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.
27,904 questions
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How to prove $\frac{f(x)}{g(x)}=\sum_{i=1}^{n}\frac{A_{i} }{x-r_{i} } $ using Bezout Identity
(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 ...
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Number of real roots of the n-th iteration of $f(x) = x^3 - 3x + 1$
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 ...
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Creative Alternatives to Vieta's formulas/Newton's identities
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 ...
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Finding all monic polynomials $P(x)$ such that $P(x+1)\mid P(x)^{2}-1$
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}...
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If the evaluation map $\varepsilon\colon R[X] \to R^R$ is injective, does every non-zero polynomial have only finitely many roots?
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 ...
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Solving quintic equations with PowerPoint shapes
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 ...
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$f(x)$ is a cubic polynomial $f(k-1)f(k+1)<0$ is NOT true for any integer $k$. Find $f(8)$
$f(x)$ 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 $f(8)$
...
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Proof of Nagell's Theorem
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 $a,b$.
...
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Dimension of the locus of multivariate homogeneous polynomials with a prescribed factorization type
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 ...
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Linear factors of polynomial in a polynomial ring
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 ...
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Is there a difference between polynomial remainders and numeric remainders? [duplicate]
I'm going through study material for the remainder theorem, and was asked to determine the remainder if $f(x)$ was divided by $g(x)$:
$$\begin{aligned}
f(x)&= x^4 + 5x^2 +2x -8\\
g(x)&= x+1
\...
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Number of factors of a polynomial through Newton polygon argument
Let the Newton polygon of $f(x) \in \mathbb{Q}_p[x]$ has a single segment of horizontal length $L$ and slope $\lambda=\frac{h}{e},~\gcd(h,e)=1$.
Is it true that $f(x)$ has $\frac{L}{e}$ many ...
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How can we prove that $\alpha>-\frac 13$ without using the Intermediate Value Theorem? [closed]
Let $\alpha$ be a real root of $$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 ...
user1713090
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Birkhoff interpolation at complex points
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 $p(x)$ ...
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Prove $x^{46}+69x+2025$ is irreducible in $\mathbb Z[x]$
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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