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

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3answers
31 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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1answer
34 views

Simple factorising $1/x$ [on hold]

$y = x - 1/x$ I want to get x on it's own, I'm lost as to how to factorise 1/x. Thanks for any help.
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4answers
53 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
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2answers
53 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
37 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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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?
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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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1answer
35 views

Find two numbers, given their greatest common divisor and least common multiple [closed]

Highest common factor (HCF) of two numbers is $20$. Least common multiple (LCM) of the same two numbers is $420$. Both numbers are higher than $50$. Find the $2$ numbers. I used factorising trees ...
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5answers
49 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
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1answer
65 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
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2answers
28 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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1answer
231 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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0answers
30 views

Please factorise this [duplicate]

a³ - b³ - c³ - 3abc = what???????
0
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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
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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
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1answer
24 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
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1answer
36 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
23 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
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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 ?
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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
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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.
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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
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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 ...
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 ...
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4answers
213 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
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1answer
36 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
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1answer
32 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
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4answers
295 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
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1answer
32 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
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1answer
23 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
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2answers
40 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
39 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
47 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 ...
0
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1answer
16 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: ...
2
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0answers
24 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
52 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
44 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 ?
0
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2answers
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 ?
1
vote
2answers
51 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
31 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 ...
0
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2answers
35 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? ...
0
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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
votes
2answers
25 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], ...
2
votes
4answers
61 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 ...
0
votes
2answers
34 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 ...