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

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3
votes
1answer
111 views

Factoring $y$ out of $y^2 + e^y$ possible?

I'm working on solving a separable first order differential equation where I get down to: $$.5y^2 + e^y = .5x^2 + e^{-x} + c$$ Is it possible to solve for $y$ here? I can't think of a way to factor ...
1
vote
1answer
133 views

Quick way to find the highest multiplicity of a divisor of a number?

Not sure if worded properly. For instance, the highest multiplicity of 2 in 60 is 2 because the prime factorization of 60 is 2^2*3*5. For 16, the highest multiplicity of 2 would be 4, etc. Is there ...
4
votes
2answers
802 views

Factorize polynomial over $GF(3)$

I want to factorize $x^{11}-1$ over $GF(3)$ but I'm stuck at $(x-1)(x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1).$ I have tried to do it trial and error but failed. Is $$ ...
1
vote
2answers
2k views

Notation : What is the meaning of the (mod n) in factoring algorithms?

Pretty much every thing is in the title, really! I'm trying to come up with an efficient algorithm to factorize large integer as an homework for a parallel programming course. I've seen a few pages ...
2
votes
0answers
93 views

Factors of a polynomial in several variables

Fix an embedding of $\overline{\mathbb{Q}}$ into $\mathbb{C}$. Suppose you have a polynomial in several variables, with algebraic coefficients: $P\in \overline{\mathbb{Q}}[z_1, \ldots, z_n]$. Also ...
4
votes
1answer
187 views

Help with basic high school math. What happens to $j$?

I know my math is very rusty, actually, its always been that way. but I need help with this. The question below has me stumped. I've tried to show the steps I went through to get the answer. Please ...
0
votes
2answers
135 views

What are the specifics and the possible outputs of Pollard's Rho algorithm?

I'm trying to implement a simple prime factorization program (for Project Euler), and want to be able to use Pollard's Rho algorithm. I read the Wikipedia, wolfram|alpha, and planet math explanations ...
31
votes
4answers
1k views

Could G. H. Hardy make a product of two primes so big he couldn't find out which?

This question of course began as a slightly irreverent play on the riddle, "Can God make a stone so big He can't lift it?" Then I wondered about the answer. Is it possible to exhibit a number that is ...
0
votes
1answer
91 views

Beginners question: Factorised expression, not sure how book got the answer

I have an expression: $2a^3 / 3a^2 * 6a^5$ and my book gives the answer: $4a^6$ I get where the $a^6$ is from but how is the $4$ worked out?
2
votes
3answers
129 views

How to express $\frac{x^3+4x^2-1}{(x^2+1)^2}$ as a polynomial plus a proper fraction, using long division?

I'm trying to express $$\dfrac{x^3+4x^2-1}{(x^2+1)^2}$$ as a polynomial plus a proper fraction, using long division but I don't know how to do that. It'd be cool if you can solve this. Thanks.
1
vote
1answer
176 views

What are the steps for factorizing $1 - ab - cd + abcd$?

I can see that $1 - ab - cd + abcd$ factors to $(1-cd)(1-ab)$ but only because I tried a lot of different factors in a trial and error method, so it took me a while. I was wondering what the pattern ...
2
votes
2answers
149 views

For $x_1,x_2,x_3\in\mathbb R$ that $x_1+x_2+x_3=0$ show that $\sum_{i=1}^{3}\frac{1}{x^2_i} =({\sum_{i=1}^{3}\frac{1}{x_i}})^2$

Show that if $ x_1,x_2,x_3 \in \mathbb{R}$ , and $x_1+x_2+x_3=0$ , we can say that: $$\sum_{i=1}^{3}\frac{1}{x^2_i} = \left({\sum_{i=1}^{3}\frac{1}{x_i}}\right)^2.$$
1
vote
2answers
532 views

equations with lcm and gcf

If $p$, $q$, and $r$ are three different odd prime numbers what are the $\operatorname{lcm} (2pq, 2pr, 2qr)$ and $\operatorname{gcf} (2pq, 2pr, 2qr)$. Anyone have any suggestions on how to even start ...
4
votes
4answers
2k views

How can we prove that among positive integers any number can have only one prime factorization?

I have read right from school that prime factorization is unique, but have never found proof for this. Can someone show me the proof?
0
votes
1answer
85 views

How to solve for x

$W(4d^2 (1-x^2)^2) = abc^3x \sqrt{(\pi^2 (i-x^2)^2 + 16 x^2) }$ I have to find x ,i have the values of all other constants , I tried to separate it using partial fraction but I am stuck. a=3 b=4 c=7 ...
3
votes
5answers
724 views

Number of Solutions of $3\cos^2(x)+\cos(x)-2=0$

I'm trying to figure out how many solutions there are for $$3\cos^2(x)+\cos(x)-2=0.$$ I can come up with at least two solutions I believe are correct, but I'm not sure if there is a third.
1
vote
1answer
155 views

Interesting prime factorization function divisibility problem [duplicate]

Possible Duplicate: Is the set of all numbers which divide a specific function of their prime factors, infinite? Let the function $f(n) =(p_1^{a+1}-1)(p_2^{b+1}-1)... $ where $n$ is an ...
5
votes
2answers
858 views

Why is integer factorization considered to be in NP if a quantum computer can compute a factorization in polynomial time?

Sorry if this seems off topic, the cstheory guys told me it was off topic over there, and sent me here. Shor's algorithm on a quantum computer can solve an integer factorization problem in polynomial ...
4
votes
3answers
1k views

Factors of a number?

I'm studying for the SAT on collegeboard.org, and I came across the problem: What is the least positive integer that has the same number of positive factors as 175? ...
0
votes
1answer
116 views

Suitable Loss function for Order preserving Factoring of a matrix?

(Old-Question) Given a $n\times n$ symmetric matrix $X$, I would like to factor it using a vector $c$ of size $n \times 1$ such that: $\sum_{i,j} [X_{ij} \cdot c_i\cdot c_j]$ is minimum. How can I ...
3
votes
1answer
692 views

Factorize a Symmetric matrix as an 'Approximation' with an outer product.

(deprecated-taken back based on discussion(OLD)) What is a good way to factor a symmetric matrix $X$ as an outer product of two vectors $u$ and $v$. i.e, Find two vectors $u$ and $v$ such that ...
2
votes
4answers
205 views

Factoring a polynomial

I am trying to factor the following polynomial: $$ 4x^3 - 8x^2 -x + 2 $$ I am trying to do the following: $ 4x^2(x - 2)-x+2 $ but I am stuck. Thanks for your help. edit: correction.
2
votes
2answers
151 views

Factoring 4 terms polynomial

I am trying to factor the following polynomial: $$ 8x^3 -4x^2y -18xy^2 + 9y^3 $$ $$ (a-b)^3 = a^3 -3a^2b + 3ab^2 - b^3 $$ Thanks
0
votes
1answer
425 views

Exponential equation factoring

I've empiricaly produced this exponential equation to express a graphical representation : $$ y = \left(a^x + (bx)²\right) \left((1-10^{-x}) x\right) $$ I know the constants $a$ and $b$. ...
2
votes
1answer
113 views

Subset of bits of factors of integer

Is there any information on the internet concerning analysis of subsets of bits of the (unknown) factors of any given integer n? Me being unskilled with phrasing things properly for google has given ...
1
vote
0answers
94 views

Is there a name for a number whose factors' exponents are all prime?

For instance, 864, whose factorization is 2^5 x 3^3.
5
votes
2answers
174 views

elegant way to show $P= t^{1024}+t+1$ is reducible in $\mathbf{F}_{2}[t]$

This is homework exercise: $$P=t^{1024} + t + 1 , R = \mathbf{F}_{2}[t] \Rightarrow P \ \text{reducible in R}$$ I wanted to show this analogous to how a book shows it (book shows it with other ...
1
vote
1answer
1k views

General method for factorizing matrix determinants

I'm learning how to factorize determinants of a square matrix in school, but we haven't learnt a general method to do that, besides 'creating zeros'. So I thought maybe I'll ask here if someone does ...
1
vote
1answer
196 views

Unique factorization less than 100

How do I approach this problem using unique factorization?... How many numbers are product of (exactly) $3$ distinct primes $< 100$? edit: Just to add to that, How does unique factorization ...
0
votes
3answers
116 views

finite fields factorization

Let $\mathbb{F}_2$ be the finite field with two elements. Let $f(x) = x^6+x^4+x+1$ be in $\mathbb{F}_2[x]$. If $f(x)$ is irreducible, give a reason. If it is not irreducible, determine a factorization ...
1
vote
2answers
720 views

How do I factor a polynomial function with a degree higher than 2 without guessing numbers of $\frac{p}{q}$?

I have an equation $f(x)=x^4+4x^3+2x^22-x+6$. In the past I was taught to factor it by getting the zeros by getting $p/q$, and start guessing zeros, and plugging them into the function. Once I got one ...
5
votes
6answers
8k views

How do you factor $x^3-3x^2+3x-1$?

$$x^3-3x^2+3x-1?$$ I know this may seem trivial, but I, for the life of me, I cannot figure out how to factor this polynomial, I know that the root is $$(x-1)^3=0$$ because of wolframalpha, but I ...
1
vote
1answer
80 views

Factorizing Composites

Say $N=AB$ where $A$ and $B$ are primes. We write: $$A=a+x,\qquad B=a-x.$$ That is, $$a=\frac{A+B}{2};\qquad x=\frac{A-B}{2};$$ $A$ and $B$ are odd numbers. Therefore $A+B$ and $A-B$ are even. And ...
28
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
5
votes
2answers
671 views

A theorem about prime divisors of generalized Fermat numbers?

A theorem of Édouard Lucas related to the Fermat numbers states that : Any prime divisor $p$ of $F_n=2^{2^n}+1$ is of the form $p=k\cdot 2^{n+2}+1$ whenever $n$ is greater than one. Does anyone ...
42
votes
16answers
43k views

What is a real world application of polynomial factoring?

The wife and I are sitting here on a Saturday night doing some algebra homework. We are factoring polynomials and we both had the same thought at the same time: when are we going to use this? I feel ...
16
votes
4answers
552 views

Are polynomials of the form : $ f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a$ irreducible over $\mathbb{Z} $?

Is it true that polynomials of the form : $ f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a$ where $\gcd(n+1,k+1)=1$ , $ a\in \mathbb{Z^{+}}$ , $a$ is odd number , $a>1$, and ...
2
votes
3answers
192 views

How to factor $2x^2 - 8y^2$

How to factor $2x^2 - 8y^2$ ? So far I got it down to $$2(x^2 - 4y^2),$$ but it's not the answer; I don't think it's factored enough.
2
votes
1answer
140 views

Is $ x^2+ax+a$ irreducible over ring $\mathbb{Z}$ of integers?

How to prove that polynomials of the form : $P(x)= x^2+ax+a$ , where $a \in \mathbb{Z^{+}}$ \ $ \left \{ 4 \right \} $ are irreducible over ring $\mathbb{Z}$ of integers ? Eisenstein's ...
5
votes
2answers
229 views

Is an algebraic formula for the number of cyclic compositions of n known?

From Wikipedia: In January 2011, it was announced that Ono and Jan Hendrik Bruinier, of the Technische Universität Darmstadt, had developed a finite, algebraic formula determining the value of p(n) ...
1
vote
3answers
146 views

Adjunction of a root to a UFD

Let $R$ be a unique factorization domain which is a finitely generated $\Bbbk$-algebra for an algebraically closed field $\Bbbk$. For $x\in R\setminus\{0\}$, let $y$ be an $n$-th root of $x$. My ...
2
votes
1answer
64 views

Factor the binomial $36 m^2 - \frac{25}{4} $

OK, I got a new one... it's $36 m^2 - \frac{25}4 $ and I got: $(18m-\frac{5}2 )(18m+\frac{5}2 )$ although that is incorrect... where did I go wrong?
2
votes
1answer
174 views

How does the rational root test work?

I've doing some research in wikipedia, where appears this example:$$3x^{3} - 5x^2 + 5x - 2 = 0$$the rational solution must be among the numbers symbolically indicated by: $$±\frac{1,2}{1,3}$$ So ...
0
votes
2answers
994 views

Factor By Grouping 3rd Degree Polynomial

Just to be upfront, this is a homework question, I already know the answer, but I can't figure out how to get there or the logic behind the hint, which is really what I'm after. Please don't solve it ...
1
vote
1answer
93 views

How do I factor this kind of equations?

I was doing some integration by partial fractions exercises and I found this equation:$$\int_{0}^{1}\frac{x^{3}+1}{x^{4}+4x+3},$$ and I don't know how to factor that in order to compute the partial ...
1
vote
0answers
264 views

Factoring multivariate polynomials

If I have a multivariate polynomial $P[X_1,\dots,X_n]\in \mathbb{R}[X_1,\dots, X_n]$, is there a polynomial time algorithm to factor the polynomial into irreducible polynomials $\in ...
0
votes
0answers
60 views

What is this special type of factor called?

I'm wondering if there's a special term for the following: The (special factor) of a number $x$ is a pair of numbers that multiply to give $x$ but has the smallest difference compared to other ...
1
vote
0answers
71 views

lower bounds for maximum computing times for integer factorisation

Supposing that n were known to have two prime factors, and that the computer had a database of all the primes $<\sqrt{n}$. Then, unless n is square, one factor would be $<\sqrt{n}$. If an ...
1
vote
1answer
213 views

Polynomial Factorization

I was handed $x^3-x^2+x-2=0$ to factor, but I'm not sure how. I tried all the methods I know of--which, at the time of writing, are limited by my precalc math background (I'm working on that...). Is ...
0
votes
0answers
56 views

Figuring $x$ where $x = \max \{ \operatorname{round}(7),0 \}$

I'm a software developer and do very little with math but I have been called upon to incorporate a math function into my app I'm developing. I'm stumped as to how to figure $x$ where $x = \max \{ ...