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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19 views

A question in numerical range

Suppose ${A_j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a complex ...
2
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2answers
24 views

Quadratic form has non-trivial zero?

For each of the following quadratic forms, determine whether the form has a non-trivial zero (we do not need to exhibit it): $f(x, y, z) = 2x^2 + 3y^2 - 6z^2$; $g(x, y, z) = 2x^2 + 3y^2 - ...
2
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2answers
41 views

coefficients that make $p(x)=(a_1x+b_1)^3+(a_2x+b_2)^3+(a_3x+b_3)^3+(a_4x+b_4)^3-x$ a constant

Find integers $a_i$ and $b_i$, $i=1,2,3,4$, such that $p(x)$ is a constant function: $p(x)=(a_1x+b_1)^3+(a_2x+b_2)^3+(a_3x+b_3)^3+(a_4x+b_4)^3-x$ I don't even know if such coefficients exist or not. ...
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0answers
20 views

Number of solutions of “pseudo-polynomials”.

Let's call an equation of the form: $$ \mathcal P=\sum_{i\geq 0} a_ix^{k_i} $$ A pseudopolynomial if it satisfies: There exists a $q\in \Bbb N$ such that $a_q,a_{q+1},...$ are zero. $a_i,k_i \in ...
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0answers
46 views

What are the roots of the polynomial $x^{3}+3x-2\pi$ $?$

By using Dedekind's sign change rule , I can tell this polynomial $$x^{3}+3x-2\pi$$ has one real root. But I want to know what that root is and what the factorization of ...
6
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2answers
62 views

$a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$.

I was curious if there are quadratic equations where $a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$. So trivially if $c=0$, $a$ and $b$ can be arbitrary, and if either $a$ or $b$ is ...
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1answer
18 views

How to derive formula for second-order divided difference from a naive calculation?

I'm trying to study polynomial interpolation for my numerical analysis course, but I've become absolutely stuck in proving deriving the second order divided difference for myself: Assuming we have ...
0
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1answer
20 views

Only finitely many reduced cubics with given discriminant?

I call a cubic form $x^3+ax^2+bx+c$ (with $a,b,c\in{\mathbb Q}$) reduced when $a=0$. The discriminant is then $-4b^3-27c^2$. For a fixed $t\in{\mathbb Q}$, is there an algorithm to decide if there ...
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1answer
25 views

How to factorize polynomials

How can I factorize the polynomial $$-a(b-c)(x-b)(x-c)-b(c-a)(x-a)(x-c)-c(a-b)(x-a)(x-b)?$$ Please show me the whole process step by step in an easy way.
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0answers
32 views

How is Buchberger algorithm a generalization of the Euclid GCD algorithm?

It is said in many places (for example, on the Wikipedia article for Buchberger's algorithm) that Buchberger's algorithm to find Groebner basis is a generalization of Euclid's GCD algorithm. This is ...
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5answers
594 views

How can this expression be simplified?

How do I factorize $$a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)?$$ I've tried it in different ways but failed. Wish some one could help solving it out.
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0answers
35 views

How to compute the correct residue without change of modulus?

Suppose $F=1857$, $G=2017$, and we want to compute $ FG \equiv 1606$ mod $(2^{11} -1)$. Also let $t =2^4$, then $F = 1+4t+7t^2$ and $G = 1+14t+7t^2$, where the most significant digits are bounded ...
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0answers
18 views

Can any integer Y be represented by a polynomial sum of an integer x - with binary coefficients?

Can any integer Y be represented by a polynomial sum of a positive integer x, where the polynomial coefficients can only be 0 or 1? And can you represent any base 10 integer Y as an integer number in ...
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1answer
17 views

How to find the number of positive or negative real factors by using descartes' sign rule?

I have a problem about finding the number of positive or negative factors of polynomial. Here is the question: How many positive or negative real factors from this polynomial? ...
2
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2answers
42 views

$\frac{d^{100}}{dx^{100}}\left[\frac{f(x)}{g(x)}\right]=\frac{p(x)}{q(x)}$,then find the degrees of the polynomials $p(x)$ and $q(x)$

Let $g(x)=x^3-x$,and $f(x)$ be a polynomial of degree $\leq100$.If $f(x)$ and $g(x)$ have no common factor and $\frac{d^{100}}{dx^{100}}\left[\frac{f(x)}{g(x)}\right]=\frac{p(x)}{q(x)}$,then find the ...
2
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2answers
49 views

Firing Solution on a Moving Target

I need to calculate the 3-component $\vec V$, which is the gun barrel vector needed to hit a target moving at a constant velocity. To find this information I'll also need to find $t$ which is the time ...
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1answer
34 views

Product and intersection of ideals in a polynomial ring

I want to show that in the polynomial ring $K[X,Y,Z,W]$ (where $K$ is a field) the equality $(X,Y)\cap(Z,W)=(XZ,XW,YZ,YW)$ holds. Obviously RHS is contained in LHS. How to show that LHS is ...
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1answer
33 views

Irreducible linear set of quadratics over $\Bbb F_p$

Given $a,b\in\Bbb F_p$, denote $$S(a,b)=\big\{(a+\beta)x^2+(b-\beta)x+1\in\Bbb F_p[x]:\beta\in\Bbb F_p\big\}.$$ Denote $$S(a,b)_\mathrm{red}=\big\{g(x)\in S(a,b):g(x)\text{ is reducible}\big\}.$$ ...
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0answers
64 views

How to factor $81x^2+16y^6$? [on hold]

How to factor $81x^2+16y^6$? I know it is elementary but I just have no idea. Thanks!
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0answers
37 views

Possible proof strategy for Sendov conjecture?

Sendov's conjecture says that if all roots of a polynomial lie within the unit disk, then for every root, there exists a critical point at a distance at most one from the root. I read that Sendov ...
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1answer
22 views

Approximation of continuous functions by Bernstein polynomials

Recently a professor show me the following heuristic to provide approximations of continuous functions by polynomials: Let $P_n(x) = \sum_{k=0}^{n} {n \choose k} f(\frac{k}{n}) x^k (1-x)^{n-k}$. ...
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1answer
22 views

Solve for $x$ in the following inequality

$2x+x^3+7\gt0$ I have no idea what to do here, because my level of knowledge goes up to binomials and such. Thank you
11
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1answer
91 views

Why is this approximation of polynomial root so accurate?

I have an engineering problem where I have to find the smallest positive real root of a polynomial in $x$: $$Ax^5+Bx^3 - C = 0$$ Instead of solving numerically, I want simple approximative formulas ...
1
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1answer
30 views

how to show a polynomial inequality

How to show something like this? $$ \frac{[1-(1-p)^n]^2}{np} > 1- (1-p)^n - \frac{n-1}{n} [1-(1-p)^{n-1}]$$ where $0<p<1$, and $n \geq 2$ is an integer.
1
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1answer
16 views

verification that simplification in textbook is incorrect

I am using a textbook that asks for the following expression to be simplified: ${9vw^3 - 7v^3w - vw^2}$ The answer given is: ${8vw^2 - 7v^3w}$ which seems wrong because I did not think you could ...
0
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1answer
20 views

Show that a set of polynomials are linearly independent in the complex space

I have been trying the solve the following question without any success: Let $\lambda_1, \lambda_2, \lambda_3$ be three distinct complex numbers and define the polynomials $m(\lambda), m_1(\lambda), ...
4
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2answers
83 views

Considered as functions $\Bbb C \to \Bbb C$, are polynomials open maps?

If $p$ is a polynomial, is it true that $p(A)$ is open for every open $A\subseteq\Bbb C$? I really don't know how to approach this. I'm fairly certain that they are closed maps, though.
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1answer
7 views

Walking through the reduction of a cumulative probability function to a polynomial

Setup Define $P(p)$ as follows: $$ P(p) = \sum_{N_1-\phi \cdot N_2 \geq \theta} {n_1 \choose N_1} {n_2 \choose N_2} p^{N_1 + N_2}q^{n_1 + n_2 - N_1 - N_2}. $$ Here, $$ q = 1 - p. $$ The sum is ...
3
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1answer
57 views

Is there a nice expression for $f(x) = (1+x)(1+x^2)(1+x^3)\cdots$

While I was solving a problem, I stumbled upon this function $$f(x) = (1+x)(1+x^2)(1+x^3)\cdots$$ I tried to write out the first few products but I couldn't recognize any meaningful pattern. Is there ...
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0answers
23 views

A question on matrix norm

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
2
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3answers
66 views

Why does the a*c cheat work when factoring trinomials?

When factoring a trinomial, in the form $ax^2 + bx + c$, I am told that one can multiply $a$ and $c$ which gives a product whose factors add to $b$. So if I have $2x^2 + 5x -3$ that gives me $-6$. ...
2
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1answer
28 views

How to create a ring in MAGMA with relations?

I'm using MAGMA221 and would like to create a ring over $GF(2)$ with respect to a list of relations. Here's what I have so far: $\mathtt{Z:=GF(2);} \\\mathtt{P<x,y,z>:=PolynomialRing(Z,3);}$ ...
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0answers
10 views

What's set of two variables over field? [on hold]

What's The set of polynomial ring $F_{4}[x,y]$ ? and What's primitive variables of $GF(4)$?($F_4$ is a finite field)
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0answers
22 views

Using Magma to solve a multivariate polynomial system with parameters

I want to solve a system of multivariate polynomials with parameters. Mathematically, the ground field is F = Q(a, b, c, …), the field of rational functions. The polynomials are in F[x,y,z,…]. I ...
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3answers
28 views

Finding remainder

Okay I saw this one on a test so here it goes: A polynomial of (degree > 3) when divided by $ (x-1)^2$ and $x-3$ leaves the remainder $2x+1$ and $15$ respectively. The remainder when it is divided by ...
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1answer
14 views

How to resolve polynomials with just two terms where one of them is a constant

1) I have the information that to resolve: $$ x^n = 3*x^{n-4} $$ 2) I would need to solve $$ x^4 = 3 $$ 3) Whose positive value would be: $$ x = 3^{1/4}. $$ 4) Which is then: $$ x = \sqrt[3]{4}. $$ ...
10
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1answer
132 views

Does $f(x) \in \mathbb{Z}[x]$ irreducible, imply $f(2x)$ also irreducible?

It is well known that $f(x+a) \in \mathbb{Z}[x]$ is irreducible when $f(x)$ is irreducible. I was wondering whether $f(2x)$ and in general whether $f(nx)$ is irreducible as well. Though it is ...
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1answer
41 views

Left remainder when dividing by $x-b$

Give a polynomial $p(x) = a_0 + a_1 x + ... a_n x^n \in \mathcal R[x]$ ($\mathcal R $ is any ring with unity), the book says when dividing $p(x)$ by $x-b \quad (b\in \mathcal R)$, the left remainder ...
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1answer
32 views

Linear algebra: generalize from characteristic $0$ a problem about polynomial coefficients.

Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ...
2
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0answers
47 views

Find a polynomial equation satisfied by $\phi$

I'm solving the following exercise from my class notes: Let $A=k[x^2,y^2]$ and let $M=k[x^2,xy,y^2]$ ($k$ a field). Show that $M$ is an $A$-module. Define $ \phi :M \to M$ by $\phi(m)=xym$. Find ...
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4answers
45 views

Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain

Here is an exercise(p.129, ex.1.15) from Algebra: Chapter 0 by P.Aluffi. Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain. The implication part makes no problems, ...
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2answers
33 views

Determine polynomial whose roots are a linear combination of roots of another polynomial

Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the polynomial $p(x)=x^3+5x^2+7x+11$. Find a polynomial whose roots are $\frac{\alpha_1+\alpha_2}{2}, \frac{\alpha_2+\alpha_3}{2}, ...
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0answers
15 views

Efficient ways to find a single root of a multivariate polynomial system to arbitrary precision

I am looking for a practical and efficient way to compute, to arbitrary precision, a single root of a multivariate polynomial system (over $\mathbb{Q}$). It seems like the fancy methods compute all ...
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0answers
22 views

Showing polynomials as products of roots

How do I show rigorously that any polynomial $a_nx^n+a_{n-1}x^{n-1}+...a_1x+a_0$ can be written as $a_n(x-b_1)(x-b_2)...(x-b_n)$ for real $a_i$ and real or complex $b_i$
2
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1answer
117 views

Polynomial tending to infinity

Take any polynomial $(x-a_1)(x-a_2)\ldots(x-a_n)$ with roots $a_1, a_2,\ldots,a_n$ where we order them so that $a_{i+1}>a_i$ is increasing so $a_n$ is the biggest root. It doesn't matter whether ...
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0answers
74 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
0
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0answers
16 views

Maximum degree of a polynomial [duplicate]

What is the maximum degree of a polynomial of the form $\sum_{i=0}^n a_i x^{n-i}$ with $a_i = \pm 1$ for $0 \leq i \leq n, 1 \leq n$, such that all the zeros are real? I have no idea where to start.
3
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4answers
109 views

Find a Polynomial in $x-\frac1x$

Given that $x^n - (1/x^n)$ is expressible as a polynomial in $x - (1/x)$ with real coefficients only if $n$ is an odd positive integer, find $P(z)$ so that $P(x-(1/x)) = x^5 - (1/x)^5.$ To start, I ...
1
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0answers
14 views

Under what condition given $(x_1, y_1\cdot r_1),…,(x_n, y_n\cdot r_n)$ we can interpolate polynomial $T$ that has specific random root?

We know given $(x_1, y_1),...,(x_n, y_n)$ we can interpolate a polynomial $P$ of of degree at most $n-1$. Let us define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is at most $n-1$, ...
0
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0answers
27 views

Taylor expansion for the roots of real polynomials

Consider a (real) polynomial $\mathcal{P}$ in the variable $x$ whose coefficients are themselves polynomials in the parameter $\lambda$. I am searching a taylor expansion in $\lambda$ for the roots ...