For questions about finding factors of e.g. integers or polynomials

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1answer
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Methods for factorising polynomials into irreducibles over finite fields

I was given a problem recently, part of whose solution was to factorise $x^{15}+1$ in $\mathbb F_2[x]$. It turns out that the factorisation is $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
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91 views

proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
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1answer
47 views

Factorising polynomials over $\mathbb{Z}_2$

Is there some fast way to determine whether a polynomial divides another in $\mathbb{Z}_2$? Is there some fast way to factor polynomials in $\mathbb{Z}_2$ into irreducible polynomials? Is there a ...
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0answers
37 views

Solving “finding value” using factorisation

Even my teacher could not do this.This is a question extracted from the mathematics challenge board created by my friend in school.This is the question Given $x-y=3$,find the value of ...
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1answer
393 views

Reducible polynomials in $\mathbb{Z}[X]$

Let $(a_n)_{n\geq 1}$ be a strictly increasing sequence of integers and $k$ an integer different from $0$. There exists among the polynomials $$ (X-a_1)(X-a_2)\cdots(X-a_n)+k,\ n\geq 1 $$ only ...
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1answer
35 views

Laplace transform (Simple factorization)

The question require me to find the inverse of Laplace transform. In the first line of solution, how does it go from LHS to RHS? Does it simply apply partial fractions?
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2answers
41 views

Irreducibility and factoring in $\mathbb Z[i], \mathbb Z[\sqrt{-3}]$

In $\mathbb Z[i]$, prove that $5$ is not irreducible. In $\mathbb Z[\sqrt{-3}]$, factor $4$ into irreducibles in two distinct ways. I am completely stumped on how to do this. I really need all ...
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2answers
40 views

Questions from Dixon Factorization Paper

I read the first page of Asymptotically Fast Factorization of Integers, and have a few questions. Quotes from the paper are formatted as blockquotes (>). Following Legendre, we know that there ...
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1answer
24 views

Diagonally dominant matrix for Cholesky?

I have a $10^6 \times 10^6$ dense SPD matrix, which I am called to invert, by using Cholesky factorization. However, I came across this statement: We start with the Cholesky and LU ...
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1answer
41 views

Simple Fermat Factorization Example

I'm trying to understand this example from wikipedia's Fermat's factorization method. For example, to factor N = 5959, the first try for a is the square root of 5959 rounded up to the next ...
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5answers
48 views

Factorising trigonometric functions

In order to factorise $x^2-1$ one way of thinking about it would be to set it equal to zero and solve to get $x=1$ and $x=-1$. You can then write $x^2-1=(x+1)(x-1)$ Can we do the same with ...
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1answer
50 views

Find Eigen values of given matrix with nonfactorable polynomial

I'm having trouble finding the Eigen values for this matrix: $$ A =\begin{pmatrix} 0&1&-2 \\ 1&3&0 \\ -2&0&5 \end{pmatrix} $$ I did $A - \lambda I $ and ended up with this ...
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3answers
52 views

Simplify the Complex Fraction

I am having trouble with the following complex fraction. I have simplified everything for the most part, but I am stuck on the last part and need to know what I have to do next. ...
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5answers
96 views

Help understanding how to factor completely $x^3-x^2-x+1$

I need someone to help explain the steps to completely factor the problem $x^3-x^2-x+1$. Here is what I have done so far: $x^3-x^2-x+1$ to $x^3-x^2+-1(x+1)$ Since there is a ...
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2answers
53 views

Checking irreducibility of a polynomial over a finite field

A part of a coding theory course I am doing includes some questions on irreducible polynomials. I have a question with solution but am worried I have interpreted it incorrectly. So for $\mathbb F_5$ ...
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1answer
42 views

Factor polynomial with irrational roots using quadratic equation

If I want to factor the polynomial $x^2 + 3x + 1$, I thought I could use the quadratic formula to find that its roots are $\dfrac{-3\pm\sqrt{5}}{2}$. Then, since those are both negative values, take ...
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3answers
62 views

Confused regarding the answer of a problem based on locus.

I have a question on locus which goes like this. $A(5,3)$ and $B(3,-2)$ are two fixed points. Find the equation of the locus of $P$, so that the triangle $PAB$ is 9. Now the loci of the point $P$ ...
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1answer
28 views

Is the ideal $(2,x+1)$ principal in $\mathfrak{o}_K$?

I'm trying to show (although I don't know if the statement is even correct) that the ideal $\mathfrak{p}_2$ is not principal, where $$\mathfrak{p}_2:=(2,x+1) \text{ in the ring of integers } ...
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1answer
34 views

Minor help in the factorisation techniques used in lec notes

If anyone could help me see how $$ A(\xi,\eta)=a\,\xi_x^2+2\,b\,\xi_x\,\xi_y+c\,\xi_y^2=0 $$ is turned into $$ \frac{1}{a} \left[a\,\xi_x+\left(b-\sqrt{b^2-ac}\right)\xi_y\right] ...
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3answers
76 views

Factorize Trigonometric Equation: $ 3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0 $

I have a problem with the following trigonometric equation: $$ 3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0 $$ It's from the book Engineering Mathematics 7th edition by Stroud. The book is giving the ...
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1answer
35 views

Have you seen these integer factorization algorithms before?

I have two algorithms for finding two factors, $p$ & $q$, of a number $N$. The algorithms are (hopefully) obviously related. The pseudo-code for them follows: Algorithm 1 ...
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3answers
70 views

When can I divide both sides of an equation if one side is zero

Where K is some positive Integer For the following examples: $$ K(a+b)(p+q)=0 $$ $$ Ka^2+Kbx+Kc=0 $$ Can I just divide both sides of the equation by K (dividing into 0 on the right) and effectively ...
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3answers
232 views

Multiplication partitioning into k distinct elements

Let's say I have a list with the prime factors of a number $n$ and their corresponding exponents. Is there a formula to calculate how many multiplications with $k$ distinct factors of $n$ are ...
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4answers
78 views

Find all integers n such that the quadratic $5x^2 + nx – 13$ can be expressed as the product of two linear factors with integer coefficients.

I am unsure of how to approach this problem. I have thought about using the Rational root theorem, but I am unsure if this answers the question being asked. Using the theorem, I get $\frac{p}{q} = ...
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2answers
221 views

Factoring a 5 term polynomial

I am struggling to factor $n^4 + 4n^3 + 8n^2 + 8n +4$. I have tried grouping the terms a couple of times, but got nowhere. What am I missing?
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0answers
76 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
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1answer
40 views

Finding the factors of integer $x$ and its square

What is the the theorem or property that says that $\forall{}x\in\Bbb Z$, the set of all integers, $x^2$ has the same factors as $x$, twice?
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2answers
350 views

Factorising a cubic equation

Factorise $9x-x^3$ completely. It's simple but I'm never seem to get it right; I've got $(x-1)(-x+9)x$.
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5answers
56 views

Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
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1answer
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Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$.

I have to find the form of n i.e. whether n is even or odd and whether it is multiple of 2 or 3 such that: $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$ What I tried: $$x^2 + x + 1 = (x + 1)^2 - ...
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2answers
30 views

Factorizing expressions

I am having trouble solving this problem $81f^2- \dfrac{9}{e^2}$. How do you begin when solving this problem? Do you move $f^2$ by replacing the $9$ and vice versa and does the minus change to plus?
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2answers
315 views

Is this olympiad-like question about remainders an open problem?

Suppose that we are given two positive integers $x$ and $y$ such that $$x \mod p \leqslant y \mod p$$ for each prime number $p$. (Here, $x \mod p,\; y \mod p$ stand for the least non-negative ...
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31 views

Please factorise this [duplicate]

a³ - b³ - c³ - 3abc = what???????
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4answers
56 views

How can I factorize this quadratic expression

Going by the exercises of a book I have been factorizing quadratic equations the following way, let's say I have: $$ {x^2 - 7x + 12 = 0} $$ I know that $$ {a \times b = 12 \\ \text{ and } \\ a + b ...
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2answers
68 views

How should you go about simplying cubic polynomial: $y(x) = x^3+12x^2+21x+10$

Claim: $$y(x) = x^3+12x^2+21x+10$$ Can be factored into $$(x+1)^2(x+10)$$ But what is the quickest way to see this?
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How is pollard rho different from normal factorization?

As far as I understand, pollard rho factorization generates random sequence of numbers, say x1, x2, x3 ... and then checks if x(i) - x(i-1) divides the number. If it does then it is a factor. How is ...
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1answer
51 views

Factoring and solving a cubic polynomial

When can we not use synthetic division to solve for a cubic polynomial? For example we can use synthetic division to solve $-t^3 -4t^2 +20t +48$. When I can't use synthetic division what are my other ...
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1answer
29 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
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1answer
46 views

Prove that a matrix is positive definite

I've never really done any factoring with multiple variables in an equation. I tried looking around for examples, but couldn't really find a solid one. Here is the equation I am trying to factor $$ ...
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2answers
26 views

Basic help with factoring

I am having a small problem recalling how to factor with exponents and roots. For example, I understand $\sqrt{16t^2+4t^4}$=$2t\sqrt{4+t^2}$ But I have issues when it is factoring not with a square ...
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2answers
48 views

how to factorize $x^2+10yz-2xz-2xy-3y^2-3z^2$?

How to factorize $$x^2+10yz-2xz-2xy-3y^2-3z^2$$ It is expanded and we should make them into parts and factorize each part individually. the last answer is $$(x+y-3z)(x-3y+z)$$ but how to get it ?
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3answers
53 views

how to factorize $(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)$?

how to factorize $(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)$? this is one of my hard questions. I know it is related to $(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc$ but I don't know how to factorize it.
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4answers
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Simplifying $\frac{3(a^{1/4}+4)}{2a-32a^{1/2}}$

I have a fraction $\frac{3a^{1/4}+12}{2a-32a^{1/2}}$ which I have factored out into $\frac{3\left(a^{\frac{1}{4}}+4\right)}{2a-32a^{\frac{1}{2}}}$, but checking out W|A I also get that there ought to ...
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2answers
85 views

How to factorize $x^4+2x^2-x+2$?

look at this: $$x^4+2x^2-x+2$$ How to factorize it? It should be changed to be in the form of standard factorization formulas.
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3answers
97 views

How to factorize $(x-2)^5+x-1$?

This is a difficult problem. How to factorize this? $$(x-2)^5+x-1$$ we can't do any thing now and we should expand it first: $$x^5-10x^4+40x^3-80x^2+81x-33$$ but I can't factorize it.
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On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

(Note: This has been cross-posted to MO.) A positive integer $N$ is said to be perfect if $\sigma(N) = 2N$, where $\sigma(x)$ is the sum of the divisors of $x$. An odd perfect number $N$ is said to ...
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2answers
26 views

Asymptotic upper bound on number of solutions to $ab \equiv n \pmod m$

Does anyone know a rough upper bound on the number of solutions to $ab \equiv n \pmod m$ when $n$ and $m$ are given and $a<m$, $b<m$, $n<m$? Specifically, I want to know how the number of ...
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Splitting field of a cubic polynomial understanding

The cubic polynomial $f(x) = x^3+px+q\in K[x]$ has 3 roots $a_1,a_2,a_3\in \mathbb C$ Hence, the splitting field extension $L=K(a_1,a_2,a_3)$ $\delta=(a_1-a_2)(a_1-a_3)(a_2-a_3)\in L$ since ...
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4answers
255 views

How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials?

How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials? The link I've given to the theorem uses elaborate wordings using 'rings', ...
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1answer
42 views

How to factor $\frac{27}{125}a^6b^9-\frac{1}{64}c^{12}$

I'm stuck with the following: $\frac{27}{125}a^6b^9-\frac{1}{64}c^{12}$ My idea was/is the following: $\frac{3^3a^6b^9}{5^3}-\frac{c^{12}}{8^2}$ Trouble is that I don't know where to go from ...