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Questions tagged [factoring]

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

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Selecting a finite field for polynomial factorization

This is a very specific question about factoring standard polynomials in the field of rationals. I have a module written which does the whole chain of square-free factoring to distinct-degree ...
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4answers
52 views

Factoring $(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)$ and use the result to solve an equation

I managed to prove that $(x+a+b+c)$ is a factor of $$(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)$$ Then I was asked to use the result to solve $$(x+2)(x-3)(x-1)+4=0$$ I know by comparison, $a=2, b=-3, c=-1$, ...
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2answers
41 views

How to compute gcd of two polynomials efficiently

I have two polynomials $A=x^4+x^2+1$ And $B=x^4-x^2-2x-1$ I need to compute the gcd of $A$ and $B$ but when I do the regular Euclidean way I get fractions and it gets confusing, are you somehow able ...
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batch factoring

The table below illustrates a snapshot of results for a potential way to factor particular integers. Each C value denotes a particular "curve" of the method. The index value logged to the respective ...
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0answers
24 views

Matrix Equation AB = C with constraints, where both A and B are unknown.

As the title reads, I have the matrix equation: $$ AB = C $$ With the constraint that all elements in B are greater than $0$ and less than $1$, and the last element of each row is $1$. A and B can be ...
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1answer
42 views

Find the smallest integer n > 1 such that the product of the factors of n is $n^{15}$ [on hold]

Find the smallest integer n > 1 such that the product of the factors of n is $n^{15} $
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2answers
51 views

Solving a polynomial by grouping and factoring - why does this answer have $\pm3i$?

I am asked to solve for x in the polynomial using factoring and grouping: $5X^3+45X=2X^2+18$ My working: $5X^3-2X^2+45X-18$ $X^2(5X-2)+9(5X-2)$ $(X^2+9)(5X-2)$ So: $X^2+9=0$ $X^2=-9$ $X=i\sqrt{...
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8answers
68 views

Why $a^3 - b^3 = (a – b) (a^2 + ab + b^2)$?

Why $a^3 - b^3$ is equal to $(a–b) (a^2 + ab + b^2)$, How to calculate it. if I multiply $(a-b)(a-b)(a-b)$ then I have got $(a-b)(a^2+b^2)$ and it is not the same as above.
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1answer
17 views

Recognizing a Factoring Pattern (Pt. 2)

I am trying to identify a pattern in the following set of equations; $N_{-1}=1$ $N_{0}=2y$ $N_{1}=2y^2+z$ $N_{2}=2y^3+3yz$ $N_{3}=2y^4+5y^2 z+z^2$ $N_{4}=2y^5+7y^3 z+4yz^2$ $N_{5}=2y^6+9y^4 z+...
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1answer
43 views

Recognising a Factoring Pattern

I am trying to identify a pattern in the following set of equations; $$N_0=y$$ $$N_1=y^2+z$$ $$N_2=y^3+2yz$$ $$N_3=y^4+3y^2z+z^2$$ $$N_4=y^5+4y^3z+3yz^2$$ $$N_5=y^6+5y^4z+6y^2 z^2+z^3$$ Essentially, ...
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2answers
47 views

Sum of factors of odd numbers

Would I be justified in saying that a number $N$, that is the product of the first $k$ odd primes, would have the largest sum of factors than all odd numbers less than $N$? ex. if $k = 4; N = 3 \cdot ...
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1answer
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$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in $ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$

This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...
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3answers
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Factorisation of $\frac{1}{u^2+3u-1}$

So, I need to factor the expression $\frac{1}{u^2+3u-1}$ First I find the roots $x_1=\frac{-3+\sqrt{13}}{2}$ and $x_2=\frac{-3-\sqrt{13}}{2}$ then I have $\frac{1}{(2x+3+\sqrt{13})(2x+3-\sqrt{13})}$ ...
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4answers
36 views

Polynomial factorization in $\mathbb{R}$ and $\mathbb{Z}_{[n]}$

I've the following polynomial: $$ a(x) = x^6 + x^5 + 2x^3 - 3x^2 +x -2 \in \mathbb{K}[x] $$ Set $\mathbb{K} = \mathbb{R}$. A factorization of $a(x)$ is: $$ a(x) = (x^2 + 1)^2(x-2)(x+1) $$ Now set ...
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1answer
74 views

Name for this method of factoring quadratic and are there any textbooks that describe it?

I remember learning this method of factoring quadratics in middle school or high school, but looking for a name or more information on it leads me to dead ends. Given: $ax^2+bx+c=0$ $d*e=a*c$ $d+...
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6answers
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Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$

Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$ By factor theorem we know that $(x-a)$ is a factor of $f(x)$ if $f(a)=0$. In this case, $f(x)=x^n -nx +n-1 \implies f(1)=0$ Hence we conclude that ...
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5answers
120 views

How to factorized this 4th degree polynomial?

I need your help to this polynomial's factorization. Factorize this polynomials which doesn't have roots in Q. $ \ f(x) = x^4 +2x^3-8x^2-6x-1 $ P.S.) Are there any generalized method finidng 4th ...
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0answers
42 views

Factoring Polynomials into Galois Conjugate Linear Factors

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial of degree $n$. Question: Is there a nice way to describe the set of polynomials $f(x)$ (not necessarily monic) that can be factored as $f(x)= \prod_{i = ...
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2answers
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Easy ways of finding small prime divisors

Are there any "relatively simple" ways of seeing whether a number is divisible by a small prime not 3. For example, one simply sums the digits of n modulo 3 and if they sum to zero, then n is ...
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2answers
51 views

Algebra - Factoring Quadratic Equations

I'm in algebra and this problem was under the lesson Factoring to Solve Quadratic Equations. The problem is the following: The product of two consecutive numbers is 14 less than 10 times the ...
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1answer
40 views

Prove $ a^n+b^{n} \geq a^{k}b^{n-k} + a^{n-k}b^{k} $

can someone please help me to prove this following inequality. for $a,b > 0$ and $n,k > 1$ $ a^n+b^{n} \geq a^{k}b^{n-k} + a^{n-k}b^{k} $ Thanks in advance
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4answers
473 views

Further factorisation of a difference of cubes?

We know that a difference of cubes can be factored using $a^3-b^3=(a-b)(a^2+ab+b^2)$. How do we know that the quadratic can't be factored further. For example, $$ 27x^3-(x+3)^3=((3x-(x+3))(9x^2+3x(x+...
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1answer
30 views

How to factor this? [closed]

How do factor this expression: $-3(-4)^{n-1} + 4(-4)^{n-2}$? The answer was: $(-4)^{n-2} [(-3)(-4) + 4]$. But, I don't know how to get to the answer. Please show the work.
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determinants of factorised matrices [duplicate]

Say we have a $3{\times}3$ matrix $M$ and we factorise to $\frac{1}{x} P$ why do we get $\det M$ = $\frac{1}{x^{3}} \det P$?
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5answers
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How to factorize the polynomial in the ring $\mathbb{Z}_5[x]$

I need to factor the polynomial $x^5+3x^4+x^3+x^2+3$ into the product of the irreducible ones in the ring $\mathbb{Z}_5[x]$. The problem is I don't see any whole roots (I tried every possible divisor ...
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4answers
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How do you apply one prime factorisation to another?

I have a question in my paper, Express 4225 as the product of its prime factors in index notation. That was easy to answer, but the next question is express the square root of 42250000 using prime ...
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1answer
36 views

RSA - Quadratic Sieve - Relation Building - How to choose $b$ in $F(b)$?

On p153 of Hoffstein Pipher & Silverman's book "Intro. to Mathematical Cryptography", the list of numbers: $$F(a),F(a+1),F(a+2),...,F(b)$$ is stated. Where $N$ is number to be factored, $F(T)=...
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2answers
53 views

Help me solve this precalculus algebra expression!

Expression and my attempt at solution: $$\frac{3ab}{c^{-1}}:\left(\frac{b}{c^{-1}}+\frac{a}{c^{-1}}-\frac{a}{b^{-1}}\right)-\frac{(a-1)a^{-1}+(b-1)b^{-1}+(c+1)c^{-1}}{a^{-1}+b^{-1}-c^{-1}}=$$ $$\frac{...
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1answer
33 views

Can someone explain how does step 1 render to step 2?

(1)$(n+1)!−1+(n+1)×(n+1)!$ (2)$=(1+n+1)×(n+1)!−1$ (3)$=(n+2)×(n+1)!−1 $ (4)$=(n+2)!−1$ I understand how step 4 derived from 3, but I am confused on how does step 2 derived from step 1? thank you
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1answer
35 views

Ways of factorising into coprime factors

How many ways can we factorize a number, say $676$, into $2$ coprime factors? I tried this by factorizing $676$, and then counting by hit and trial, which worked just fine. Is there another, more ...
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3answers
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$a*x^2+b*x+c $ with integer a,b,c can always be factored with integer (!) coefficients if its discriminant is not irrational?

Does this assertion really hold? My math teacher said so. I cannot give a counterexample so far but I doubt the assertion.
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Factors of successors of powers

Given some $a < 100$ and $b < 10$, is there a fast way to find the number of factors of $(a^b)+1$? What about $(a^b)+2$ or $(a^b)+3$? If this isn't possible, are there any special cases that are?...
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2answers
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Final steps in the General Number Field Sieve

I try to understand the General Number Field Sieve, based on Michal Case paper "A beginner's guide to the general number field sieve". I was able to reproduce some results from the example, i.e. $$\...
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1answer
54 views

Method of characteristics $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + 2\frac{\partial^2 u}{\partial x \partial y}=0$

I know how to solve problems with form like this (via method of characteristics): $$a(x,y) u_{x}+b(x,y)u_y=c(x,y).$$ But I got this problem: $$ \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}...
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1answer
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RSA - statements re $\phi(n)$, $\lambda(n)$ and $n$, given $n$,$|n|=1024bits$, and $e=65537$

Assume $n=pq$, with $p,q$ primes, $e=65537$, and length of $n$, $|n|=N=1024$ bits = 309 decimal digits. $p,q$ are unknown. I am trying to understand the information sourced from Wikipedia page on RSA ...
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1answer
21 views

Simple question about relative primes in entire rings.

Let $A$ be an entire ring. Let $a,b\in A$. Does The g.c.d. of $a,b$ is a multiplicative unit. $\Rightarrow$ $\langle a,b\rangle=A$ hold? If yes, how can I proof it?
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1answer
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Factorization of $x^p-1$ modulo $p^n$

What are the (monic) divisors of the polynomial $x^p-1$ in the ring $(\mathbb{Z}/p^n\mathbb{Z})[x]$? For $n = 1$, the ring $(\mathbb{Z}/p\mathbb{Z})[x]$ is a UFD, and we have $x^p - 1 = (x-1)^p$. ...
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3answers
145 views

Factorise $1+x^2$

How do I factorise this expression? $$1+x^2$$ An attempt: complete the square $(1-x)(1+x).$ teacher said no. $x(1/x+x)$ again teacher said no. She said is related to solving this $x^2+1=0$. I ...
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1answer
110 views

A Dedekind domain without prime elements

We know examples of non Noetherian Prüfer domains, which do not contain any irreducible elements. On the other hand, a Dedekind domain (not being a field) always contains irreducible elements since ...
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How can I find irreducible polynomials in $Z_{77}[x]$?

I could find all reducible polynomials in $Z_3[x]$. But $3$ is a very small number. I'm interested in finding irreducible polynomials in not very small $N=pq$, where $p,q$ are primes. I don't have ...
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0answers
32 views

Find minimum GCD of a pair of elements in an array

Given an list of elements, I have to find the MINIMUM GCD possible between any two pairs of the array in least time complexity. Example Input list=[7,3,14,9,6] ...
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1answer
43 views

Factoring $9788111$ via Gaussian elimination over $\mathbb F_2$

I am trying to follow page 142 to page 144 of An Introduction to Mathematical Cryptography by Hoffstein, Pipher & Silverman, where they give an example using Gaussian elimination over $\mathbb F_2$...
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1answer
42 views

Writing a Matrix as a sum of squares

Currently I am attempting to write a matrix as a sum of squares. The matrix in question is as follows: $$ \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix}...
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2answers
67 views

A question for the preparation of internationals [closed]

Given the real numbers $a$ and $b$, for which it is true that $$a^3+b^3+3ab=1$$, evaluate $a+b$. I tried working this question out, by factorizing, but I didn't manage to reach a conclusion, which ...
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1answer
32 views

A discrepancy in understanding another formula. [closed]

If I have: $$\phi_{n}(z) = \chi_{n}(A(z)),(z \in \mathbb{C}, |z| = 1) $$ and $$\chi_n(A(z))=z^n+z^{n-1}z^{-1}+\cdots zz^{-n+1}+z^{-n},$$ Why $z^m - z^{-m} = \phi_{m} - \phi_{m-1}$?
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2answers
117 views

Find the positive root of $100^{2}=x^{2}+ \left( \frac{100x}{100+x} \right)^{2}$

I was struggling with this problem: $$100^{2}=x^{2}+ \left( \frac{100x}{100+x} \right)^{2}$$ It came up when i was developing a solution to a geometry problem. I've already checked in Mathematica ...
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3answers
36 views

For what values of $n$ is $10^8 < n! < 10^{12}$?

This question is from the introduction of a handbook on combinatorics. So far, the material covered contains: counting problems solved by using trees, Pascal's triangle,... definition of $n$ ...
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2answers
54 views

Does the degree of a polynomial give the number of roots?

I am aware of the fundamental theorem of algebra, i.e., the degree of a polynomial is the number of roots of the polynomial. For example, $x^2 - 9 = 0$ would have two solutions: $x=3$ and $x=-3$. ...
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1answer
19 views

Find the value of D.

The complex numbers $1+i$ and $1+2i$ are both roots of the equation $x^5-6x^4+Ax^3+Bx^2+Cx+D=0$, where $A, B, C, D \in R$ Find the value of D. My attempt: The given equation will have 5 roots (...
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1answer
47 views

Solving differential equation by factorization

I'm trying to solve the following differential equation: $ (x^{2} -1) y'' - (4x^{2} -3x-5) y' + (4x^{2} -6x -5) y = e^{2x}$ By the method of factorizing operators, so I rewrite the differential ...