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

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3answers
96 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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0answers
37 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 ...
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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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0answers
18 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 ...
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4answers
238 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
41 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 ...
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1answer
37 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 ...
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4answers
306 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 ...
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1answer
42 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 ...
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1answer
28 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 ...
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2answers
42 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?
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0answers
48 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, ...
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1answer
51 views

Examples of prime ideals?

Could anyone provide me with very simple examples of prime ideals (that is,principle ideals in the ring of integers which are generated by a prime), explaining me the way they are generated? The ...
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1answer
20 views

How many pickups $K$ should I do to have a $p$% of probability of picking up a divisor of $n$ (if exists) in the interval $[2..\lfloor n/2\rfloor]$?

I am trying to understand if it makes sense an algorithm to decide if a given number $n$ is possibly prime or not by using the divisor function bound defined by professor Jeffrey Lagarias as: ...
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0answers
26 views

About factoring trinomials over $\mathbb{Z}$

We were taught in school an algorithm to factor a trinomial of the form $$x^2\pm bx\pm c$$ with $b,c\in \mathbb{Z}^+$. Assuming the best scenario (that the polynomial has both roots in ...
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2answers
74 views

Factorization of huge integer

I have to factorize the integer $n = 2^{214313833}-1$. Obviously this is not a prime, because $214313833 = 9623 \cdot 22271$, so $n_1=2^{9623}-1$, $n_2=2^{22271}-1$ are divisor of $n$, though $n\neq ...
2
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2answers
56 views

Methods for Factoring Cubics

I am looking for some advice and tips/help about something. I am in calculus now and have been doing well but I recently realized to a bit of my own embarrassment that I am still not fully comfortable ...
1
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1answer
46 views

how do I factor this $6a^2+ 70ab$?

how to factor $6a^2+ 70ab$ ? I got this: $$6a^2+ 70ab = 2a ( 3a + 35b ) $$ Is the factorization complete ?
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33 views

I need help with this factorization problem?

How can I factorize this problem: $1-8xy-x^2-16y^2$ I noticed that there are common terms, but how should I proceed ?
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2answers
57 views

Why Can't we Factor Invertible Elements?

I'm currently studying Herstein's Algebra; specifically, UFDs and the abstract notion of factorization. This is perhaps more of an intuitive question than one with a hard answer. We define ...
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3answers
32 views

Find the speed of a jet given the time of travel back and forth

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
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2answers
39 views

Converting from factored to standard form: why is this answer wrong?

Converting the equation $$y=-2(x-2+\sqrt{5})(x-2-\sqrt{5})$$ to standard form seems to give $$-2x^{2\space }+3.528x+6.4171392.$$ My handout tells me that the answer is different. What is wrong here? ...
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0answers
29 views

How do I solve this quadratics problem? [duplicate]

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
2
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2answers
26 views

Showing that an element is prime in $\mathbb{Z}$[i]

Let p be a prime integer, and suppose p = a2 + b2 has NO integer solution. The exercise asks that if p = a2 + b2 has no solution, then p is a prime in the set of Gaussian integers $\mathbb{Z}$[i], ...
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4answers
66 views

How to factor quadratics $(x^2 + 4x + (-357) = 0)$

I need to find $2$ factors of $-357$, which add up to $4$. Obviously one number is positive and the other is negative. I understand this and I know the factors can be $21$ and $-17$; but, how do I ...
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2answers
37 views

Polynomial factorisation for unique factor domain

Suppose $R$ is a UFD and $f \in R[X]$ such that $\deg f > 0$ and $f$ has a root $\alpha \in R$. Show that $f = (X - \alpha) g$ for some $g \in R[X]$. (Suggestion: Write $f = a_0 + a_1 X + \dotsc ...
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2answers
48 views

LU Factorization - Linear Algebra

LU-factorization My solution: Am I on the right path? Or am I completely off?
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1answer
43 views

What is a good example of an algorithm that is hard to parallelise?

When I have 10 computers, the factorization of a number doesn't scale along. I am not sure how much faster it would go compared to a single computer, but not 10 times faster like one would expect. ...
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1answer
22 views

Irreducibility of polynomials of a certain kind

Let us look at factorization over the integers of polynomials of the form $x^n+n$. For the first few values of $n$ we get $x+1$ - irreducible $x^2+2$ - irreducible $x^3+3$ - irreducible $x^4+4$ - ...
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1answer
26 views

The linear factor of the polynomial

Recently I've started to study polynomials, when I found out about the remainder and factor theorems as a way to avoid long polynomial division I couldn't understand the reason for every linear factor ...
2
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2answers
64 views

Quadratic Polynomial factorization

This could be primary school stuff. But I want to ask it. In factoring $x^2+bx+c$ (i.e. $a = 1$ in $ax^2+bx+c$), we find $m$ and $n$ such that $m+n = b$ and $mn=c$. We can reason this well as ...
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1answer
18 views

Find the $LDL^{T}$ factorization of $A$ when in the range of the positive definite

I am trying to find the $LDL^{T}$ factorization of the following matrix $$ A = \begin{bmatrix} 1 & b \\ b & 4 \end{bmatrix} $$ when $b$ is in the range of positive definiteness. I have ...
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2answers
43 views

Proof of irreducibility in Z[x] when its reduction mod p has known factors

Problem: Show that if a polynomial $f(x)$ in $\mathbb{Z}[x]$ of degree $n$ has no rational root, but for some prime $p$ its reduction mod $p$ has irreducible factors of degrees $1$ and $n - ...
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1answer
129 views

Factor $x^{14}+8x^{13}+3$

I need to factor this over the rationals, and there is a hint to use reduction mod3. The reduction is $x^{14}+2x^{13}=x^{13}(x+2)$, but I know it has no rational roots (they would have to be $\pm 3$ ...
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0answers
40 views

Factorizing a quartic polynomial over an arbitrary field

The following is a problem from Artin: How might a quartic polynomial $x^4+bx^2+c$ factorize over an arbitrary field F? Explain with reference to the polynomials $f(x)=x^4+4x^2+4$ and ...
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0answers
22 views

Factorising Complex Polynomial with Complex Coefficients

I have tried to factorise the polynomial in question 19 by using the factor theorem to find other factors, however this has been unsuccessful thus far. Seeing as the conjugate root theorem does not ...
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1answer
28 views

Expression factoring question

This is from a simple book explaining differentiation to the uninitiated and I don't understand the factoring. Can anyone help me understand how equation 3 is derived? Thanks Let $y = x^{-2}$ Then ...
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3answers
96 views

Keep factoring and concatenating to get a prime?

Keep factoring and concatenating,starting from $2$ until we get a prime. $$2=2$$ $$22=2*11$$$$22211=7*19*167$$ $$22211719167=?$$ ...and so on (the prime factors are arranged from smaller to larger ...
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1answer
41 views

Principal ideal domain with finitely many ideals

Let $aR$ be a nonzero ideal in a PID $R$. Show that $R/aR$ is a ring with only finitely many ideals. Honestly, I do not know how to start. Appreciate any tips.
2
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1answer
49 views

Are there exception cases when you are bringing an exponent out of a logarithm?

The domain of a logarithm $\log(x^2)$ is $D:x\in(-\infty,0)\cup(0,\infty)$. But if I use the identity $\log(a^b)=b\log(a)$ and do: $\log(x^2)=2\log(x)$ the domain becomes $D: x\in(0,\infty)$ The ...
2
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2answers
138 views

Extracting factor from quadrinomial

As I'v learned about polynomials, I run into this quadrinomial: $$x^3+300x^2+30000x-953125 = 0$$ I've been studied how to factor this quadrinomial but didn't quite understand how it's done, here is ...
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0answers
35 views

When looking at the mod as binary value

Look at the next value: $$617*947 = 584299$$ 617, 947 are prime values. I want to see what are all the possible solutions for the next equation, for $k=4$: $$(a\mod k)(b\mod k) = 584299\mod k$$ ...
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2answers
66 views

How to simplify a fraction like this one?

$$\frac{x^2-3x+1}{x-3}$$ Is there a rule for factorizing polynomials in the numerator?
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1answer
34 views

Fourier transform doubting factorization

I have to find the fourier transform for $$ {1\over 1+16t^4} $$ I guess going there is a better way to solve it than going throug the integral but I'm not even sure if the factorization i made is ...
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1answer
47 views

Break down $x^4 + 5x^2 +5$

How do I break down the function in the title even further? I think that I need to use a square root somewhere, but I'm not certain.
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3answers
44 views

Quadratic formula question: Missing multiplying factor of A?

I have a very simple problem which must have a simple answer and I was wondering if anyone can point out my error. I have the following quadratic equation to factor: $2x^2+5x+1$ Which is of the ...
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1answer
62 views

How to solve inequality problem without factoring or quadratic equation

I'm tutoring someone, and I'm stuck on one of her problems. The equation is $\sqrt{x+14}\le x-16$. She hasn't been taught the quadratic formula or how to factor these problems yet. Is there a way to ...
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4answers
271 views

Factorizing exponential equation

I have this equation: $$2^{2x}−3⋅2^x−10=0$$ Could someone explain how you factorise it to be: $$(2^x+2)(2^x−5)$$
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3answers
88 views

HCF of two integers (a,b) =0

Suppose $hcf(p,q) = 0$ , is it even possible to prove that $p=q=0$? My answer to that is $x = hcf\left(\frac ph,\frac qh \right)$. We have to prove that $x=0$. Since $x$ is a common of $\frac ph$ ...
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1answer
25 views

Factoring terms-laws of exponents

Given the exponent terms $x^m+x^{2m}$, how would it look like if we factor out $x^m$?