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

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Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
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5answers
105 views

Derivation of factorization of $a^n-b^n$

How does one prove that: $$a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\dots+a^2b^{n-3}+ab^{n-2}+b^{n-1}\right)$$ Better yet, why is $a^n-b^n$ divisible by $a-b$? I would very much appreciate some ...
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2answers
52 views

Is $\gcd(n, \lfloor\sqrt{n}\rfloor!)$ a solution to the factoring problem?

The factoring problem: Factor $n=pq$ given only $n$ where $p$ and $q$ are primes and $0<p<\sqrt{n}<q<n$ I found that $$\gcd(n, \lfloor\sqrt{n}\rfloor!) = p$$ Would this be considered a ...
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0answers
36 views

On number of different factorizations over integers of a number field

Let $K$ be a finite field extension of the rational numbers and let $\mathcal{O}_K$ denote its ring of integers. If a rational integer $n$ factors into two distinct ways into irreducible elements in ...
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2answers
63 views

Show that $P(X) -X$ divides $P(P(X))-X$

Let $P$ be a polynomial in $R[X]$. Then show that $P(X) -X$ divides $P(P(X))-X$
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1answer
26 views

Why is it the case that the common factors of $x$, $y$ are also common factors of $x + y$

Why is it the case that the common factors of $x$, $y$ are also common factors of $x + y$? For example, $10$, $4$ share factors $2$, $1$. $10 + 4 = 14$ which has factors $2$, $1$ Similarly, $100$, ...
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1answer
48 views

Struggling to prove that if $n$ is a non zero integer, and $m > 0 \mid n$ then $m \leq |n|$

i need to prove that if $n$ is a non zero integer, and $m > 0$ and $m \mid n$ ($m$ divides $n$), then $m \le |n|$. I feel like i can do it by a combination of proof by contradiction and cases (ie ...
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2answers
40 views

Factorize $15a^2b^3 - 5a^3b^2 - 10a^4b$

I'm having problems with the factorization of $15a^2b^3 - 5a^3b^2 - 10a^4b$. I get stuck on step 2. 1: $15a^2b^3 - 5a^3b^2 - 10a^4b$ 2: $5a^2b (3b^2 - ab - 2a^2)$ How do I factorize $3b^2 - ab - ...
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3answers
230 views

Factor $x^4+1$ over $\mathbb{R}$

Factor $x^4+1$ over $\mathbb{R}$ Well, I read this question first wrongly, because the reader is about complex analysis, I did it for $\mathbb{C}$ first. I got. $x^4+1=(x-e^{\pi i/4 })(x-e^{3 ...
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4answers
136 views

Factoring $x^3 - x^2 - x + 1$

I'm familiar with basic forms of polynomial factoring like completing the square or factoring (e.g. finding that $x^2+x-6=(x+3)(x-2)$, but I'm currently working on integration of rational functions by ...
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2answers
87 views

Unique number of numbers multiplied together

I'm sure this has been asked before, but how many unique numbers can be made from multiplying $4$ numbers, each between $1$ and $100$? My guess is the all numbers from $1$ to $100^4$ except those ...
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0answers
29 views

Bounding parameters for Shor's Algorithm

I am working on a program for classical implementation of Shor's algorithm to factor products of small primes, just to better understand how it works. The program will conduct order finding using ...
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2answers
95 views

Factoring $x^n + y^n$ over the integers

Here's what i know (or think i know) about the factoring. For integer $n> 1 $ 1) If $n$ is a positive power of $2$ then it is irreducible. 2) If $n$ is an odd prime then $$x^n + y^n = (x + ...
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2answers
56 views

How to factor cubics having no rational roots

$$-8x^3 +8x -3 = 0$$ I've already tried the possible roots of $\pm 1$ and $3$ using the rational roots test, but none of these help break it down into something more workable. How do I solve this ...
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3answers
37 views

Derivative Confusion

I am confused about something. In derivation we learnt that; a^x = a^x . lna Now the question that comes to mind is what is the difference when we have: a^3 =
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1answer
105 views

Sums and differences of distinct factors

Given $k, n \in \mathbb{N}$, let $\tau_{k}(n)$ denote the $k$th positive factor of $n$ in strictly increasing order. For example, $\tau_{1}(6) = 1; \tau_{2}(6) = 2; \tau_{3}(6) = 3; \tau_{4}(6) = 6$. ...
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2answers
110 views

Solve the equations $z^2 + (2 - 2i)z + 2i = 0 $ by completing the square

I tried solving this thing by completing the square and I always end up with something like this $(z^2 + (2 - 2i)z - 2i) + 2i + 2i = 0 $ and it doesn't seem like to me that you can factor the part in ...
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1answer
37 views

Factoring a matrix out of linear matrix equation

I'm having a bit of trouble following a solution in a textbook, one step in particular. I have the equation $(Z + tV)^{-1}$ where $Z$, $V$ are matrices and $t$ is a scalar. $Z$ is positive definite, ...
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64 views

Probability distribution of the (size of the) smallest prime factor

Related question: Expected smallest prime factor Background: Given a toolbox of factorization algorithms (like trial division, ECM, quadratic sieve, GNFS) and a set of large composite numbers, I'm ...
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38 views

Mixed Signed Factorising

I need hep factoring mixed signed expressions. I know how to factorise but I'm getting really confused on to which side to put the negetive sign on etc. E.G. $1)$ $x^2 - 10x + 16$ My Steps: $1)$ I ...
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1answer
678 views

Factoring using the 'Criss-Cross' method

my teacher taught our class how to factorize using the criss cross method, and I did not understand what she tried to communicate to the class. The equation I am trying to solve as an example is $7x^2 ...
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5answers
1k views

Factoring a hard polynomial

This might seem like a basic question but I want a systematic way to factor the following polynomial: $$n^4+6n^3+11n^2+6n+1.$$ I know the answer but I am having a difficult time factoring this ...
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3answers
640 views

Factoring the quintic polynomial $x^5+4x^3+x^2+4=0$

I am trying to factor $$x^5+4x^3+x^2+4=0$$ I've used Ruffini's rule to get $$(x+1)(x^4-x^3+5x^2-4x+4)=0$$ But I don't know what to do next. The solution is $(x+1) (x^2+4) (x^2-x+1) = 0$. I've ...
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2answers
76 views

What is wrong with this equations?

so our math teacher told us how 2 * 2 = 5 today and we were like :O I thought really hard to disprove this but it seems correct, would someone please tell me how is this possible! proving 2 * 2 = 5 ...
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3answers
181 views

Factoring quartic equation

$$x^4+6x^2+25=0$$ How could I factor it into $(x^2-2x+5)(x^2+2x+5)=0$? I got the result looking into horrible formulas on wikipedia, but I suppose there's a much easier way. Could you help me?
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1answer
49 views

Odd Number of Factors in Prime Factorization during Euler Function evaluation

I am trying to calculating $\phi(150)$ Prime factorization of $150 = 2\cdot3\cdot5^2$ So i can find $$\phi(150)= 150\cdot(1-\frac{1}{2})\cdot(1-\frac{1}{3})\cdot(1-\frac{1}{5})$$ So in case say on ...
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1answer
52 views

What is the solution? Factoring and computing the equation.

If you will be gracious enough to answer, the equation is currently: $$10^x + 15^{x-1}= 20,$$ What is the value of $2x^2$? Please list all steps, if you don't mind. To follow up, what is the name of ...
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66 views

Factorization of a degree three polynomial

So I was doing some Vector Calculus homework and was working with Lagrange Multipliers, but then I came across a polynomial that I either forgot how to factor or never learned. I plugged it into ...
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2answers
55 views

If I remove the premise $a\neq b$ in this question, will the statement still be true?

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
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1answer
50 views

Simplifying expressions - factoring or expanding?

The term "simplify" has always confused me. What does "simplifying" mean? More specifically, which is more "simple" - a fully factored expression or a fully expanded one?
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27 views

Proper factors and subsets of integral domains

We want to prove that if $R$ is an integral domain (with identity element $1_R$), then $a$ is a proper factor of $b$ (a proper factor meaning, there exists $c$ in $R$ such that $b = ac$, and $c$ is ...
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1answer
25 views

Factoring polynomial with two variables

Factoring: $$f_{n+1}\geq r^{n-2}+r^{n-3}$$ Factoring out a common term of $r^{n-3}$ from line (2), we get: $$f_{n+1}\geq r^{n-3}(r+1)$$ I don't understand how to factor out r^(n-3) to get ...
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2answers
49 views

How to factor $ x^3 − 2ax −a $

I have: $$ x^3 -2k^2x - k^2 = 0 $$ Now, I want to factor it so that I can easily solve for $x$, as you can see I cannot directy use quadratic formula. AFAIK, I am expecting 3 terms here like ...
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2answers
51 views

Factor 9 terms with 3 variables into 4 expression

I just got the determinant from a 4x4 matrix and the simplified version is below. $$ det(M) = \begin{vmatrix} 2k-mw^2 & -k & 0 & 0 \\ -k & 2k-mw^2 & -k & 0 \\ 0 & -k ...
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1answer
63 views

Factorise: $a^4-b^4+c^4-d^4-2(a^2c^2-b^2d^2)+4ac(b^2+d^2)-4bd(a^2+c^2)$

Factorise: $a^4-b^4+c^4-d^4-2(a^2c^2-b^2d^2)+4ac(b^2+d^2)-4bd(a^2+c^2)$. My working: $(a^4-2a^2c^2+c^4)-(b^4-2b^2d^2+d^4)+4ac(b^2+d^2)-4bd(a^2+c^2)$ ...
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2answers
67 views

How do I use factoring to calculate this limit?

For the life of me, I cannot figure out how to factor this to solve the limit. It's probably very simple, but I need help. $$\lim_{t\to a}\frac{t^5-a^5}{t^3-a^3}$$ If you can't see the image for some ...
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1answer
50 views

Complexity of factoring non-squarefree numbers

Consider the two numbers $N_1=p_1\cdot p_2$ and $N_2=p_1^2\cdot p_2$, where $p_1$ and $p_2$ are primes. Is there any factoring algorithm that can factor $N_2$ faster than the asymptotically fastest ...
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131 views

Solve $t^4+4 t^3+6 t^2+4 t-32 t^{1/4}+1 = -16 $

I'm trying to solve the following equation: $$(t+1)^4 - 32 t^{\frac{1}{4}}=-16 $$ where t $\geq 0$, which is equivalent to $$t^4+4 t^3+6 t^2+4 t-32 t^{\frac{1}{4}}+1 = -16 $$ Wolfram Alpha tells that ...
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3answers
57 views

Factoring a polynomial in a field into irreducible

Factor $x^3 + 2x + 3$ into irreducible polynomials in $\mathbb{Z} _5 [x]$ This polynomial has 2 zeros mod 5: x = 2 and x = 4. But these only give me a 2 degree polynomial $x^2 - 4$ and I don't know ...
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0answers
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Which factors determine whether a set of variables are suitable for factor analysis?

Which factors determine whether a set of variables are suitable for factor analysis? I am looking as much for an explanation of the question as a tentative answer to it. So grateful for any help on ...
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2answers
62 views

Factorization of a polynomial

I need to find the roots of this polynomial $$2x^2-x^4-x=0.$$ I know that it is necessary the factorization to obtain $$-x(x-1)(x^2+x-1)=0.$$ I asked to factorize my polynomial to Mathematica. The ...
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2answers
146 views

Prime factorization: easiest way?

For prime factorization, is there another way of doing it, distinct from dividing the number by a series of primes (starting by the smallest)? Couldn't we also pick the same series of primes and ...
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1answer
22 views

Factoring difference of squares?

So I have a quick question on factoring when something is to a power other than 2 or 3. I've got this problem: $a^4-16$ and I think that I should use the difference of squares so that I would get: ...
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2answers
56 views

How to factor $a^2-2a$

The part of the problem I'm doing has me factoring this: $a^2-2a$ and I'm at a loss on how to factor it. Would I be right in saying: $(a-1)^2$ Okay so I just ran across this part now: $a^4-16$ I'm ...
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0answers
85 views

RSA - How can I calculate $\phi$ from $n$ in small prime numbers?

Hello I would like to learn if there is a way to calculate the $\phi$ from small prime numbers. There is a way even slow like brute-force or something else like a formula to calculate $\phi$. For ...
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4answers
150 views

Determine the largest power of 10 that is a factor of $50!\,$?

How would one find the largest power of 10 which is a factor of $50!\,{}$?
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72 views

What is the line of thinking to get $[(n^2+3n+1)^2-5n(n+1)^2]$ from $(n^4+n^3+n^2+n+1)$?

There is this one little tiny step along the working that I don't quite understand, but I think it is better if I write the whole problem and solution for clarity. Problem: Factorise $5^{1995}-1$ ...
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1answer
66 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
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2answers
82 views

GCD of the already GCD

Say $a$ and $b$ are integers. $\gcd(a,b)$ is then $d$. Now if $a$ equals $dm$ for some integer $m$ and b equals $dn$ for some integer n, how come the gcd of this m and n is always 1?
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4answers
52 views

Recognizing the proper polynomial factorization to solve an indeterminate limit

I had to solve the $\lim_{x \to 3} \frac{x^3-3x^2-x+3}{x^2-x-6}$ that is indeed an indeterminate form ($\frac{0}{0}$). The approach I adopted was to factor the polinomials so that I can deviate from ...