Questions tagged [factoring]
For questions about finding factors of e.g. integers or polynomials
0
votes
0answers
13 views
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 ...
5
votes
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, $a=2, b=-3, c=-1$, ...
1
vote
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 ...
-1
votes
0answers
24 views
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 ...
0
votes
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 ...
-2
votes
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} $
0
votes
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{...
0
votes
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.
0
votes
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+...
0
votes
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, ...
0
votes
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 ...
0
votes
1answer
21 views
$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ 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 ...
1
vote
3answers
40 views
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 $\frac{1}{(2x+3+\sqrt{13})(2x+3-\sqrt{13})}$ ...
0
votes
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 ...
2
votes
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+...
2
votes
6answers
82 views
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 ...
3
votes
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 ...
1
vote
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 = ...
1
vote
2answers
52 views
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 ...
0
votes
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 ...
0
votes
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
6
votes
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,
$$
27x^3-(x+3)^3=((3x-(x+3))(9x^2+3x(x+...
-1
votes
1answer
30 views
How to factor this? [closed]
How do factor this expression:
$-3(-4)^{n-1} + 4(-4)^{n-2}$?
The answer was:
$(-4)^{n-2} [(-3)(-4) + 4]$.
But, I don't know how to get to the answer.
Please show the work.
0
votes
0answers
14 views
determinants of factorised matrices [duplicate]
Say we have a $3{\times}3$ matrix $M$ and we factorise to $\frac{1}{x} P$ why do we get $\det M$ = $\frac{1}{x^{3}} \det P$?
1
vote
5answers
68 views
How to factorize the polynomial in the ring $\mathbb{Z}_5[x]$
I need to factor the polynomial $x^5+3x^4+x^3+x^2+3$ into the product of the irreducible ones in the ring $\mathbb{Z}_5[x]$.
The problem is I don't see any whole roots (I tried every possible divisor ...
-1
votes
4answers
36 views
How do you apply one prime factorisation to another?
I have a question in my paper, Express 4225 as the product of its prime factors in index notation. That was easy to answer, but the next question is express the square root of 42250000 using prime ...
2
votes
1answer
36 views
RSA - Quadratic Sieve - Relation Building - How to choose $b$ in $F(b)$?
On p153 of Hoffstein Pipher & Silverman's book "Intro. to Mathematical Cryptography", the list of numbers:
$$F(a),F(a+1),F(a+2),...,F(b)$$
is stated. Where $N$ is number to be factored, $F(T)=...
0
votes
2answers
53 views
Help me solve this precalculus algebra expression!
Expression and my attempt at solution:
$$\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{...
1
vote
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
0
votes
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 ...
0
votes
3answers
36 views
$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.
0
votes
0answers
24 views
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?...
1
vote
2answers
46 views
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. $$\...
1
vote
1answer
54 views
Method of characteristics $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + 2\frac{\partial^2 u}{\partial x \partial y}=0$
I know how to solve problems with form like this (via method of characteristics):
$$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}...
2
votes
1answer
59 views
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 ...
0
votes
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?
1
vote
1answer
44 views
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 $x^p - 1 = (x-1)^p$.
...
-1
votes
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 ...
8
votes
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 ...
0
votes
0answers
29 views
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 ...
2
votes
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]
...
0
votes
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 $\mathbb F_2$...
0
votes
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:
$$
\begin{bmatrix}
2 & -1 & -1 \\
-1 & 2 & -1 \\
-1 & -1 & 2
\end{bmatrix}...
1
vote
2answers
67 views
A question for the preparation of internationals [closed]
Given the real numbers $a$ and $b$, for which it is true that $$a^3+b^3+3ab=1$$, evaluate $a+b$.
I tried working this question out, by factorizing, but I didn't manage to reach a conclusion, which ...
0
votes
1answer
32 views
A discrepancy in understanding another formula. [closed]
If I have:
$$\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},$$
Why $z^m - z^{-m} = \phi_{m} - \phi_{m-1}$?
4
votes
2answers
117 views
Find the positive root of $100^{2}=x^{2}+ \left( \frac{100x}{100+x} \right)^{2}$
I was struggling with this problem:
$$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 ...
1
vote
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 $n$ ...
1
vote
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 $x=-3$. ...
1
vote
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 (...
1
vote
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 ...
