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.

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Finding if roots of a polynomial function are shifted when adding another polynomial function?

Not sure if this question has an easy answer but I was wondering if there is a heuristic or some theorem that helps proving which way the roots are being shifted for an equation $f(x)=g(x)+h(x)$. For ...
Baklava Gain's user avatar
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Condition to have the convexity of the infimum of multi variate polynomial

I consider $f\in\mathbb{R}[x,u]$ a polynomial function where $x\in\mathbb{R}^n$ and $u\in\mathbb{R}^m$. I am interested in $$ g(x)=\inf_{u\in U}\{ f(x,u)\} $$ Where $U$ is a compact set. I am trying ...
coboy's user avatar
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the Variance of the number of a recurrent event in n trials ~ $\frac{\sigma^2}{\mu^3}n$

Let E be a recurrent aperiodic event. Its recurrence time has finite mean $\mu$ and variance $\sigma^2$. Let $N_n$ be the number of occurrences of E in n trials as given, then $$ E(N_n) \sim \frac{n}{\...
David Lee's user avatar
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2 votes
4 answers
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Prove that value of given polynomial can't be negative $2x^2 +5y^2 +3z^2 -6xy -2xz + 5yz$

Obviously, I tried transforming polynomial into a sum of squares. Nothing worked, there are simply not enough squared variables to match the product of them to make a square. $(x^2 -4xy +4y^2) + (x^2 -...
Stephanie V's user avatar
4 votes
2 answers
53 views

Solving an exotic trigonometric function or a sextic

I'm an aerospace engineering student and I've been doing some analytical work on an interdisciplinary problem involving orbital mechanics and electromagnetism. In the final part of my work, I ended up ...
Eric D'Antona's user avatar
2 votes
2 answers
179 views

Interesting third degree polynomial

Let $P(x)$ be a third-degree polynomial with coefficients as natural numbers, and the constant term of $P(x)$ is $1$, and the sum of the coefficients of $P(x)$ is $2020$. Prove that there exist ...
nth's user avatar
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Multiplying polynomial factorisation by a constant term [closed]

We can write every polynomial as a factorisation of its roots, when the polynomial is monic, doing this is simple. However when its not monic, the factorisation is multiplied by a constant which is ...
math_learner's user avatar
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1 answer
50 views

Finding the number of zeros of a function

Suppose we have a function that asymptotically behaves as a polynomial. For example, something like $$f(x) = \sum^N_{n=0} a_n x^n + \sum_{m=1}^M\frac{b_m}{x^m}.$$ Then, of course, for large $x$ this ...
Geigercounter's user avatar
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Approximate the function cos(x) by a polynomial of degree N using linear optimization

I want to find an approximation to the cos(x). I formulated the problem as a linear optimization problem as follows: $$ \min \sum_{i=1}^{M} e_i $$ subject to: $-(a_0 + a_1x_i + \dots + a_nx_i^{N})-...
Felipe Oliveira's user avatar
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Set of polynomials and dimension stability under addition

I consider the subset $\mathcal{U}\subset\mathbb{R}_{2}[x_1,x_2]\times \mathbb{R}_{2}[x_1,x_2]$ defined by $$ \mathcal{U} = \{(x_1,x_2)\in[0,1]^2\mapsto (ax_1,b(1-x_1)x_2 : (a,b)\in\mathbb{R}^2\} $$ ...
coboy's user avatar
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Help with proving the fundamental theorem of Algebra for complex polynomials

$\textbf{Problem:}$ Prove that for $x, a_k, b_k \in \mathbb{R}$ and $\theta, \phi_k \in [0, pi]$, that there exists a solution to the sets of equations:$$K(x, \theta) = \sum_{k=0}^n a_k x^kcos(k\theta ...
IV-301's user avatar
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If $P(x)$ is a fifth degree polynomial such that $P(x)+1$ is dividable by $(x-1)^3$ and $P(x)-1$ is dividable by $(x+1)^3$, How to find $P(x)$

I saw this question: If $P(x)$ is a fifth degree polynomial such that $P(x)+1$ is dividable by $(x-1)^3$ and $P(x)-1$ is dividable by $(x+1)^3$,find $P(x)$. I tried my best to find such function but ...
pie's user avatar
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Are there any problems about the difference between set theoretic definitions of polynomials?

I am a novice about this question, so if there is a misunderstanding then I apologize for it. As for Peano axioms, if I choose Zermelo natural numbers, and you choose von Neumann ones, then this doesn'...
categoricalequivalent's user avatar
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$f$ is irreducible if the polynomial reduced $p$ is irreducible and the degrees are the same

Let $f$ be an irreducible polynomial and $h(f)$ the polynomial with coefficients reduced modulo a prime $p$. Then if $\deg(f)=\deg(h(f))$ and $h(f)$ is irreducible then $f$ is irreducible as an ...
Xaver Wallenstein's user avatar
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1 answer
230 views

Probability of choosing a constant polynomial [closed]

Suppose we have a polynomial $$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$ where all the coefficients are whole numbers which includes Zero. Suppose we now randomly choose the values of these coefficients ...
Aarush Saharan's user avatar
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Convexity of cubic affine and quartic affine?

I am a post graduate student who is currently study the subject of convex optimization. From my class I know that affine function is convex and also $x^r$ for $r>=1$ is convex for non-negative $x$. ...
Tuong Nguyen Minh's user avatar
5 votes
4 answers
97 views

How to factor $(x-y)^5 + (y-z)^5 + (z-x)^5$

We see that polynomial is cyclic. For $x=y$, $P(y)=0$ so $x-y$ factors polynomial. Because it is cyclic we instantly know other two factor $(y-z)$ and $(z-x)$. $P(x,y,z)=(x-y)(y-z)(z-x)*N(x)$ Because ...
Stephanie V's user avatar
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0 answers
57 views

Leading term of difference of two products

$n$ is a positive integer. Let $$P_n(x)=\prod_{\substack{\mu_1,\dots,\mu_n\in\{\pm1\}\\\mu_1\dots\mu_n=1}}(x-(\mu_1{x_1}+\dots+\mu_n{x_n}))\\ Q_n(x)=\prod_{\substack{\mu_1,\dots,\mu_n\in\{\pm1\}\\\...
hbghlyj's user avatar
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Number of odd coefficients on the remainder of a polynomial division

I came across the following question a few days ago and I can't solve it no matter what I think: Let $r(x)$ be the polynomial that is the remainder when dividing $x^{2050}$ by $x^5 + x^2 + 1$. How ...
Vinicius Rispoli's user avatar
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Coefficients of invertible element in polynomial rings [duplicate]

Let $f = \sum_{i=0}^n f_i x^i$ - an element of $R[x]$, where $R$ is a commutative ring. I want to prove that if $f$ is invertible if and only if $f_0$ is invertible and the rest of the coefficients $...
Irene's user avatar
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42 views

Find the difference of $4b^3 + 6b - 7$ and $-12b^2 + 11b + 5$. [closed]

Trying to make some sense of this seemingly easy problem: Find the difference of $4b^3 + 6b - 7$ and $-12b^2 + 11b + 5$. Trying to prepare for my Algebra 1 final, which is a week from Thursday, and ...
YourLordJoyBoy's user avatar
2 votes
1 answer
12 views

About nonnegative polynomials

Do there exist real polynomials $P(x)$ and $Q(x)$ with nonnegative coefficients such that $$\left(\sum\limits_{k=0}^{20} x^k\right)^2=(x-3)^2P(x)+(x+4)^2 Q(x)?$$ I asked this question here, but I ...
Dattier's user avatar
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29 views

Method for solving polynomial system without multilinear form?

I am an engineer who is currently working with some network optimization problem during my post graduate study. During my study time, I see that sometimes I need to look for solution of polynomial ...
Tuong Nguyen Minh's user avatar
1 vote
1 answer
31 views

Deriative of algebraic implicit function is algebraic

Let $P(x,y)$ be a real polynomial with $P(0,0)=0$, and assume that $\partial_y P(x,y)$ is nonzero over $[-1,1]^2$. Then the equation $P(x,y)=0$ defines an implicit function $y=f(x)$ near $0$. In this ...
Tongou Yang's user avatar
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(sub)monoids of the positive integers under multiplication, with density $0$ in the positive integers, are always multiplicative norms of rings?

Consider integer polynomials of type $"x"$ where we take as imput nonnegative integers. With these nonnegative integer imputs we strictly generate a subset of nonnegative integers ; the set $X$. The ...
mick's user avatar
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3 votes
5 answers
180 views

Coefficient of $x^{21}$ in $(1+x+x^2+\dots+x^{10})^4$

Find the coefficient of $x^{21}$ in $(1+x+x^2+\dots+x^{10})^4$ I tried splitting the terms inside the bracket into two parts $1+x+\dots+x^9$ and $x^{10}$, and then tried binomial theorem, but that ...
math_learner's user avatar
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1 answer
77 views

How to make polynomials with so many exponents and variables within said terms easier to solve [closed]

Look guys, I want to level with you. I am reaching the end of my 4th-5th algebra 1 or MAT 101 class in college, and this is one of the classes I need to graduate altogether so it, in of itself, is ...
YourLordJoyBoy's user avatar
2 votes
0 answers
46 views

Prove that a given symmetric set of bilinear equations has only symmetric solutions.

Conjecture: Let $n \in \mathbb{N}$ and let $x,y \in \mathbb{R}^n$. Suppose that, for all $k \in [n] = \{1,2,\cdots,n\}$, we have $$ \sum_{i=0}^{k-1} x_{i+1} y_{k-i} = 1 \tag{1}$$ and $$ \sum_{i=0}^{n-...
Thomas's user avatar
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2 votes
1 answer
40 views

How to factor $k^2(k+1)^2 -c(k+1)^2 +k^2c +c^2 +c$

Basically I need to prove that $k^2(k+1)^2 -c(k+1)^2 +k^2c +c^2 +c = [k(k+1) -c]^2$. When I first saw polynomial I thought that it could’t be factored into anything meaningful, let alone a square of ...
StrongBegginer01's user avatar
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1 answer
53 views

What's the maximum and minimum value of $P(-2)$?

Consider a quartic polynomial $$P(x) = ax^4+bx^3+cx^2+dx+e$$ Where $a, b, c, d, e\in \{0, 1, 2\}$ and $0, 1, 2$ are chosen at least once. What's the maximum and minimum value of $P(-2)$? If we were to ...
Noether's user avatar
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0 answers
15 views

Bounding a ratio of parametrized sums, uniformly

Let $\gamma_{p}\ge0$ be a sequence such that $\sum_{p\ge2}\gamma_p^2p^k<\infty$ for every $k\in\mathbb{N}$. Define: $C_k(s)=\sum_{p\ge2}\gamma_p^2s^pp^k$ for $s\in\left(0,1\right]$. I want to show ...
Sunroot's user avatar
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1 vote
1 answer
65 views

How to solve the roots of the following function numerically?

I have the following function $$f(n) = \frac{\sqrt{n}+ \sqrt{n+ 16 (\sqrt{n}+1)}}{8} - \frac{1}{2}\left(\sqrt[3]{n + \sqrt{n^2 - \frac{1}{27}}}+ \sqrt[3]{n - \sqrt{n^2 - \frac{1}{27}}}\right).$$ It is ...
SHASHANK RANJAN's user avatar
1 vote
0 answers
38 views

Integrality of a quotient

Consider two positive integers $m$ and $n$ with $m>n$. I would like to prove that the quotient $$\prod_{k=0}^{n-1}\dfrac{X^{2^m-2^k}-1}{X^{2^n-2^k}-1}$$ is a polynomial in fact. What I did is to ...
joaopa's user avatar
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0 answers
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Why does $\text{Im} \ q(x) = (\text{Im} \ a_0) + (\text{Im} \ a_1)x + \cdots + (\text{Im} \ a_{n-2}) \ x^{n-2}$?

In the proof of result 4.16 of Linear Algebra Done Right (see here: https://linear.axler.net/LADR4e.pdf#page=142), Axler writes $$ 0 = \text{Im} \ q(x) = (\text{Im} \ a_0) + (\text{Im} \ a_1)x ...
Paul Ash's user avatar
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1 vote
3 answers
87 views

IF $a+b+c=0$ prove that $ \frac{a^5 +b^5 +c^5}{5}=\frac{a^3 + b^3 +c^3}{3}\frac{a^2 + b^2 +c^2}{2}=abc\frac{a^2 +b^2 +c^2}{2} $

First thing that I did is prove that $abc\frac{a^3 + b^3 +c^3}{3} \times \frac{a^2 + b^2 +c^2}{2}=abc\times\frac{a^2 +b^2 +c^2}{2}$ I did this by using identity that I needed to prove before $a^3 +b^3 ...
Stephanie V's user avatar
0 votes
1 answer
32 views

Irreducible polynomials with complex root.

I need to show that if $f$ and $g$ are irreducible in $\mathbb{Q}$[$x$] and they share a common complex root, then there is $a \in \mathbb{Q}$ such that $f = a . g$. What I thought: Call $u \in \...
pdrlmdd's user avatar
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1 vote
1 answer
37 views

Real coefficients polynomial $P_n(\alpha)=0,|\alpha|=1\implies \alpha^{n+1}=1$

Let $0<a_n\leq a_{n-1}\leq\cdots\leq a_1\leq a_0$ and $$P_n(z)=a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0.$$ If there exists $\alpha\in\mathbb C$ with $|\alpha|=1$ such that $P_n(\alpha)=0$, then $\...
Riemann's user avatar
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1 vote
1 answer
36 views

Upper bound of depressed quartic

For the depressed quartic equation $x^4-ax^2-bx-c=0$, $a,b,c>0$, is there some (relatively easy) way to find an upper bound of the positive root in terms of $a,b$ and $c$? I am aware of Ferrari's ...
Luis's user avatar
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-2 votes
1 answer
127 views

If $f\in \mathbb{Q}[x,y]$ factors in some field extension of $\mathbb{Q}$, then it also factors in some finite extension of $\mathbb{Q}$? [closed]

If $f(x,y)\in \mathbb{Q}[x,y]$ factors non-trivially (as a product of two non-constant polynomials) in some field extension of $\mathbb{Q}$, then does it also factor non-trivially in some finite field ...
Cezar's user avatar
  • 117
4 votes
0 answers
50 views

Approximation using polynomials with integer coefficients

The classic Weierstrass Theorem states that the set of polynomials are dense in $C[0,1]$ equipped with $|| \cdot ||_{\infty}$. Bernstein's proof of Weierstrass Theorem gives an explicit form of ...
Yixuan Huang's user avatar
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0 answers
46 views

Hermite Polynomial and its Expectation

Currently, I'm stuck to some statement in a paper (in chapter 8: Nonlinear Model, from page 26 ~27). Although this topic generally covers statistics and machine learning theory, my main question is ...
jason 1's user avatar
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0 answers
26 views

Generalized Factor and Remainder Theorems

We know about the factor and remainder theorems for univariate polynomials. My question is does a similar analogue exists for multivariate polynomials as well ?
PlayerUnknown1098's user avatar
2 votes
1 answer
46 views

Randomly Generating Real-Rooted Polynomial Equations

I need a simple function to generate real-rooted polynomial functions to demo my Desmos Aberth-Ehrlich rootfinding implementation. My current function is as follows: Let $n \in \mathbb{Z}^+$ be the ...
James Baw's user avatar
5 votes
1 answer
60 views

Is there a unique irreducible polynomial relating two algebraic numbers beyond their minimal polynomials?

Given two algebraic numbers a,b over $\mathbb{Q}$, with $p(x) \in \mathbb{Z}[x]$ being the minimal polynomial of $a$, and $q(y) \in \mathbb{Z}[y]$ the minimal polynomial of $b$, suppose that there ...
Edo's user avatar
  • 51
1 vote
1 answer
82 views

Sequence $f(k)=\underbrace{\sqrt{m+\sqrt{m+\sqrt{m+\cdots}}}}_{\text{$k\ m$'s}}-\underbrace{\sqrt{m-\sqrt{m-\sqrt{m-\cdots}}}}_{\text{$k\ m$'s}}$

Define $$f(k)=\underbrace{\sqrt{m+\sqrt{m+\sqrt{m+\cdots}}}}_{\text{$k$ $m$'s}}-\underbrace{\sqrt{m-\sqrt{m-\sqrt{m-\cdots}}}}_{\text{$k$ $m$'s}}$$ Given $m$ and $k$ are integers such that $m\ge1$ and ...
Sankalp Kumar Jha's user avatar
0 votes
1 answer
80 views

How is it possible that polynomial $x^2 +x +2$ can't be written as binomial square but can give number that is square of some number?

$$ P(x) = x^2 + x + 2 = \bigl(x + \tfrac{1}{2}\bigr)^2 + \tfrac{7}{4} $$ This function/polynomial doesn't have real roots or $4$ as constant but for $n = 1$, $P(x) = 4$. I got confused when I was ...
Stephanie V's user avatar
0 votes
0 answers
42 views

Prove an inequality about symmetric polynomials

$x_i\geq0,\sigma_1=\sum x_i,\sigma_2=\sum x_ix_j,\sigma_3=\sum x_ix_jx_k,...,\sigma_n=x_1x_2...x_n,$. They are elementary symmetric polynomials about $x_i$, Prove:$$(\frac{\sigma_n}{C_n^n})^{\frac{1}{...
MathNoob's user avatar
  • 329
7 votes
1 answer
170 views

Factorization and irreducibilty for $x^n-2x^m+1$ trinomials.

I have encountered a weird phenomenon while trying to solve a problem on Reddit. Here is the phenomenon. Let $a>b \in \mathbb{N}$ and $p_{(a,b)} = x^a - 2x^b + 1$ It seems that if $gcd(a,b,c,d) = 1,...
Vatsa Srinivas's user avatar
0 votes
2 answers
75 views

Prove that $Q(x)=Q(x-1)$ must be constant

This one is subproblem from "Find all polynomial P(x) which satisfies equation $xp(x-1)=(x-3)p(x)$ By trying $x=0,1,2$ you get that $P(x)=x*(x-1)(x-2)*Q(x)$ By inserting new equation into ...
Stephanie V's user avatar
4 votes
1 answer
201 views

Weird solution for functional equation

The functional equation i am dealing with is as follows $$f: R \rightarrow R, f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17, \forall x \in R$$ I am to find the function $f(x)$ I tried various ...
koiboi's user avatar
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