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

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1answer
40 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
46 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
votes
2answers
136 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 ...
1
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0answers
34 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 $k=4$: $$(a\mod k)(b\mod k) = 584299\mod k$$ ...
0
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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?
1
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1answer
30 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 ...
-1
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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
43 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 ...
1
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1answer
60 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
269 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
74 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
23 views

Factoring terms-laws of exponents

Given the exponent terms $x^m+x^{2m}$, how would it look like if we factor out $x^m$?
2
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1answer
40 views

Factoring algebraic expressions of three variables

I want to factor $$bc^2+ab^2+a^2c-b^2c-ac^2-a^2b$$ Using Wolfram, I know it's factored into $$-(a-b)(a-c)(b-c) = (b-a)(a-c)(b-c)$$ However, I don't think I ever got taught how to simplify such ...
1
vote
1answer
54 views

Integer factorization with sieving

I am trying to solve the Integer Factorization problem using the sieving method, and I was wonder if there been a study in this area and if there more on this topic that I can read? Note, I am not ...
0
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2answers
152 views

what numbers multiply to 1 but add to negative 4

I have math hw on writing quadratic equations. You have to write them based on the parabola given in vertex form standard form and intercept/factored form. For the intercept form one step is to find a ...
1
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0answers
26 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
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1answer
54 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
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3answers
120 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
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2answers
21 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
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2answers
44 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
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0answers
34 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 ...
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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
110 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
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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
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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
37 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: ...
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3answers
37 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?
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3answers
76 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 ...
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4answers
75 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
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2answers
53 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 ...
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0answers
15 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 ...
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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
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1answer
84 views

Factorizable huge semiprime

I'm trying to understand how the number decimal The correct decimal number is: ...
-4
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2answers
98 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
50 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
38 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
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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
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1answer
48 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 ...
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0answers
35 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
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2answers
114 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
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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 ...
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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
174 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
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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 ...
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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
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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 $5 (y+3) (y-3)$