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

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0
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1answer
13 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 ...
0
votes
1answer
23 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 ...
4
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5answers
44 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 ...
-2
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2answers
60 views

What are two numbers that could multiply up to 373248 and add up to 20? [on hold]

Example: $x+y=20$ $xy=373248$ I need to know what $x$ and $y$ would be. Thanks for helping!
2
votes
1answer
38 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 ...
1
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2answers
23 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. ...
0
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5answers
74 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 ...
0
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2answers
38 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 $\mathbb F_5$ ...
0
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0answers
32 views

Name for this possible mathematical structure? [on hold]

I'm thinking of a latent variable as an nth (each representing a variable) dimensional object - it can either be for a correlation matrix, or for a factor structure in factor analysis - that's ...
1
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1answer
29 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
25 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 $P$ ...
1
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1answer
24 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 } ...
-1
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0answers
24 views

Minor help in the factorisation techniques used in lec notes

If anyone could help me see how is turned into Thanks in advance!
3
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3answers
64 views

Factorize Trigonometric Equation

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 ...
0
votes
1answer
30 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 ...
0
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3answers
40 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 ...
1
vote
3answers
66 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 ...
4
votes
4answers
55 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} = ...
0
votes
2answers
54 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?
1
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0answers
38 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 ...
1
vote
0answers
31 views

Finding the factors of integer x and its square

What is the the theorem or property that says that $\forall{}x\in{}Z$, the set of all integers, $x^2$ has the same factors as $x$, twice?
4
votes
2answers
310 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 $(x-1)(-x+9)x$.
0
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5answers
51 views

Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
4
votes
1answer
78 views

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 - ...
1
vote
2answers
29 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?
10
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1answer
239 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 ...
0
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0answers
30 views

Please factorise this [duplicate]

a³ - b³ - c³ - 3abc = what???????
0
votes
4answers
48 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 ...
0
votes
2answers
59 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?
0
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0answers
22 views

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 ...
0
votes
1answer
33 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 ...
1
vote
1answer
25 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, ...
2
votes
1answer
37 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 $$ ...
2
votes
2answers
24 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 ...
5
votes
2answers
47 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 ?
0
votes
3answers
47 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.
0
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4answers
28 views

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 ...
3
votes
2answers
83 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.
1
vote
3answers
94 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.
0
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0answers
33 views

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 ...
0
votes
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 ...
1
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0answers
17 views

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 ...
6
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4answers
216 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', ...
2
votes
1answer
37 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 ...
1
vote
1answer
33 views

polynomial inverse in rings understanding

This problem and solution are in the book. I need help understanding the solution. Problem: Let u be a root of the polynomial $x^3+3x+3$. In $\mathbb Q(u)$, express $(7-2u+u^2)^{-1}$ in the form $a ...
7
votes
4answers
296 views

Why are there four solutions to $x^2-2x-8=0$ in $\mathbb{R}$? Or am I wrong?

It might be a very trivial question to ask but why do we get four different solutions for a quadratic equation using these two methods? $x^2-2x-8=0$ We see that factors are $(x-4)$ and $(x+2)$ so ...
0
votes
1answer
38 views

In the general number field sieve, do we need to know whether powers of elements in the algebraic factor base divide an element $a+b\theta$?

I'm reading this paper trying to implement the number field sieve. http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.219.2389 Let $\theta$ be the root of some monic ...
0
votes
1answer
24 views

Factoring ln functions

can anyone tell me how the following factoring ends in $\ln x - \frac{ln 2}{2}$ Original $\frac{\ln x}{\ln 2} - \frac{\ln 2}{ln 2}$ Work shown from Professor $\frac{1}{ln 2} (\ln x - \frac{ln ...
0
votes
2answers
41 views

Factor the polynomial $x^4 + 2x − 4$ in $\mathbb{Z}_5[x]$.

I'm confused as to how this is different from factoring in the reals? Would I start this by writing $x^4+2x-4 \equiv 0 \pmod 5$? What changes?
0
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0answers
44 views

How does the factor command on the TI-89 works?

So to put my question in context, I am working on the following problem. Let $N=1291233941$. Eve's magic box tells her the following three encryption/decryption pairs for $N$: $$(1103927639, ...