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

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2answers
42 views

Factor the polynomial $x^4 + 2x − 4$ in $\mathbb{Z}_5[x]$.

I'm confused as to how this is different from factoring in the reals? Would I start this by writing $x^4+2x-4 \equiv 0 \pmod 5$? What changes?
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0answers
51 views

How does the factor command on the TI-89 works?

So to put my question in context, I am working on the following problem. Let $N=1291233941$. Eve's magic box tells her the following three encryption/decryption pairs for $N$: $$(1103927639, ...
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1answer
52 views

Examples of prime ideals?

Could anyone provide me with very simple examples of prime ideals (that is,principle ideals in the ring of integers which are generated by a prime), explaining me the way they are generated? The ...
0
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1answer
22 views

How many pickups $K$ should I do to have a $p$% of probability of picking up a divisor of $n$ (if exists) in the interval $[2..\lfloor n/2\rfloor]$?

I am trying to understand if it makes sense an algorithm to decide if a given number $n$ is possibly prime or not by using the divisor function bound defined by professor Jeffrey Lagarias as: ...
2
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0answers
29 views

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 ...
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2answers
76 views

Factorization of huge integer

I have to factorize the integer $n = 2^{214313833}-1$. Obviously this is not a prime, because $214313833 = 9623 \cdot 22271$, so $n_1=2^{9623}-1$, $n_2=2^{22271}-1$ are divisor of $n$, though $n\neq ...
2
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2answers
62 views

Methods for Factoring Cubics

I am looking for some advice and tips/help about something. I am in calculus now and have been doing well but I recently realized to a bit of my own embarrassment that I am still not fully comfortable ...
1
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1answer
46 views

how do I factor this $6a^2+ 70ab$?

how to factor $6a^2+ 70ab$ ? I got this: $$6a^2+ 70ab = 2a ( 3a + 35b ) $$ Is the factorization complete ?
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2answers
33 views

I need help with this factorization problem?

How can I factorize this problem: $1-8xy-x^2-16y^2$ I noticed that there are common terms, but how should I proceed ?
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2answers
59 views

Why Can't we Factor Invertible Elements?

I'm currently studying Herstein's Algebra; specifically, UFDs and the abstract notion of factorization. This is perhaps more of an intuitive question than one with a hard answer. We define ...
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3answers
37 views

Find the speed of a jet given the time of travel back and forth

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
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2answers
41 views

Converting from factored to standard form: why is this answer wrong?

Converting the equation $$y=-2(x-2+\sqrt{5})(x-2-\sqrt{5})$$ to standard form seems to give $$-2x^{2\space }+3.528x+6.4171392.$$ My handout tells me that the answer is different. What is wrong here? ...
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0answers
30 views

How do I solve this quadratics problem? [duplicate]

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
2
votes
2answers
28 views

Showing that an element is prime in $\mathbb{Z}$[i]

Let p be a prime integer, and suppose p = a2 + b2 has NO integer solution. The exercise asks that if p = a2 + b2 has no solution, then p is a prime in the set of Gaussian integers $\mathbb{Z}$[i], ...
2
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4answers
67 views

How to factor quadratics $(x^2 + 4x + (-357) = 0)$

I need to find $2$ factors of $-357$, which add up to $4$. Obviously one number is positive and the other is negative. I understand this and I know the factors can be $21$ and $-17$; but, how do I ...
0
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2answers
39 views

Polynomial factorisation for unique factor domain

Suppose $R$ is a UFD and $f \in R[X]$ such that $\deg f > 0$ and $f$ has a root $\alpha \in R$. Show that $f = (X - \alpha) g$ for some $g \in R[X]$. (Suggestion: Write $f = a_0 + a_1 X + \dotsc ...
4
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2answers
50 views

LU Factorization - Linear Algebra

LU-factorization My solution: Am I on the right path? Or am I completely off?
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1answer
48 views

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. ...
0
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1answer
22 views

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 $x^4+4$ - ...
0
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1answer
27 views

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 ...
2
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2answers
64 views

Quadratic Polynomial factorization

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 ...
0
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1answer
19 views

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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2answers
43 views

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 - ...
4
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1answer
129 views

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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0answers
40 views

Factorizing a quartic polynomial over an arbitrary field

The following is a problem from Artin: How might a quartic polynomial $x^4+bx^2+c$ factorize over an arbitrary field F? Explain with reference to the polynomials $f(x)=x^4+4x^2+4$ and ...
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0answers
26 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 ...
1
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1answer
29 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 ...
0
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3answers
102 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 ...
0
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1answer
43 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
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 ...
2
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2answers
142 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
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 $k=4$: $$(a\mod k)(b\mod k) = 584299\mod k$$ ...
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
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 ...
-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
45 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
68 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 ...
0
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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: $$(2^x+2)(2^x−5)$$
0
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3answers
102 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 $\frac ph$ ...
0
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1answer
27 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
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 ...
1
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1answer
58 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
355 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] ...
0
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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 ...
4
votes
3answers
133 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
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
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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. ($x^5-1$) = ($x-1$) * ...
0
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0answers
37 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
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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 ...