# Tagged Questions

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

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### Basic help with factoring

I am having a small problem recalling how to factor with exponents and roots. For example, I understand $\sqrt{16t^2+4t^4}$=$2t\sqrt{4+t^2}$ But I have issues when it is factoring not with a square ...
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### how to factorize $x^2+10yz-2xz-2xy-3y^2-3z^2$?

How to factorize $$x^2+10yz-2xz-2xy-3y^2-3z^2$$ It is expanded and we should make them into parts and factorize each part individually. the last answer is $$(x+y-3z)(x-3y+z)$$ but how to get it ?
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### how to factorize $(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)$?

how to factorize $(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)$? this is one of my hard questions. I know it is related to $(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc$ but I don't know how to factorize it.
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### Simplifying $\frac{3(a^{1/4}+4)}{2a-32a^{1/2}}$

I have a fraction $\frac{3a^{1/4}+12}{2a-32a^{1/2}}$ which I have factored out into $\frac{3\left(a^{\frac{1}{4}}+4\right)}{2a-32a^{\frac{1}{2}}}$, but checking out W|A I also get that there ought to ...
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### How to factorize $x^4+2x^2-x+2$?

look at this: $$x^4+2x^2-x+2$$ How to factorize it? It should be changed to be in the form of standard factorization formulas.
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### How to factorize $(x-2)^5+x-1$?

This is a difficult problem. How to factorize this? $$(x-2)^5+x-1$$ we can't do any thing now and we should expand it first: $$x^5-10x^4+40x^3-80x^2+81x-33$$ but I can't factorize it.
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### On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

(Note: This has been cross-posted to MO.) A positive integer $N$ is said to be perfect if $\sigma(N) = 2N$, where $\sigma(x)$ is the sum of the divisors of $x$. An odd perfect number $N$ is said to ...
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### Asymptotic upper bound on number of solutions to $ab \equiv n \pmod m$

Does anyone know a rough upper bound on the number of solutions to $ab \equiv n \pmod m$ when $n$ and $m$ are given and $a<m$, $b<m$, $n<m$? Specifically, I want to know how the number of ...
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### Why are there four solutions to $x^2-2x-8=0$ in $\mathbb{R}$? Or am I wrong?

It might be a very trivial question to ask but why do we get four different solutions for a quadratic equation using these two methods? $x^2-2x-8=0$ We see that factors are $(x-4)$ and $(x+2)$ so ...
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### In the general number field sieve, do we need to know whether powers of elements in the algebraic factor base divide an element $a+b\theta$?

I'm reading this paper trying to implement the number field sieve. http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.219.2389 Let $\theta$ be the root of some monic ...
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### About factoring trinomials over $\mathbb{Z}$

We were taught in school an algorithm to factor a trinomial of the form $$x^2\pm bx\pm c$$ with $b,c\in \mathbb{Z}^+$. Assuming the best scenario (that the polynomial has both roots in $\mathbb{Z}$),...
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### LU Factorization - Linear Algebra

LU-factorization My solution: Am I on the right path? Or am I completely off?
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### What is a good example of an algorithm that is hard to parallelise?

When I have 10 computers, the factorization of a number doesn't scale along. I am not sure how much faster it would go compared to a single computer, but not 10 times faster like one would expect. ...
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### Irreducibility of polynomials of a certain kind

Let us look at factorization over the integers of polynomials of the form $x^n+n$. For the first few values of $n$ we get $x+1$ - irreducible $x^2+2$ - irreducible $x^3+3$ - irreducible $x^4+4$ - ...
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### The linear factor of the polynomial

Recently I've started to study polynomials, when I found out about the remainder and factor theorems as a way to avoid long polynomial division I couldn't understand the reason for every linear factor ...
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This could be primary school stuff. But I want to ask it. In factoring $x^2+bx+c$ (i.e. $a = 1$ in $ax^2+bx+c$), we find $m$ and $n$ such that $m+n = b$ and $mn=c$. We can reason this well as follows:...
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### Find the $LDL^{T}$ factorization of $A$ when in the range of the positive definite

I am trying to find the $LDL^{T}$ factorization of the following matrix $$A = \begin{bmatrix} 1 & b \\ b & 4 \end{bmatrix}$$ when $b$ is in the range of positive definiteness. I have ...
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### Proof of irreducibility in Z[x] when its reduction mod p has known factors

Problem: Show that if a polynomial $f(x)$ in $\mathbb{Z}[x]$ of degree $n$ has no rational root, but for some prime $p$ its reduction mod $p$ has irreducible factors of degrees $1$ and $n - 1$...
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### Factor $x^{14}+8x^{13}+3$

I need to factor this over the rationals, and there is a hint to use reduction mod3. The reduction is $x^{14}+2x^{13}=x^{13}(x+2)$, but I know it has no rational roots (they would have to be $\pm 3$ ...
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### 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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### 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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### 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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### 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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### 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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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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### 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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### 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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### 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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### 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.
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 ...
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 ...