z) )z$, so $p_1(z) = -z$, $p_2(z) = z(z-1)$, $p_3(z) = -z + 3z^2 - z^3$. Are they well-known? Do they have ...
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Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
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419 views

A problem on Lagrange interpolation polynomials

Based on a previous question, I had the following conjecture and was wondering if anyone knew how to prove it or find a counterexample. Consider the polynomial $$ ...
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561 views

Show that this polynomial is positive

Consider the following polynomial in two variables : $$ Q(k,x)=27x^6 - 144kx^4 + 80k^2x^3 + 240k^2x^2 - 192k^3x + (64k^4 - 128k^3) $$ Then for any integer $k \geq 5$, the polynomial $Q(k,.)$ (in one ...
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906 views

Remainder term of Lagrange Interpolation Polynomial

Suppose $x_0,x_1,\ldots,x_n$ are $n+1$ distinct numbers in the interval $[a,b]$ and $f\in C^{n+1}[a,b]$. Then for each $x$ in $[a,b]$, there is a number $\xi$ in $(a,b)$ such that $$f(x) = P(x) + ...
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1answer
59 views

A problem with polynomials.

This is a problem from a test in my course in analytic functions. I didn't manage to solve it. Could you please give me a hint? The problem is: Calculate the third root of the sum of the coefficients ...
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391 views

On rank of a matrix whose entries are polynomials

(I took courses on linear algebra, but I don't know anything about $R$-modules or such things.) How do you define the rank of a matrix whose entries are polynomials in $K[X]$? If you assign some ...