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

learn more… | top users | synonyms

1
vote
0answers
27 views

By considering z[sqrt[-2]] show that x^2+2=y^3 only has two integer solutions [duplicate]

By considering z[sqrt[-2]] show that x^2+2=y^3 only has two integer solutions, (+/-5,3) I can see that N(x+i Sqrt[2])=y^3, I think x+i Sqrt[2] is prime in z[i Sqrt[2]] so y^(3/2) must be x+i Sqrt[2] ...
0
votes
1answer
58 views

Modified version of Eisenstein's irreducibility criterion

I have an assignment to extend/modify (and of course prove it) Eisenstein's criterion as follows: Let $f(x)=\sum a_ix^i\in\mathbb{Z}[x]$ with $n\ge 2$ and let $p$ be a prime such that $p\mid a_i$ for ...
4
votes
3answers
128 views

When is the number of $N$'s factors $1 + \sqrt{N}$?

(Answer: Only $N = 4$ and $N = 16$.) The following question arose in a course for pre-service and in-service elementary school teachers: For what $N \in \mathbb{N}$ is it the case that the ...
0
votes
2answers
23 views

Change of factorization in extension field

I have to factorize the polynomial ($x^8-x$) in $\mathbb{F}_{2}$. I found the following factorization: ($x^8-x$) = x*($x+1$) * ($x^3$+x+1)* ($x^3$+$x^2$+1). But now I change to the $\mathbb{F}_{4}$. ...
2
votes
2answers
45 views

Factoring Polynomials in Fields

I always have problems to factorize polynomials that have no linear factors any more. For example ($x^5-1$) in $\mathbb{F}_{19}$. It's easy to find the root 1 and to split it. ($x^5-1$) = ($x-1$) * ...
0
votes
0answers
36 views

Symmetric mod game

$N$ is a big integer value, with only two non trivial factors. Value $k$ will be Symmetric mod if and only if, all the possible nontrivial factors of $N$ will be ...
0
votes
0answers
47 views

“Factorizations” of $a^3+b^3+c^3+mabc$?

It is easy to see that $$a^2+b^2+c^2+ab+bc+ca=\frac{1}{2}((a+b)^2+(b+c)^2+(c+a)^2),$$ $$a^2+b^2+c^2+2(ab+bc+ca)=(a+b+c)^2,$$ $$a^2+b^2+c^2-ab-bc-ca=\frac{1}{2}((a-b)^2+(b-c)^2+(c-a)^2),$$ and ...
3
votes
4answers
112 views

How do you factorise $x^3z - x^3y - y^3z + yz^3 + xy^3 - xz^3$?

I'm trying to factorise $$ x^3z - x^3y - y^3z + yz^3 + xy^3 - xz^3 $$ into four linear factors. By plugging it into WolframAlpha I've learned that it's $$-(x-y)(x-z)(y-z)(x+y+z)$$ My question is: ...
2
votes
0answers
52 views

Multiplication Sieving

Background I made another improvement to Fermat's sieve factorization, by merging the sieves groups of $4,3,5,7,11,13,17$ in two one big group. This method allows me to reduce the values that I need ...
0
votes
2answers
26 views

FULLY Factoring this “polynomial”

So, after a series of step, I was left with (4x-12x), now I need to factor that out completely. So I thought about taking a 4x from each term, and having 4x(_ - 3), however, since I took out a 4x, ...
2
votes
1answer
24 views

Grouping and Factoring Polynomials

Alright, so I have $x^3 + 2x^2y - 4x - 8y$. I've learned that I need to group them together, so I chose to group $2x^2y - 4x$ and $x^3 - 8y$ When taking out their GCF, the numbers left in the ...
2
votes
2answers
41 views

Factoring Trinomials: Dealing with Variables

I'm current working with Trinomials, doing things such as $2w^2 + 38w + 140$. I know how to solve this, however, I encountered a different type of problem, where the last term has a variable in it: ...
0
votes
3answers
38 views

Difference of Squares (Factoring)

I have no idea whatsoever how to factor out this: $$ab^2 - a$$ I know how to solve $a^2 - 4$: $$(a-2)(a+2)$$ But in that case, how can I factor out the $- a$ part?
0
votes
3answers
79 views

Maximal ideal of the ring $\mathbb{R}[x]$

Prove that ideal $M:=(x^2+1)\mathbb{R}[x]$ is a maximal ideal in the ring $\mathbb{R}[x]$. Which field is isomorphic to $\mathbb{R}/M$? Please, help me to solve this problem. I have an exam tomorrow ...
1
vote
4answers
77 views

Prove that if $a +b\sqrt{c}$ is a root of a polynomial in $\mathbb Z$ then $a-b\sqrt{c}$ is also a root of a polynomial in $\mathbb Z$.

Prove that if $a +b\sqrt{c}$ is a root of a polynomial in $\mathbb{Z},$ then $a-b\sqrt{c}$ is also a root of a polynomial in $\mathbb{Z}$. a,b, and c are all integers and c is not a perfect square. I ...
0
votes
2answers
55 views

ring of polynomials and factorization

These are standard facts: $R$ field implies $R[x]$ is a Euclidean domain $R$ is a UFD implies $R[x]$ is a UFD $R$ is an integral domain if and only if $R[x]$ is an integral domain My questions ...
0
votes
0answers
16 views

Divisors of a number in a given range

I'm working on a problem and wondered if there was a clever way to do it. The general form of the problem is like this: given $\ell_1,\ell_2,$ and $N$, find all divisors $d$ of $N$ with $\ell_1\le ...
3
votes
1answer
44 views

Factoring Polynomials (x^4) - using completing squares [duplicate]

Exercise 6. By viewing the polynomials as a difference of two squares, factorise the following polymomials. $x^4+x^2+1$, $x^4+3x^2+4$, $x^4+4$. To solve difference of two squares ...
2
votes
1answer
180 views

By viewing the polynomials as a difference of two squares, factorise the following polynomials.

By viewing the polynomials as a difference of two squares, factorise the following polynomial: $$x^4+x^2+1.$$ I searched but couldn't find a way to solve this Edit: By using Hans Lundmark hint, I ...
-1
votes
5answers
112 views

How can I factorize $x^{10}+x^5+1$? [closed]

How can I factorize $x^{10}+x^5+1$ ? hope you explain the steps :) Thanks
0
votes
1answer
87 views

Factorizable huge semiprime

I'm trying to understand how the number decimal The correct decimal number is: ...
-4
votes
2answers
112 views

How to find all the Quadratic residues modulo $p$

I want to implement Sieve improvement for Fermat's factorization method. For that I need your help answering: How to find all the Quadratic residues modulo $p$? $$\{x ~\vert~ x^2 \equiv q ...
2
votes
3answers
57 views

Need help with a limit to infinity involving a radical with indeterminate form (stuck in the factoring)

this is my first time on Math Exchange, I searched around the site and could not find a question for this math problem so I do not believe that I am asking a previously asked question, if I am please ...
1
vote
1answer
39 views

For what values of $n > 1$ and a is $x^n - a$ irreducible in $\mathbb Q[x]$?

$$x^n-a$$ So $n$ is any integer greater than 1, and $a$ is any integer. $a$ being any integer is where I am running into trouble. I have already shown and worked out a proof for this being ...
0
votes
4answers
119 views

How to factor $56x^4+18x^2-8$?

I've been trying to figure out how you solve this question but I just can't seem to understand how to factor $$56x^4+18x^2-8$$
0
votes
1answer
53 views

Sieve improvement for Fermat's factorization method

I been reading the wiki article about, Sieve improvement for Fermat's factorization method. And I don't understand the mode 16 example, I understand why $a^2$ must be $9$. But why $a$ must be $3$ or ...
0
votes
0answers
36 views

Can I simplify this function any further (beginner question)?

Beginner question! How do I simplify this function? $y = \frac{1}{a} + \frac{3}{a^2}$ I can get as far as: $y =( \frac{1}{a})(1 + \frac{3}{a})$ But I'm not sure if it's possible to simplify ...
3
votes
2answers
116 views

The ultimate formula to factor them all.

Context I am working on Integer factorization problem, I found a formula for factoring numbers, and I need your help to simplify it. First I will explain how I get there and then I present the ...
3
votes
2answers
73 views

When $\sqrt{(x+a)^2 -b}$ is an integer?

While working on integer factorization problem, I came to this: How to find for which values of $x$ the next equation is an integer? $$\sqrt{(x+a)^2 -b}$$ $a,b$ are positive known integers In ...
-1
votes
1answer
31 views

How many solutions exist in reals

Let $f(x)= x^3+3x^2+6x+2009$ and $$g(x)=\dfrac{1}{x-f(1)}+\dfrac{2}{x-f(2)}+\dfrac{3}{(x-f(3)}.$$ The number of real solutions of $g(x)=0$ is
2
votes
1answer
184 views

How do I show a cubic polynomial does not factorise?

In particular: $x^3 - 4x^2 + 4x -2 $. I would know what to do if it had one root which was an integer, however it does not. any help is much appreciated, thank you.
0
votes
1answer
38 views

Quadrinomial with large numbers factorization

I was looking for a way to factor quadrinomial with large numbers without using the remainder theorem to check each factor, for example: $$x^3+300x^2+30000x-953125$$ Is there a way to quickly find ...
1
vote
1answer
40 views

Negative factors vs positive factors

I'm learning about factoring and the lecturer show this example: $$-3x^2+12x-18$$ For start he factor this polynomial like: $$3(-x^2+4x-6)$$ So far so good but now he said: In some ...
0
votes
2answers
59 views

Factorization of $x^8-x$ over $F_2$ and $F_4$

How can I factorize $x^8-x$ over the fields $F_2$ and $F_4$?
0
votes
1answer
27 views

Expansion of polynomial

Expand the following: $-4(5x - 3) ^2$ As for this one, factorise : $5(y^2 - 45) $ Can't it just be $5(y^2 - 9)$? Why is it $5 (y+3) (y-3)$
2
votes
2answers
51 views

When factoring polynomials over the {reals,rationals,integers}, can one get stuck with an incorrect partial factorization?

Suppose I have a polynomial $A$, which I factorize as $A=BC$ (where $B$ and $C$ are polynomials with integer, rational or real coefficients). When factoring $B$ as $B=B'$ and $C$ as $C=C'$ (to ...
0
votes
2answers
44 views

factoring polynomials with 3rd degree or higher

I was searching for a method that would allow me to factor polynomials like this one $x^3 - 13x^2 +(14+4y)x + 8y=0$ I failed, I've only found how to factor by grouping or long division with already ...
1
vote
2answers
88 views

Factorizing $(x+a)(x+b)(x+c)$

I was solving questions related to polynomial factorization. I have learnt the remainder and factor theorems, and some basic identities. There was a question like this one: $$p(x)=x^3+8x^2+19x+12$$ ...
3
votes
1answer
42 views

Does there exist a natural number $a$ such that $a^2+1$ is divisble by $9$?

Can the above question be solved? Or can it be proved that it can not be solved? What is the best approach to solving such questions?
0
votes
2answers
16 views

How do you find the common factor of these expressions?

I have the answer in my answer book but I don't know how to work it out. $2a^2 - a - 3$ ----- $(2a - 3)^2$ ------- $4a^2 - 9$ $a^2b^2 - b^4$-------- $ab^2 + b^3$---------- $ab - b^2$ (I used ...
2
votes
1answer
42 views

If I'm factoring $2p^2+p-10$ would the answer be $p(2p+5) -2(2p+5)$?

If I'm factoring $2p^2+p-10$ would the answer be $p(2p+5) -2(2p+5)$? And to check would I just distribute and see if it matches up to the original problem?
2
votes
1answer
25 views

Factors of a polynomial over $\mathbb{C}$

For one of the problems on my algebra homework I am asked to find the zeros of $p(t)=t^5+t+1$ over $\mathbb{C}$. I have factored it into $$ p(t) = (t^2+t+1)(t^3-t^2+1).$$ We can compute the zeros of ...
1
vote
1answer
92 views

$n$ is a divider of $c$ if and only if $n = 2(c \mod (n-1)) - (c \mod(n-2)) + 2$

While working on Integer factorization problem I came to this conclusion: If and only if $n$ is a divider of $c$ $$c\mod n = 0$$ Than $$n = 2(c \mod (n-1)) - (c \mod(n-2)) + 2$$ c,n are positive ...
0
votes
1answer
37 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
vote
2answers
57 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 ...
2
votes
2answers
147 views

Integer factorization: What is the meaning of $d^2 - kc = e^2$

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. ...
1
vote
1answer
45 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? $$a^2-c=b^2$$
2
votes
0answers
66 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
vote
1answer
60 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 ...
0
votes
1answer
49 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 ...