Questions tagged [polynomials]

Question 1

(I'm actually learning calculus.)Before I started working on this problem,I went to read this proof: Partial Fractions Proof I think I understand what the proof tried to do(And I can complete some ...

Question 2

Given, $$f(x) = x^3 - 3x + 1$$ I was solving a problem to find the number of distinct real roots of the composite function $f(f(x)) = 0$. By analyzing the graph of $f(x)$, we can observe the local ...

Question 3

Vieta's formulas are well known. $$\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}\left(\prod _{j=1}^{k}r_{i_{j}}\right)=(-1)^{k}{\frac {a_{n-k}}{a_{n}}}$$ For example, the sum of the roots of ...

Question 4

Math olympiad polynomial problem: Find all monic polynomials $P(x)$ such that $$P(x+1)\mid P(x)^{2}-1$$ My attempts to solve this problem: $P(x_{1}+1)=0\Rightarrow P(x_{1})=-1,1$ $x_{1}+1\mid P(x_{1}...

Question 5

Let $R$ be a (commutative, unital) ring. We then have a homomorphism of rings $\varepsilon \colon R[X] \to R^R$, given by associating to a polynomial the function it induces. EDIT: More precisely, for ...

Question 6

We are attempting to measure roots of quintic equations in Powerpoint. This is a mathematical curiosity inspired by Dr. Zye's recent video on making flags in Powerpoint. If you are interested in the ...

Question 7

$f(x)$ is a monic cubic polynomial and $f(k-1)f(k+1)<0$ is NOT true for any integer $k$. Also $f'\left(-\frac{1}{4}\right)=-\frac{1}{4}$ and $f'\left(\frac{1}{4}\right)<0$. Find value of $f(8)$ ...

Question 8

There's this theorem by T. Nagell which states: Given two non-constant polynomials $P,Q \in \mathbb{Z}[X]$, there exists infinitely many primes $p$ which divide $P(a), Q(b)$ for some integers $a,b$. ...

Question 9

In the parameter space of homogeneous polynomials I would like a formula for the dimension of the locus of polynomials with a given factorization pattern. Let $$S^d(\mathbb{R}^m)$$ denote the real ...

Question 10

What is the fastest known method to split $f \in \mathbb{Z}[x]$ into linear factors mod $g \in \mathbb{Z}[y]$, assuming this happens? Since that is not very clear, here is an example of what I'm ...

Question 11

I'm going through study material for the remainder theorem, and was asked to determine the remainder if $f(x)$ was divided by $g(x)$: $$\begin{aligned} f(x)&= x^4 + 5x^2 +2x -8\\ g(x)&= x+1 \...

Question 12

Let the Newton polygon of $f(x) \in \mathbb{Q}_p[x]$ has a single segment of horizontal length $L$ and slope $\lambda=\frac{h}{e},~\gcd(h,e)=1$. Is it true that $f(x)$ has $\frac{L}{e}$ many ...

Question 13

Let $\alpha$ be a real root of $$x^7 - 3 x^4 - x^3 + 3 x + 1 = 0.$$ How can we prove that $\alpha>-\frac 13$ without using the Intermediate Value Theorem ? It is easy to prove $\alpha > -\frac ...

Question 14

Does anyone know any results regarding existence of a solution to a Birkhoff interpolation problem at complex points? In particular, are there conditions for when there exists a real polynomial $p(x)$ ...

Question 15

I was told to work in $\mathbb F_{23}$, and also show it has a linear factor $\mathbb Z_5$ Write $f(x)=x^{46}+69x+2025$. We begin by supposing that $f=gh$ for some $g,h \in \mathbb Z[x]$. First, $g$ ...

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Questions tagged [polynomials]

How to prove $\frac{f(x)}{g(x)}=\sum_{i=1}^{n}\frac{A_{i} }{x-r_{i} } $ using Bezout Identity

Number of real roots of the n-th iteration of $f(x) = x^3 - 3x + 1$

Creative Alternatives to Vieta's formulas/Newton's identities

Finding all monic polynomials $P(x)$ such that $P(x+1)\mid P(x)^{2}-1$

If the evaluation map $\varepsilon\colon R[X] \to R^R$ is injective, does every non-zero polynomial have only finitely many roots?

Solving quintic equations with PowerPoint shapes

$f(x)$ is a cubic polynomial $f(k-1)f(k+1)<0$ is NOT true for any integer $k$. Find $f(8)$

Proof of Nagell's Theorem

Dimension of the locus of multivariate homogeneous polynomials with a prescribed factorization type

Linear factors of polynomial in a polynomial ring

Is there a difference between polynomial remainders and numeric remainders? [duplicate]

Number of factors of a polynomial through Newton polygon argument

How can we prove that $\alpha>-\frac 13$ without using the Intermediate Value Theorem? [closed]

Birkhoff interpolation at complex points

Prove $x^{46}+69x+2025$ is irreducible in $\mathbb Z[x]$

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