Questions tagged [factoring]
For questions about finding factors of e.g. integers or polynomials
3,559 questions
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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 it possible to simplify a sum of these two cube roots by completing the cube?
Let $x$ be a real number. Consider the following expression:
$$
\sqrt[3]{\frac{x^{3} - 3x + \left(x^{2} - 1\right)\sqrt{x^{2} - 4}}{2}} + \sqrt[3]{\frac{x^{3} - 3x - \left(x^{2} - 1\right)\sqrt{x^{2} -...
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Factorization of a little higher degree polynomial over $p$-adic number field
Let $K=\mathbb{Q}_p(t)$ be the finite extension of the $p$-adic number field $\mathbb{Q}_p$, where $t=p^{1/13}$. With the help of Newton polygon argument, it seems the polynomial $$f(x)=(x^{p^2}-t^2)^...
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What is the Upper Bound on the Period for Factoring by Shor's Algorithm?
Say I built a nuclear bomb quantum computer in my garage. Now I want to factor all the things.
We are told that Shor's algorithm can do it because $f(x)$ happens to be periodic. Veritasium gives the ...
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Show $ \lim_{x\to\infty} (\sqrt[3]{x-1}-\sqrt[3]{x+1})=0 $
Show $$\lim_{x\to\infty} (\sqrt[3]{x-1}-\sqrt[3]{x+1})=0. $$
I did something but I dont know if is correct and I have 2 questions
$$ \lim_{x\to\infty} (\sqrt[3]{x-1}-\sqrt[3]{x+1})=0 $$
$$ \lim_{x\to\...
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Is there a way to simplify an implicit function so that $x$ and $y$ will be on separate sides of the equation?
I have an implicit equation $(x^2+y^2)^2=a^2(x^2-by^2)$, where $a$ and $b$ are variables. I have tried to take this implicit function and separate the $x$ and $y$ values, but there is an unfactorable ...
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Factoring $s^3 + 2s^2 + s +1$ [duplicate]
I'm looking to factor a polynomial using any known method. The equation with a polynomial is:
$$s^3 + 2s^2 + s +1 = 0 \tag{1}$$
For this polynomial, I got factors as $(s+1.7549)((s+0.122)^2 + 0.555)$ ...
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Factoring $(a + b + c)^3 - a^3 - b^3 - c^3$
I am doing I. M. Gelfand's "Algebra" problem 122 e), factoring
$$(a + b + c)^3 - a^3 - b^3 - c^3$$
So my solution is following:
$$\begin{align}
(a + b + c)^3 - a^3 - b^3 - c^3 &= (a + b)^...
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On Chern roots and the Chern polynomial
Background
I am having trouble reconciling Chern classes with Chern roots when using the Vieta formulas. The setup is the following:
$E\xrightarrow{\pi}B$ is a vector bundle of rank $r$.
Its Chern ...
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Can norm $N_{L/K}(X)$ of a polynomial $g(X) \in L[X]$ gives a factor of $f(X)$ in $K[X]$, when $g(X)$ is a factor of $ f(X)$ in $L[X]$?
Let us consider $L/K$ is a finite Galois extension and $G = \mathrm{Gal}(L/K)$ is the corresponding Galois group.
Assumptions:
$f(X)$ is a monic polynomial in $K[X].$
$g(X)$ is a monic factor of $f(...
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$p(x)$ irreducible is not equivalent to $p(x + r)$ irreducible over a division ring
If $R$ is a commutative ring, $p(x)$ is a polynomial over $R$ and $r$ is an element of $R$, then $p(x)$ is irreducible if and only if $p(x + r)$ is irreducible.
If I understand it correctly, the proof ...
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Why am I winning always? A coincidence perhaps?
Erdős and Pólya are playing a game in which initially the number $10^6$ is written on a blackboard. If the current number on the board is $n$, a move consists of choosing two different positive ...
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Generalization of new Galois factoring idea
I have detailed a new factoring idea involving Galois cubic fields here https://mathoverflow.net/questions/497222/factoring-special-form-numbers-via-galois-cubic-polynomial.
The method involves ...
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Factorize $x^8+x^7+1$ over $\Bbb Q[x]$
It seems a folklore problem to see $x^{3k+2}+x^{3k+1}+1$ is reducible over $\Bbb Q[x]$. It could be concluded as a special case of $\left.\Phi_m\left(x\right)\big| x^{mk} \big[\Phi_m\left(x\right) - 1\...
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Factorising quadratic polynomials (no quadratic formula allowed!) [duplicate]
I am trying to factorise $2x^2+5x-12$ from Stewart's Calculus Early Transcendental diagnostic test part A (Algebra). The question is to factorise $2x^2+5x-12$ in the form of $(ax+b)(cx+d), a,b,c,d\in \...
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