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

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14
votes
1answer
842 views

Finding smallest and largest prime factor of $\frac{200!}{180!}$

I originally started off by listing all the primes: $p<200$ then trying to calculate the prime factorisation of each (which I realise is a silly thing to do) I believe there must be a simpler way ...
6
votes
4answers
8k views

How many positive integers are factors of a given number?

I've been trying to find / generate a formula for the following problem: Given a number, how many positive integers are factors of this number. In practice, you could simply build a table as such ...
3
votes
2answers
24 views

irreducibility of polynomials made by perturbation from a polynomial

Suppose $f(x)\in\mathbb{Z}[x]$ with $\text{deg}f=2n,n\in\mathbb{Z_+}$ and $f_m(x):=f(x)+ mx^n $ for each integer $m\in\mathbb{Z}$. Let us define a number $P_f$: ...
1
vote
5answers
88 views

Help understanding how to factor completely $x^3-x^2-x+1$

I need someone to help explain the steps to completely factor the problem $x^3-x^2-x+1$. Here is what I have done so far: $x^3-x^2-x+1$ to $x^3-x^2+-1(x+1)$ Since there is a ...
0
votes
3answers
44 views

How do I factorise the following expression?

How do I go from the left expression to the right one? $$ (2-x)^2 \cdot (-2-x) - (-2-x) = - (x+2)(x-3)(x-1) $$ I'm guessing that I have to solve the third degree equation. What are the steps for ...
2
votes
0answers
48 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
1
vote
3answers
42 views

Simplify the Complex Fraction

I am having trouble with the following complex fraction. I have simplified everything for the most part, but I am stuck on the last part and need to know what I have to do next. ...
0
votes
2answers
25 views

Finding value of $m$ such that such that the polynomial is factorized

A polynomial $2x^2+mxy+3y^2-5y-2$ Find the value of $m$ much that $p(x)$ can be factorized into two linear factors
1
vote
1answer
39 views

Finding the factors of integer $x$ and its square

What is the the theorem or property that says that $\forall{}x\in\Bbb Z$, the set of all integers, $x^2$ has the same factors as $x$, twice?
0
votes
1answer
32 views

Minor help in the factorisation techniques used in lec notes

If anyone could help me see how $$ A(\xi,\eta)=a\,\xi_x^2+2\,b\,\xi_x\,\xi_y+c\,\xi_y^2=0 $$ is turned into $$ \frac{1}{a} \left[a\,\xi_x+\left(b-\sqrt{b^2-ac}\right)\xi_y\right] ...
1
vote
0answers
58 views

Factorisation of large polynomials and Galois theory

As I understand it, one of the consequences of Galois theory is that there is no way of expressing the solutions to a general polynomial of degree 5 or higher in terms of radicals. Would a theory that ...
0
votes
3answers
95 views

Is it possible to factor $x^2-6x+7$ over $\mathbb{R}$

None of the online calculators seem to give me an answer. I am trying to find the values for x. How do I do this again? $$x^2-6x = -7$$ Then what?
-2
votes
4answers
72 views

What is the decomposition of $x^4+x^3+x^2+x+1$. [closed]

What is the decomposition of $$x^4+x^3+x^2+x+1.$$ It seems that there is a special way to decompose this, I couldn't find it. It will be great that if you help me about it, thanks. I am asking for ...
0
votes
5answers
44 views

How does this seemingly-trivial simplification work?

In a section on inductive proofs in the book Modelling Computing Systems: Mathematics for Computer Science (Muller, Struth) there is a simplification that is assumed to be trivial, but that I can't ...
1
vote
2answers
28 views

Factoring Gaussian integers

How do I factor the elements $2, 3$ and $5$ of the ring $\mathbb{Z}[i]$? Are they not primes, that is $ 2=2 \times 1$, etc? (an exercise from Vinberg's Algebra).
1
vote
2answers
580 views

Factor $x^6 +5x^3 +8$

I wanted to know, how can I factor $x^6 +5x^3 +8$, I have no idea. Is there any method to know if a polynomial is factored. Just some advice will do. Help appreciated. Thanks.
1
vote
1answer
55 views

Dixon's Factorization Method Modulo Question

Looking at Wikipedia's example for Dixon's Factorization Method, it shows the following. We will try to factor N = 84923 using bound B = 7. Our factor base is then P = {2, 3, 5, 7}. We then search ...
5
votes
1answer
920 views

Factorization of $x^7-1$ into irreducible factors over $GF(4)$

I need to find cyclotomic cosets depending on $n=7$ and $q=4$ and find the factorization of $x^7-1$ into irreducible factors over $GF(4)$. Thanks for any advice.
1
vote
1answer
28 views

Polynomial Rings

I know that an Euclidean ring is a principal ring and this last is a factorial ring. I also know that if $B$ is a field, then $B[x]$ is Euclidean. While, for instance $\mathbb{Z}[x]$ is not a ...
1
vote
1answer
38 views

Methods for factorising polynomials into irreducibles over finite fields

I was given a problem recently, part of whose solution was to factorise $x^{15}+1$ in $\mathbb F_2[x]$. It turns out that the factorisation is $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
0
votes
1answer
40 views

Write ${2-i} \over {-1-5i}$ in factored form?

Trying to help someone study for a final exam. My background: Calculus 1, Statistics. I haven't done this kind of math in ages. Considering $i$ is $\sqrt{-1}$, I was thinking rationalize the ...
6
votes
1answer
45 views

Gauss's lemma: More than a stepping stone on the way to proving $R[x]$ is a UFD when $R$ is?

I'm reviewing my abstract algebra a bit. Currently looking at UFDs. In this context, Gauss's lemma (or part of it, at least) says that the product of two primitive polynomials over a UFD is primitive. ...
0
votes
0answers
16 views

Understanding SQUFOF

SQUFOF (square forme factorization) is an algorithm for factorizing numbers. As I understand, it is an improvement of Fermat's factorization. Fermat factorization assume that a composite number $N = ...
5
votes
1answer
68 views

Theoretical way to prove no positive integer $n$ exists such that $n+3$ and $n^2+3n+3$ are both perfect cubes.

I have to prove that for any positive integer $n$ at least one of $n+3$ and $n^2+3n+3$ is not a perfect cube. Is there a methodical way to solve this problem? I managed to solve it by contradiction, ...
0
votes
1answer
29 views

Residue class ring $\mathbb{Z}[x]$/I and $\mathbb{Z}[x]$/J

$I = \left\lbrace \sum_{i=1}^{n} a_ix^i : n \in \mathbb{N}, a_1, ..., a_n \in \mathbb{Z} \right\rbrace$ beeing an ideal of $\mathbb{Z}[x]$ with polynomials without a constant term and $J = ...
1
vote
2answers
58 views

How do I quickly factorize quadratic equations?

Whenever I have to factorize an equation I usually just look for the common factors and then just work form there. However, I was wondering whether there is a quicker way to get the factorized form. ...
0
votes
5answers
35 views

How to know if equation can be solved by factorising before trying?

So, I have core 1 test tomorrow and there is a lot of solving of quadratic equations without calculator and my weakest point is the time I waste in trying to factorise and equation but then it ends up ...
3
votes
3answers
47 views

What is the remainder $ax+b$ when a cubic polynomial $P(x)$ is divided by $x^2-1$?

If a cubic polynomial $P(x)$ with real coefficients has remainder 3 when divided by $x-1$ and remainder -7 when divided by $x+1$, What is the remainder $ax+b$ when divided by $x^2-1$? I see that ...
3
votes
0answers
83 views

proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
3
votes
1answer
89 views

Reducible polynomials in $\mathbb{Z}[X]$

Let $(a_n)_{n\geq 1}$ be a strictly increasing sequence of integers and $k$ a non-zero integer such that for some $N\in \Bbb Z^+$, the polynomial $$ p_{N}(x)=(X-a_1)(X-a_2)\cdots(X-a_N)+k $$ ...
2
votes
1answer
43 views

Factorising polynomials over $\mathbb{Z}_2$

Is there some fast way to determine whether a polynomial divides another in $\mathbb{Z}_2$? Is there some fast way to factor polynomials in $\mathbb{Z}_2$ into irreducible polynomials? Is there a ...
1
vote
0answers
31 views

Solving “finding value” using factorisation

Even my teacher could not do this.This is a question extracted from the mathematics challenge board created by my friend in school.This is the question Given $x-y=3$,find the value of ...
0
votes
1answer
27 views

Laplace transform (Simple factorization)

The question require me to find the inverse of Laplace transform. In the first line of solution, how does it go from LHS to RHS? Does it simply apply partial fractions?
1
vote
2answers
41 views

Irreducibility and factoring in $\mathbb Z[i], \mathbb Z[\sqrt{-3}]$

In $\mathbb Z[i]$, prove that $5$ is not irreducible. In $\mathbb Z[\sqrt{-3}]$, factor $4$ into irreducibles in two distinct ways. I am completely stumped on how to do this. I really need all ...
1
vote
1answer
27 views

Questions from Dixon Factorization Paper

I read the first page of Asymptotically Fast Factorization of Integers, and have a few questions. Quotes from the paper are formatted as blockquotes (>). Following Legendre, we know that there ...
4
votes
5answers
44 views

Factorising trigonometric functions

In order to factorise $x^2-1$ one way of thinking about it would be to set it equal to zero and solve to get $x=1$ and $x=-1$. You can then write $x^2-1=(x+1)(x-1)$ Can we do the same with ...
0
votes
1answer
15 views

Diagonally dominant matrix for Cholesky?

I have a $10^6 \times 10^6$ dense SPD matrix, which I am called to invert, by using Cholesky factorization. However, I came across this statement: We start with the Cholesky and LU ...
0
votes
1answer
26 views

Simple Fermat Factorization Example

I'm trying to understand this example from wikipedia's Fermat's factorization method. For example, to factor N = 5959, the first try for a is the square root of 5959 rounded up to the next ...
2
votes
1answer
41 views

Find Eigen values of given matrix with nonfactorable polynomial

I'm having trouble finding the Eigen values for this matrix: $$ A =\begin{pmatrix} 0&1&-2 \\ 1&3&0 \\ -2&0&5 \end{pmatrix} $$ I did $A - \lambda I $ and ended up with this ...
4
votes
1answer
82 views

Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$.

I have to find the form of n i.e. whether n is even or odd and whether it is multiple of 2 or 3 such that: $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$ What I tried: $$x^2 + x + 1 = (x + 1)^2 - ...
0
votes
2answers
46 views

Checking irreducibility of a polynomial over a finite field

A part of a coding theory course I am doing includes some questions on irreducible polynomials. I have a question with solution but am worried I have interpreted it incorrectly. So for $\mathbb F_5$ ...
2
votes
2answers
161 views

Is the polynomial $x^3 + 2x^2 + 1$ irreducible in $\mathbb{Z}_{17}[x]$?

Is the polynomial $x^3 + 2x^2 + 1$ irreducible in $\mathbb{Z}_{17}[x]$? It seems this polynomial is reducible. How can I factor this? Thanks!
8
votes
3answers
950 views

A theorem about prime divisors of generalized Fermat numbers?

A theorem of Édouard Lucas related to the Fermat numbers states that : Any prime divisor $p$ of $F_n=2^{2^n}+1$ is of the form $p=k\cdot 2^{n+2}+1$ whenever $n$ is greater than one. Does anyone ...
1
vote
1answer
32 views

Factor polynomial with irrational roots using quadratic equation

If I want to factor the polynomial $x^2 + 3x + 1$, I thought I could use the quadratic formula to find that its roots are $\dfrac{-3\pm\sqrt{5}}{2}$. Then, since those are both negative values, take ...
3
votes
3answers
2k views

About the factors of the product of prime numbers

If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using ...
0
votes
3answers
32 views

Confused regarding the answer of a problem based on locus.

I have a question on locus which goes like this. $A(5,3)$ and $B(3,-2)$ are two fixed points. Find the equation of the locus of $P$, so that the triangle $PAB$ is 9. Now the loci of the point $P$ ...
1
vote
1answer
26 views

Is the ideal $(2,x+1)$ principal in $\mathfrak{o}_K$?

I'm trying to show (although I don't know if the statement is even correct) that the ideal $\mathfrak{p}_2$ is not principal, where $$\mathfrak{p}_2:=(2,x+1) \text{ in the ring of integers } ...
3
votes
3answers
69 views

Factorize Trigonometric Equation

I have a problem with the following trigonometric equation: $$3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0$$ It's from the book Engineering Mathematics 7th edition by Stroud. The book is giving the ...
0
votes
1answer
32 views

Have you seen these integer factorization algorithms before?

I have two algorithms for finding two factors, $p$ & $q$, of a number $N$. The algorithms are (hopefully) obviously related. The pseudo-code for them follows: Algorithm 1 ...
0
votes
3answers
47 views

When can I divide both sides of an equation if one side is zero

Where K is some positive Integer For the following examples: $$ K(a+b)(p+q)=0 $$ $$ Ka^2+Kbx+Kc=0 $$ Can I just divide both sides of the equation by K (dividing into 0 on the right) and effectively ...