Questions tagged [polynomials]

For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.

Usually, polynomials are introduced as expressions of the form $\sum_{i=0}^dc_ix^i$ such as $15x^3 - 14x^2 + 8$. Here, the numbers are called coefficients, the $x$'s are the variables or indeterminates of the polynomial, and $d$ is known as the degree of the polynomial. In general the coefficients may be taken from any ring $R$ and any finite number of variables is allowed. The set of all polynomials in $n$ variables $X_1,\ldots,X_n$ over a ring $R$ is denoted by $R[X_1,\ldots,X_n]$. Strictly speaking this is a formal sum, because the variables do not represent any value. Nevertheless, the variables of a polynomial obey the usual arithmetic laws in a ring (like commutativity and distributivity). This makes $R[X_1,\ldots,X_n]$ a ring itself. One should note that $R[X_1][X_2]=R[X_1,X_2]$. This idea can be extended to $R[X_1,\ldots,X_n]$ in a very natural way.

An expression of the form $rX_1^{i_1}X_2^{i_2}\cdots X_n^{i_n}$ ($r\in R$) is called a term (of the polynomial). Polynomials are defined to have only finitely many terms. An expression with infinitely many different terms is generally not considered to be a polynomial, but a (formal) power series in one or more variables.

When $P\in R[X]$, $P(x)$ is the evaluation of $P$ at $x$ (pronounced $P$ of $x$, or simply $Px$). Here $x$ does not necessarily have to be an element of $R$. For $P(x)$ to be properly defined for an $x$ in some ring $S$ we need:

  • a homomorphism $\phi:R\to S$
  • the image of all coefficients of $P$ under $\phi$ should commute with $x$.

Evaluation is now simply performed by replacing all coefficients $r_i$ of $P$ by $\phi(r_i)$ and all appearances of $X$ by $x$. This quite naturally gives an expression that is well defined as an element of $S$. The concept of evaluation is naturally extended to $R[X_1,\ldots,X_n]$.

24386 questions
7
votes
1 answer

Is it possible that $|z+\sum_{i\not=1} a_i z^i| <1$ for some $a_i \in \mathbb{C}$ and for all $|z|=1$?

I wonder that whether there exists a complex polynomial of the form $$ P(z)= z+\sum_{i\not=1} a_iz^i, a_i,z\in \mathbb{C},$$ (i.e. its first order term has coefficient 1) s.t. its modulus is less than 1 on $|z|\leq 1$, i.e. $$ |P(z)|<1,\forall…
Brian Ding
  • 1,776
  • 9
  • 15
7
votes
1 answer

Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\setminus\{0\}$?

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 $uf + vg \in k[x]\setminus\{0\}$? I can do it for specific polynomials, but I'm struggling to structure a coherent proof. Any hints would be greatly…
Matt
  • 2,233
  • 5
  • 22
  • 31
7
votes
3 answers

Evaluate this Trigonometric Expression: $\sqrt[3]{\cos \frac{2\pi}{7}} + \sqrt[3]{\cos \frac{4\pi}{7}} + \sqrt[3]{\cos \frac{6\pi}{7}}$

Evaluate $$ \sqrt[3]{\cos \frac{2\pi}{7}} + \sqrt[3]{\cos \frac{4\pi}{7}} + \sqrt[3]{\cos \frac{6\pi}{7}}$$ I found the following $\large{\cos \frac{2\pi}{7}+\cos \frac{4\pi}{7} + \cos \frac{6\pi}{7}=-\dfrac{1}{2}}$ $\large{\cos…
user196761
7
votes
1 answer

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 descents. The Eulerian polynomials satisfy a standard…
7
votes
1 answer

how can I find the smallest integer $n$ such that a polynomial divides $x^{n}-1$

I have a simple question.. Assume that I have an arbitrary polynomial $f$ in $F_q[x]$. Is there a practical way to find the smallest integer $n$ for which $f$ divides $x^n-1$ ? A small example would be appreciated. Thanks, in advance. -I have my…
7
votes
2 answers

System of 4 quartic equations

\begin{align*}a &=\sqrt{4+\sqrt{5+a}},\\ b &=\sqrt{4-\sqrt{5+b}},\\ c &=\sqrt{4+\sqrt{5-c}},\\ d &=\sqrt{4-\sqrt{5-d}}.\end{align*} Compute $abcd$. I set up each as a quartic and got $a^4-8a^2-a+11=0$ and similar equations for $b,c,d$, but am…
rk_347
  • 695
  • 4
  • 13
7
votes
3 answers

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 multiply the polynomial by $\frac{1+x}{1+x}$;…
Logan Serman
  • 669
  • 1
  • 6
  • 13
7
votes
3 answers

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 variable $x$) seems to be always positive (i.e,…
Ewan Delanoy
  • 58,700
  • 4
  • 61
  • 151
7
votes
2 answers

Real roots of a polynomial of real co-efficients , with the co-efficients of $x^2 , x$ and the constant term all $1$

Can all the roots of the polynomial equation (with real co-efficients) $a_nx^n+...+a_3x^3+x^2+x+1=0$ be real ? I tried using Vieta's formulae $\prod \alpha=\dfrac {(-1)^n}{a_n}$ , $(\prod \alpha )(\sum \dfrac 1 \alpha)=(-1)^{n-1}\dfrac 1 a_n$ , so…
user123733
7
votes
3 answers

Show that $\sqrt [3]{2}-\sqrt [3]{4}$ is algebraic

How do I show, step by step, that $\sqrt [3]{2}-\sqrt [3]{4}$ is a root of $x^3+6x+2$? Start with $x=\sqrt [3]{2}-\sqrt [3]{4}$ do not use the cubic, the cubic is given for convenience. ( This is example 4.1.3 from Introductory ANT by…
nilo de roock
  • 2,578
  • 2
  • 23
  • 42
7
votes
5 answers

Why do we choose cubic polynomials when we make a spline?

Good morning, I want to learn more about cubic splines but unfortunately my class goes pretty quickly and we really only get the high level overview of why they're important and why they work. To me it's clear why we dont use linear functions, we…
John
  • 139
  • 1
  • 1
  • 9
7
votes
1 answer

Irreducibility of an integer polynomial with exponents in linear sequence?

Let $b$ and $n$ be two positive integers. Is there are a general result which tell us when the polynomial $$1+x^{b}+x^{2b}+x^{3b}+\cdots+x^{nb}$$ is irreducible over the integers? I know that $$1+x+\cdots+x^{n-1}$$ is irreducible if and only if…
6
votes
1 answer

Quadratic polynomial with alternate negative value

Let $f(x)=-x^2+ax+b$, where $a,b\in\mathbb{R}$. Suppose there exist distinct integers $m,n$ such that $f(m)=-n^2$ and $f(n)=-m^2$. Prove that there are infinitely many pairs of integers $x,y$ such that $f(x)=-y^2$ and $f(y)=-x^2$. [Source: Russian…
simmons
  • 2,531
  • 8
  • 24
6
votes
0 answers

generalization of geometric series $ \sum_{k=0}^\infty x^{p(k)} $

I have been playing around with infinite series and I wondered if it is possible to find an expression for the series: $$ \sum_{k=0}^\infty x^{p(k)} $$ as a generalization of geometric series. $p(k)$ is an arbitrary polynomial.
Redundant Aunt
  • 11,320
  • 2
  • 18
  • 63
6
votes
2 answers

Methods for determining which roots of a polynomial are inside of the unit circle?

Let's say I have a polynomial such as $$p(x) = x^4 + bx^3 + cx^2 + bx + 1.$$ I strongly suspect that, for any parameters, there are always two roots inside the unit circle and two roots outside of the unit circle. What tools can I use to determine…
user157227
  • 1,906
  • 11
  • 28
1 2 3
99
100