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

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1answer
23 views

How to factorize this.

We just started calculus and busy with limits. we were told that use a limit as long as it does not make the equation undefined. So the question is: $\displaystyle \lim_{x\to 0} \dfrac{2x}{x^2+x}$ ...
-1
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1answer
15 views

Factorisation of polynomials in C [on hold]

How can I factorize this term in $\mathbb{C}$? Any further explanation will be appreciated! $$z^2-3z+4$$
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2answers
48 views

Irreducible polynomials of the form $x^n - q$

Is there any easy way to see the following? Let $n\in\mathbb{N}$. Let $a,b\in\mathbb{R}$ s.t. $a < b$. Then, there exists $q\in\mathbb{Q}$ s.t. $a<q<b$ and $p(x) = x^n - q$ is irreducible ...
1
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1answer
23 views

Which factors determine whether a set of variables are suitable for factor analysis?

Which factors determine whether a set of variables are suitable for factor analysis? I am looking as much for an explanation of the question as a tentative answer to it. So grateful for any help on ...
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0answers
40 views

Integer factorization: What is the meaning of $a^2 - kc = b^2$

I found an interesting behavior when placing the integer factorization problem in to geometry, I call it pyramid factoring. Lets assume we have $c$ boxes and we want to order them in to rectangle. ...
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0answers
42 views

If Integer Factorization is in $P$ — what are the implications? [on hold]

If Integer Factorization is in $P$ Beside the bad news of the wide use, RSA cryptosystem, becoming insecure, is there any good news for such an algorithm to be found? For the seek of this question, ...
3
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0answers
213 views
+50

Integer Factorization: Possible progress

I build an algorithm for solving Integer Factorization problem when the number to factor is selected by multiplication of two prime numbers with the same bit count. So far I only been able to factor ...
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1answer
37 views

Is it easier to find $a^2-8c=b^2$ than $a^2-c=b^2$

I found a way to factor numbers if I find: $$a^2-8c=b^2$$ Where $c$ is the number I want to factor Is it easier than searching for the next equation? $$a^2-c=b^2$$
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2answers
239 views

finding residue with $\oint_C \frac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$

I am doing the integral $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$, and I am trying to find the residue at the pole $3i$;I am unsure how to do this. Could I factor $z^2 + 9$ further?
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2answers
24 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
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3answers
17 views

Factors of polynomial not passing the Bezout's identity test

When factoring $x^3 - 2x^2 - 4x - 8$ the result you get is $(x-2)(x^2 - 4)$ or $(x-2)^2 (x+2)$ , meaning that the mentioned polynomial is divisible by each of these factors. When using the Bezout's ...
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2answers
138 views

Number of Factors of 6

factor of 6 is 1,2,3,6, or factor of 6 is 1,2,3,6,-1,-2,-3,-6 Which one is correct?
2
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0answers
26 views

Does this approach for factorizing RSA numbers help in any way?

I was thinking about why factorizing RSA numbers is so hard. When humans perform any kind of maths manually, they often employ various 'tricks' that get them closer to the answer. Some are based on ...
1
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1answer
35 views

Integer factorization simplification

I found a small improvement to the brute force algorithm for the Integer Factorization. Please tell me if there is a point to investigate it more or there are better similar ideas. I found that if ...
2
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2answers
123 views

prime factors of number with a particular form

I try to factorize this huge number $2^{(3^{(5^7)})} +7^{(5^{(3^2)})}$ .but i have no idea,the only thing i know is that it's not divisible by 7 and 11. can you help me find some prime factors of ...
0
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1answer
35 views

Factoring polynomial $x^3−2x^2−4x−8$ that fails Bezout's identity test

I usually factor 3rd degree polynomial in two steps. First, I find all the divisors of the last, coefficient-free part of the polynomial (in this case that's 8) and try (applying Bezout's identity) to ...
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0answers
25 views

When $\sqrt{(bce + ae + 1)^2 - 4bed} $ is integer?

While been working on my factoring algorithm I came to this: $$\sqrt{(bce + ae + 1)^2 - 4bed} = x$$ $a,b,c,d$ are known positive integers $e$ positive prime integer How can I find all the $e$ ...
4
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2answers
103 views

How do I simplify this expression about factorization?

I am trying to simplify this $$\frac{9x^2 - x^4} {x^2 - 6x +9}$$ The solution is $$\frac{-x^2(x +3)}{x-3} = \frac{-x^3 - 3x^2}{x-3} $$ I have done $$\frac{x^2(9-x^2)}{(x-3)(x-3)} = ...
1
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3answers
28 views

What is $c$ in $\left\lfloor\frac{a}{bc}\right\rfloor=d$

As part of my attempts to solve integer factorization problem. I came to this equation: $$\left\lfloor\frac{a}{bc}\right\rfloor=d$$ $a,b,c,d$ are positive integer values $\frac{a}{b}$ is an integer ...
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3answers
37 views

Can every polynomial be factored into constant and linear complex factors?

That is, can any polynomial, $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1+a_0$, be expressed $b_0\left(x + b_1\right)\left(x + b_2\right)\ldots \left(x + b_n\right)$ where $b_i \in \mathbb{C}$?
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3answers
75 views

How do I solve this, first I have to factor $ 2x\over x-1$ + $ 3x +1\over x-1$ - $ 1 + 9x + 2x^2\over x^2-1$?

I am doing calculus exercises but I'm in trouble with this $$\frac{ 2x}{x-1} + \frac{3x +1}{ x-1} - \frac{1 + 9x + 2x^2}{x^2-1}$$ the solution is The only advance that I have done is factor $ ...
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0answers
34 views

When $\frac{-1436+y\pm \sqrt{(1436-y)^2 - 4\cdot480547}}{-2}$ is integer

While working on my algorithm I came to this problem: $$x_{1,2} = \frac{-b-a+y \pm \sqrt{(b + a-y)^2 - 4c}}{-2}$$ $$x_1 = \frac{c}{x_2}$$ $a,b,c$ are positive known integers $y$ is positive integer ...
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0answers
112 views
+50

Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
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2answers
32 views

Why $(10^ab+c)^{4d+1}-c \mid 10$?

I came across the following equation: $$x=(10^ab+c)^{4d+1}-c$$ Why is $x$ a multiple of $10$ for any natural number values for $a$, $b$, $c$ and $d$? The only progress I made was that $a$ could be ...
0
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2answers
43 views

Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$

I have to show that if $d > 2$ is a prime number, then $2$ is not prime in $\mathbb Z[\sqrt{-d}]$. The case when $d$ is of the form $4k+1$ for integer $k$ is quite easy: $2\mid ...
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2answers
21 views

Using the distributive property to factor $(5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x)$

I can't seem to understand the distributive property. Take this: $$ 5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x$$ becoming this: $$ 5^x\left(\frac 15 - 1 - 5 +25\right) $$ Help? :D
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0answers
17 views

Should this be rephrased into saying no common factors but 1?

This is from Discrete Mathematics and its Applications For the phrase "a and b have no common factors" , does that actually mean a and b have no common factors other than 1? I feel like this would ...
2
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2answers
777 views

What is “prime factorisation” of polynomials?

I have the following question: Find the prime factorization in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreducibility in $\mathbb{Z}[x]$, of ...
2
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2answers
122 views

If GCD of a list of numbers is 1, is it a necessary condition that GCD of at least one pair of numbers from the list should be 1?

Suppose our numbers are {2, 6, 3}. GCD (2, 6, 3) = 1, GCD (2, 6) = 2, GCD (6, 3) = 3, but GCD(2, 3) = 1 If GCD(a,b,c) = p, GCD(a,b) = q, GCD(b,c) = r, GCD(c,a) = s, is it possible that p = 1 and (q ...
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0answers
32 views

Integer factorization: does $az + by$ helps?

I am trying to find solutions for integer factorization problem. And particularly I am curries in RSA cracking. I came to the next equation: $$az + by = \frac{c}{x}$$ $c$ is the number that I am ...
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0answers
19 views

Factorise the following polynomial

file://localhost/var/folders/0p/frxrkc9d4_z99dy684t4_9100000gn/T/LaTeXiT-2.6.1/latexit-drag.pdf How do you factorise the above?
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2answers
58 views

Factorise the following polynomial [closed]

$$x^6+3x^4+4x^2+2$$ How do you factor this polynomial if it has no real solutions?
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0answers
17 views

length plus width equals price, factoring?

im trying to grasp what this means as I usually work with areas (L x W = A) or perimeters (2L + 2W = P)... This problems was presented to me by a colleague and i'm just trying to wrap my head around ...
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2answers
20 views

How to factor this expression

Could someone explain to me the process of factoring this $-6x(x+1)(x^2+2)^{-5/2} + 2(x^2+2)^{-3/2}$ into this $2(1-2x)(x+2)(x^2+2)^{-5/2}$?
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3answers
79 views

factorizing without removing brackets [closed]

\begin{equation} 2y(y+z)-(x+y)(x+z) \end{equation} Factorize this without removing bracket at any stage I dont know what method to use here Should I factor the brackets separately
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1answer
23 views

. Find the partial fraction decomposition of the following rational function.

Find the PFD of the folowing: $$\frac{x^6-x^5+48x^3-53x^2+99x+48}{x^5-2x^4+2x^3-4x^2+x-2}$$ Initially I used long division and got $$x+1+\frac{50(x^3-x^2+2x+1)}{x^5-2x^4+2x^3-4x^2+x-2}$$ I then used ...
13
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1answer
792 views

Finding smallest and largest prime factor of $\frac{200!}{180!}$

I originally started off by listing all the primes: $p<200$ then trying to calculate the prime factorisation of each (which I realise is a silly thing to do) I believe there must be a simpler way ...
1
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1answer
39 views

proof factoring of $x^n-1$

I have a question about the proof of the theorem stating that $$x^n-1 = \prod_{i \in I} m^{(i)}(x)$$ is a factorisation in irreducible factors of $x^n-1$ over a field $\mathbb{F}_q$. Here $m^{(i)}$ is ...
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0answers
64 views

Is reducing factoring of integers to finding a polynomial which takes a perfect square value useful?

Below we only consider numbers $N=pq$, where $p$,$q$ are primes of the form $ (6j+1)$. It's easy to show that $N + 9n^2 = d^2$. $9k^2$ is related to $(p-q)$ and $d^2$ is related to $(p+q)$. the $n$ in ...
0
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1answer
37 views

Factorization with a Primitive Factor of Polynomials

Question: Let $f,g\in\Bbb Q[x]$. Why is it that $\rm\color{#c00}{(1)}$ if $f$ is monic then $f=\frac{1}{a}f^*$ for some primitive polynomial $f^*\in\Bbb Z[x]$ and $a\in\Bbb Z$ ? ...
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1answer
61 views

Integer factorization: Single solution for integer equation

While working on my integer factorization project, I came to this: $(A + CX)(B + CY) = D$ $X,Y,A,B,C,D$ Are integer numbers $A,B,C,D > 0$ $X,Y >= 0$ $A,B,X,Y < C < D$ If $X=Y$ than $Y ...
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0answers
61 views

Reducing multivariate rational fractions to lowest terms

I wish to simplify multivariate rational fractions to a canonical form. Thanks to some very helpful mathematically inclined people who verified that my understanding of Wikipedia was correct, I'm now ...
4
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3answers
113 views

Factor $3x^2-11xy+6y^2-xz-4yz-2z^2$

This problem is from my Math Challenge II Algebra class, and it's really confusing. How can you factor something like this? Here's the question again: Factor $3x^2-11xy+6y^2-xz-4yz-2z^2$.
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2answers
24 views

Why can we write a uncontinual function continual?

Let's consider $$f(x) = \frac{(x-1)(x-2)(x-3)}{x^2-3x+2}$$ with definition $D_{f} = \mathbb{R} \setminus \lbrace 1, 2 \rbrace$. This means we are allowed to set $x$ to every value of $\mathbb{R}$ ...
2
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2answers
34 views

How to factorize polynomial in GF(2)?

I want to know how to factorize polynomials in $\ GF(2) $ without a calculator in a product of irreducible factors. For example, in my exercises, I must factorize $\ p(x) = (x^7 - 1)$ The response is ...
2
votes
1answer
54 views

Possible values of $\gcd(a+b, a\times b)$

Main Question: Let $N \in \mathbb{N}$. What are the possible values of $\gcd(a+b, a\times b)$ given that $\gcd(a,b) = N$? Fact 0. If $\gcd(a,b) = N$, then $N \leq \gcd(a+b, a\times b) \leq ...
1
vote
6answers
546 views

How to factor $x^4-7x^2-18$

I am not sure how I would factor this. The $x^4$ and $x^2$ are really throwing me off. Can someone explain how I would factor this?
0
votes
2answers
71 views

Is it possible to factor $4x^2-3$?

Is it possible to factor $4x^2-3?$ I honestly can't thing of any way to factor this, but I wanted to be sure it was, in fact, impossible to factor. EDIT: Thanks for the help in the comments. ...
0
votes
2answers
57 views

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomials over Q for any natural number n.

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomial over $\mathbb{Q}$ for any natural number n. This is an extended version of my previous question here. I know if $n=4$, ...
2
votes
0answers
74 views

Show that $x^4+n(2-n)x^2+n^2$ reducible over Q for any natural number n.

Show that $$x^4+n(2-n)x^2+n^2$$ reducible over Q for any natural number n. Here is what I did $$x^4+n(2-n)x^2+n^2=x^4+2nx^2-n^2x^2+n^2=x^4+2nx^2+n^2-n^2x^2$$ $$=(x^2+n)^2-(nx)^2$$ ...