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

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23 views

Factorising Complex Polynomial with Complex Coefficients

I have tried to factorise the polynomial in question 19 by using the factor theorem to find other factors, however this has been unsuccessful thus far. Seeing as the conjugate root theorem does not ...
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1answer
28 views

Expression factoring question

This is from a simple book explaining differentiation to the uninitiated and I don't understand the factoring. Can anyone help me understand how equation 3 is derived? Thanks Let $y = x^{-2}$ Then ...
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3answers
100 views

Keep factoring and concatenating to get a prime?

Keep factoring and concatenating,starting from $2$ until we get a prime. $$2=2$$ $$22=2*11$$$$22211=7*19*167$$ $$22211719167=?$$ ...and so on (the prime factors are arranged from smaller to larger ...
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1answer
42 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.
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1answer
49 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 ...
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2answers
140 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 ...
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0answers
36 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 $k=4$: $$(a\mod k)(b\mod k) = 584299\mod k$$ ...
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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?
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1answer
34 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 ...
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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
44 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 ...
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1answer
64 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
271 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
95 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
25 views

Factoring terms-laws of exponents

Given the exponent terms $x^m+x^{2m}$, how would it look like if we factor out $x^m$?
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1answer
41 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 ...
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1answer
57 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 ...
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2answers
269 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 ...
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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] ...
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1answer
59 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 ...
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3answers
131 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 ...
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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 $\mathbb{F}_{4}$. ...
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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. ($x^5-1$) = ($x-1$) * ...
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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 ...
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0answers
48 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 ...
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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: ...
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0answers
54 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 ...
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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, ...
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1answer
25 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 ...
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2answers
44 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
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?
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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 ...
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4answers
78 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 ...
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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 ...
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19 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 ...
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1answer
45 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 ...
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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
113 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
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1answer
89 views

Factorizable huge semiprime

I'm trying to understand how the number decimal The correct decimal number is: ...
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2answers
121 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 ...
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3answers
60 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 ...
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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 ...
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4answers
120 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 $$56x^4+18x^2-8$$
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1answer
60 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
37 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 ...
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2answers
119 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 ...
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2answers
74 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
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1answer
193 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.
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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 ...