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

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4
votes
1answer
78 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
30 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
158 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
900 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 ...
0
votes
0answers
27 views

Name for this possible mathematical structure? [on hold]

I'm thinking of a latent variable as an nth (each representing a variable) dimensional object - it can either be for a correlation matrix, or for a factor structure in factor analysis - that's ...
1
vote
1answer
25 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
25 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
24 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 } ...
-1
votes
0answers
24 views

Minor help in the factorisation techniques used in lec notes

If anyone could help me see how is turned into Thanks in advance!
3
votes
3answers
64 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
30 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
40 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 ...
1
vote
3answers
63 views

Multiplication partitioning into k distinct elements

Let's say I have a list with the prime factors of a number $n$ and their corresponding exponents. Is there a formula to calculate how many multiplications with $k$ distinct factors of $n$ are ...
0
votes
1answer
21 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 $x^4+4$ - ...
-2
votes
1answer
36 views

Simple factorising $1/x$ [closed]

$y = x - 1/x$ I want to get x on it's own, I'm lost as to how to factorise 1/x. Thanks for any help.
4
votes
4answers
55 views

Find all integers n such that the quadratic $5x^2 + nx – 13$ can be expressed as the product of two linear factors with integer coefficients.

I am unsure of how to approach this problem. I have thought about using the Rational root theorem, but I am unsure if this answers the question being asked. Using the theorem, I get $\frac{p}{q} = ...
1
vote
2answers
802 views

Fast way to solve a system of linear equations from Givens QR decomposition

I have this system of linear equations: $$ A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix} $$ $$ b= \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix} $$ I ...
10
votes
1answer
238 views

Is this olympiad-like question about remainders an open problem?

Suppose that we are given two positive integers $x$ and $y$ such that $$x \mod p \leqslant y \mod p$$ for each prime number $p$. (Here, $x \mod p,\; y \mod p$ stand for the least non-negative ...
0
votes
2answers
54 views

Factoring a 5 term polynomial

I am struggling to factor $n^4 + 4n^3 + 8n^2 + 8n +4$. I have tried grouping the terms a couple of times, but got nowhere. What am I missing?
1
vote
0answers
38 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
0answers
31 views

Finding the factors of integer x and its square

What is the the theorem or property that says that $\forall{}x\in{}Z$, the set of all integers, $x^2$ has the same factors as $x$, twice?
4
votes
2answers
310 views

Factorising a cubic equation

Factorise $9x-x^3$ completely. It's simple but I'm never seem to get it right; I've got $(x-1)(-x+9)x$.
0
votes
1answer
39 views

Find two numbers, given their greatest common divisor and least common multiple [closed]

Highest common factor (HCF) of two numbers is $20$. Least common multiple (LCM) of the same two numbers is $420$. Both numbers are higher than $50$. Find the $2$ numbers. I used factorising trees ...
0
votes
5answers
51 views

Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
1
vote
2answers
29 views

Factorizing expressions

I am having trouble solving this problem $81f^2- \dfrac{9}{e^2}$. How do you begin when solving this problem? Do you move $f^2$ by replacing the $9$ and vice versa and does the minus change to plus?
3
votes
2answers
83 views

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.
0
votes
4answers
48 views

How can I factorize this quadratic expression

Going by the exercises of a book I have been factorizing quadratic equations the following way, let's say I have: $$ {x^2 - 7x + 12 = 0} $$ I know that $$ {a \times b = 12 \\ \text{ and } \\ a + b ...
2
votes
2answers
52 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
votes
0answers
30 views
0
votes
2answers
59 views

How should you go about simplying cubic polynomial: $y(x) = x^3+12x^2+21x+10$

Claim: $$y(x) = x^3+12x^2+21x+10$$ Can be factored into $$(x+1)^2(x+10)$$ But what is the quickest way to see this?
0
votes
0answers
22 views

How is pollard rho different from normal factorization?

As far as I understand, pollard rho factorization generates random sequence of numbers, say x1, x2, x3 ... and then checks if x(i) - x(i-1) divides the number. If it does then it is a factor. How is ...
6
votes
4answers
216 views

How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials?

How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials? The link I've given to the theorem uses elaborate wordings using 'rings', ...
15
votes
5answers
3k 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 ...
0
votes
1answer
33 views

Factoring and solving a cubic polynomial

When can we not use synthetic division to solve for a cubic polynomial? For example we can use synthetic division to solve $-t^3 -4t^2 +20t +48$. When I can't use synthetic division what are my other ...
1
vote
1answer
25 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
0
votes
0answers
33 views

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 ...
2
votes
4answers
64 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 ...
2
votes
1answer
37 views

Prove that a matrix is positive definite

I've never really done any factoring with multiple variables in an equation. I tried looking around for examples, but couldn't really find a solid one. Here is the equation I am trying to factor $$ ...
2
votes
3answers
127 views

Using telescoping property to prove difference of powers

Ok so I have started working through Apostol calculus and as you can see I am stuck. The problem is that I can not see the telescoping pattern anywhere for following problem. Prove that $$a^n - b^n ...
2
votes
2answers
24 views

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 ...
5
votes
2answers
47 views

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 ?
0
votes
3answers
47 views

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.
0
votes
4answers
28 views

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 ...
1
vote
3answers
94 views

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.
0
votes
2answers
26 views

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 ...
1
vote
0answers
17 views

Splitting field of a cubic polynomial understanding

The cubic polynomial $f(x) = x^3+px+q\in K[x]$ has 3 roots $a_1,a_2,a_3\in \mathbb C$ Hence, the splitting field extension $L=K(a_1,a_2,a_3)$ $\delta=(a_1-a_2)(a_1-a_3)(a_2-a_3)\in L$ since ...
2
votes
1answer
36 views

How to factor $\frac{27}{125}a^6b^9-\frac{1}{64}c^{12}$

I'm stuck with the following: $\frac{27}{125}a^6b^9-\frac{1}{64}c^{12}$ My idea was/is the following: $\frac{3^3a^6b^9}{5^3}-\frac{c^{12}}{8^2}$ Trouble is that I don't know where to go from ...
0
votes
1answer
37 views

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 ...
1
vote
1answer
33 views

polynomial inverse in rings understanding

This problem and solution are in the book. I need help understanding the solution. Problem: Let u be a root of the polynomial $x^3+3x+3$. In $\mathbb Q(u)$, express $(7-2u+u^2)^{-1}$ in the form $a ...