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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2
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3answers
53 views

Factorize $2a^3 - b^3 - c^3$

I need to factorize the expression $2a^3 - b^3 - c^3$. I see that one zero is achieved when $a=b=c$, but I can't find the factor(s).
1
vote
0answers
35 views

How do I see if $g$ is a polynomial or not??

Let $u$ be a real valued harmonic function on $\mathbb{C}$. Let $g: \mathbb{R^2} \to \mathbb{R}$ be defined by: $$g(x,y)=\int_0^{2\pi}u(e^{i\theta}(x+iy))\sin \theta \,d\theta$$ Which of the ...
0
votes
1answer
36 views

If $2^{x + 1} < y$, then what is the largest polynomial in $x$ that cannot be an upper bound for $y$?

Update: I have posted a follow-up question here. The title says it all. If $2^{x + 1} < y$, then what is the largest polynomial in $x$ (of maximum possible degree) that cannot be an upper ...
1
vote
1answer
27 views

Proving P(x) > 0 given a condition.

$P(x)$ is a polynomial function such that, $P(1) = 0, P′(x) > P(x), ∀ x > 1. $ Prove that $P(x) > 0, ∀ x > 1.$ I was trying to do by taking the P(x) in the denominator and then ...
2
votes
1answer
31 views

Non-linear optimization programming

How many methods do we have for non-linear optimization problems, which the target function is linear but constrains are polynomial shape? Are there methods which can solve most of them? Or what ...
2
votes
2answers
35 views

Polynomial with no integer roots

This is an excercise given to a kid I am tutoring, as part of a set of problems regarding polynomials. He is currently at the last class before graduation year. Let $p$ be a polynomial in $ℤ[x]$ such ...
0
votes
1answer
37 views

Finding the general Taylor polynomial formula for $f(x)=\log\left(\frac{1+x}{1-x}\right)$

I am trying to find the general form of the Taylor polynomial for $f(x)=\log\left(\frac{1+x}{1-x}\right)$. The $log$ is of base $e$ and I have rewritten the original formula as: $\log(1+x)-\log(1-x)$ ...
1
vote
3answers
59 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
0
votes
2answers
19 views

Polynomial with bounded coefficients and real root

A polynomial with degree $2n$ has all coefficients in the range $[100,101]$ and has a real root. What is the minimum possible $n$? Degree $0$ is clearly not possible. For degree $2$, the discriminant ...
2
votes
0answers
30 views

If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?

If $\alpha$ is an algebraic element and $L \subset K$ are both field, does the polynomial ring $L[\alpha]$ is also a field? I am trying to prove that the ring of fraction $L(\alpha)$ is equal to ...
9
votes
4answers
483 views

How can I guarantee the unique positive root of this polynomial?

How can I guarantee the unique positive root of this polynomial? I have two polynomial, $$ x^{n+1} + x^n - 1 =0 $$ and $$ x^{n+1} - x^n - 1 =0 $$ respectively, where $n\in\mathbb{N}$. I have ...
0
votes
1answer
9 views

Writing the function to maximize volume or a cylinder

A rectangular piece of paper is curled into a cylinder with two open circles on each side. The perimeter of the piece of paper is 124 inches. What is a function that could be written to find the ...
0
votes
0answers
24 views

Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. ...
2
votes
4answers
47 views

Elementary symmetric polynomial task with three variables

Can anyone help me to wite this as sum or product of elementary symmetric polynomial. $$\frac xy+\frac yx +\frac xz + \frac zx +\frac yz + \frac zy =7$$ I tried to set under one fraction, but I ...
4
votes
1answer
99 views

root pattern of second degree polynomial

I'm considering the following 2nd degree polynomial for the case where the roots are complex conjugate. $ P(z) = z^2 + (f^2 + f q -2)z + (1 - f q) = (z - z_1) (z - z^*_1) $ where f and q are real ...
1
vote
1answer
40 views

Finding general form of Taylor polynomial for function $f(x)=e^{x}\sin(x)$

I am trying to find the general form the Taylor polynomial of the function $f(x)=e^{x}\sin(x)$. I have calculated the derivatives up to $5$: $$\begin{align} f^{(1)}(x)&=e^{x}\cos(x) + e^{x} ...
1
vote
2answers
31 views

Type of polynomial where leading coefficient is to the power of $6$ [duplicate]

I need to identify the type of polynomial that a polynomial is based on the power of the leading coefficient. (Example $x^2$ = quadratic, $x^3$ = cubic, $x^4$ quartic). In this case, it is $x^6$. What ...
0
votes
0answers
25 views

Is there an “easier” way to find the factors of a polynomial than using Ruffini's method?

I am a first year Mathematics student, and sometimes I have a hard time with Ruffini's method for polynomials, specially in the field of Rationals
0
votes
0answers
12 views

Concave property on elementary symmetric polynomials

Let ${\sigma _k}$ be the k-th elementary symmetric polynomial, namely ${\sigma _k}({x_1},...,{x_n}) = \sum\limits_{1 \leqslant {i_1} < ... < {i_k} \leqslant n} {{x_{{i_1}}}...{x_{{i_k}}}} $ ...
1
vote
0answers
25 views

Clarifications regarding Lagrange resolvent

I'm trying to understand the technique used by Lagrange to solve cubic and quartic equations. I have read that the Lagrange resolvent for the cubic is $$ x_1+\omega x_2+ \omega^2 x_3 $$ where ...
0
votes
0answers
17 views

Root of interpolated polynomial when y-coordinates are permuted

Hypothesis: All values and polynomials are defined over a field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit) Suppose we have $n$ pairs of $(x_i,y_i)$. As we all know, given the ...
4
votes
3answers
44 views

What is the extraneous solution of $\sqrt a=a-6$?

What is the extraneous solution of $$\sqrt a=a-6$$ The roots are $9$ and $4$. So I'm assuming that $4$ is the extraneous solution because when you plug it in to the equation you wind up with $2=-2$. ...
0
votes
1answer
26 views

Polynomial division without remainder

First, some background on what I'm actually trying to achieve: I have a reflectance filter (a discrete IIR) for use in an FDTD boundary condition: ...
0
votes
0answers
14 views

Characteristic polynomial of a graph and structure function of a graph?

The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. It is a graph invariant, though it is not complete: the smallest pair of non-isomorphic graphs ...
2
votes
3answers
18 views

Construct field Extension given that $\alpha$ has the following minimal polynomial

I apologize I have nothing to show that I have really attempted this question, but it's simply because I am struggling to get used to the terms and ideas. I randomly chose a question from a past paper ...
0
votes
1answer
39 views

Multiplicative Inverse in a $256$ Galois Field

I am working on finding the multiplicative reverse in $GF(2^8)$ using the Euclidean Algorithm but after reading multiple sources, I feel as though I am proceeding incorrectly. Using the irreducible ...
2
votes
2answers
40 views

When the Polynomial is bounded The coefficients are bounded

The actual question is as follows: Suppose you have a sequence of polynomials of degree $n$, $$P_j(x)=\sum_{k=0}^na_{jk}\,x^k\rightarrow P(x) $$ which is bounded in $n+1$ points for $x$ for all ...
0
votes
2answers
30 views

Prove there isn't an isomorphism between quotient polynomial rings

Prove there isn't an isomorphism $$\phi: {{\mathbb Q [x]} \over {I_1}} \to {{\mathbb Q [x]} \over {I_2}}$$ when $I_1=\langle x^2-2\rangle$, $I_2=\langle x^2+2\rangle$. I want to assume there is an ...
11
votes
0answers
90 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: $$f(1+x+g(x)^{2})=g(1+x+f(x)^{2})$$ Prove $f(x)=g(x)$. Rather naturally, I ...
1
vote
1answer
54 views

Gaussian polynomial identities

I'd appreciate any hints for showing that these identities are true for Gaussian polynomials. I've tried to approach the problem using basic algebra but it gets messy very quickly and I've gotten ...
0
votes
0answers
16 views

How to separate a real integer polynomial into two with real/complex roots?

If the coefficients of a polynomial p(x) are all real integers, then every root of p(x) is either 1) real or 2) a complex number whose conjugate is also a root of p(x). Is there any easy way to ...
0
votes
0answers
42 views

Solutions of a parameterised system of equations and inequations - 2 accelerating objects catching up

First, let me describe what I want to acheive : an object A is trying to catch up with an object B. They are separated by an initial distance $x_0$ but their initial speed is 0. B is accelerating at a ...
1
vote
1answer
26 views

What is the equation of the bottom half of the parabola $x + (y - 2)^2 = 0$?

A parabola has the equation: $$x + (y - 2)^2 = 0$$ I can't find the $y$ without getting the equation into some weird recursion.
2
votes
0answers
48 views

The convergence of the fixed-point iteration for solving a cubic equation

I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this: $0=Ax^3+Bx+C$ and for a given ...
2
votes
0answers
38 views

How to tell if a polynomial has exact trigonometric or logarithmic roots?

The polynomial $$ 64x^7 -112x^5 -8x^4 +56x^3 +8x^2 -7x - 1 = 0 $$ has seven roots, x = {1, $-\dfrac{1}{2}, \cos \dfrac{2n\pi}{11}$}, where n={1,2,3,4,5}. Is there any way to tell if an arbitrary ...
1
vote
0answers
49 views

Fastest way of find roots of polynomial defined over a finite field

Suppose we have polynomial $G(x)$ of degree $d$, where $d$ is a large value (e.g. $10^6$). The polynomial is defined over a finite field $\mathbb{F}_p$ for a large prime number $p$ (e.g. $p$ is ...
0
votes
0answers
24 views

Are there separable polynomials in $K[Y][X]$ with constant discriminant?

Let $A$ be a ring and $P\in A[X]$ be a monic degree $n$ polynomial. Let $Disc(P):= Res(P,P')$ be the discriminant of $P$. If $A=\mathbf Z$, then Minkowski's theorem says that there are no non ...
0
votes
1answer
21 views

Testing if a polynomial has roots within a radius/range

Is there a way to test if a high-order polynomials has any roots within a radius r of a specified point? I need this so that I can find all the complex roots of the following system for arbitrary ...
1
vote
0answers
34 views

Jacobian criterion algebraic independence

I have two polynomials $f_{1}(x,y)$ and $f_{2}(x,y)$ and I want to know if they are algebraically independent. I am using the Jacobian criterion which says that $f_{1}$ and $f_{2}$ are algebraically ...
2
votes
1answer
57 views

Commutativity of “extension” and “taking the radical” of ideals

Let $K$ be a field (not necessarily algebraically closed) and $\overline{K}$ its algebraic closure. By $K[\text{X}]$, I mean $K[X_1,...,X_n]$. Is it true that the operations of "extension" and ...
2
votes
5answers
83 views

Coefficient of $x^{50}$ in the expansion of $\prod_{n=1}^{52}{(x+n)}$

Find the coefficient of $x^{50}$ in the expansion of $$\prod_{n=1}^{52}{(x+n)}$$ I can't find a way out.
2
votes
2answers
39 views

Finding the Jordan Form of a matrix

Let $A$ be a 7*7 matrix with a single eigenvalue $q\in C$. It is know that $\rho (A-qI) = 2$ and that $\rho (A-qI)^2 = 1$. How can I find the Jordan form of A (+ the minimal polynomial)?
1
vote
0answers
31 views

Size of coefficients of polynomials that satisfy a Chebyshev-like extremal property

The famous Chebyshev polynomials satisfy many extremal properties. One of these is that they attain the largest possible derivative over the interval [-1,1] among polynomials whose absolute value over ...
1
vote
2answers
26 views

Finding roots of cubic (trig)

The question is By putting $x$ $=$ $\frac 23 cos (\theta)$ Find the exact roots of the equation in terms of $\pi$ $$ 27x^3 - 9x = 1 $$ What I have attempted: $$ ...
0
votes
1answer
14 views

S-polynomial remainder of two polynomials

In Page 90 of Ideal, Varieties and Algorithms, it shows a calculation of two polynomials being $x^3-2xy$ and $-2xy$, and shows the remainder being zero when using the S-polynomial method, but whenever ...
1
vote
1answer
18 views

Comparing a series expansion to polynomial regression

So I don't have a great background in mathematics but I have a quick and hopefully simple question for you guys. I'm a graduate student and I'm doing some polynomial regression on some thermodynamic ...
1
vote
2answers
26 views

Is there a concrete definition/formula for finding the leading coefficient of any polynomial?

Is there a concrete definition that tells one the leading coefficient of any polynomial? Using logic, I derived this formula: $$ a=\frac{\frac{d^p}{dx^p}f(x)}{p!}$$ where $f(x)$ is a polynomial, $p$ ...
0
votes
1answer
25 views

Why is the set $U$ with $p(0)=c$ only a subspace of $P_3$ when $c\in \mathbb{R}=0$ ?

I'm having trouble grasping why the set $$ U_c = \{ p \in P_3\ |\ p(0) = c \}$$ with $c\in \mathbb{R}$ only counts as a subspace of $P_3$ when $c=0$. I've been told that it wouldn't be a valid ...
1
vote
1answer
33 views

Divided differences of a polynomial

Let $f(x)$ be a polynomial of degree $ n $. Prove that the $k$-th divided difference $f[x_0, x_1,... x_k]$ is a polynomial of degree $n-k$, with respect to one of the variables $x_0, x_1, ... x_k$. ...
5
votes
2answers
90 views

Is $‎‎‎\sqrt[3]{y^3}‎‎‎$ or $\frac{x^2}{x}$ a polynomial?

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Now are $$‎‎‎\sqrt[3]{y^3}‎‎‎,\quad ...