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

learn more… | top users | synonyms

3
votes
1answer
78 views

Prove that the number $2^{2^n} + 2^{2^{n - 1}} + 1$ can be expressed as the product of at least $n$ prime factors, not necessarily distinct.

Prove that the number $$2^{2^n} + 2^{2^{n - 1}} + 1$$ can be expressed as the product of at least $n$ prime factors, not necessarily distinct. I have tried out a few smaller numbers of $n$, and I ...
1
vote
1answer
37 views

Need help factoring polynomial expression

I've started reading through a pre-calculus textbook for self-study and came across this problem in the second chapter: $$(x-2)^3-(x-2)^2$$ The final answer is $(x-2)^2(x-3)$ Everywhere I look but I ...
0
votes
2answers
31 views

Program to find Shortest Symbolic Expression

I was solving a physics problem using lots of replacements for some expressions to get a manageable equation. Finally I got my answer, but it looked more like a waterfall than an actual result : ...
1
vote
2answers
46 views

If we do not know a number's factors, what is the algorithm (if there is one) to write it as a difference of two squares?

For example, if we have a number like 29873412895, is there an algorithm that can find it as a difference of two squares? Or must you need the factors of the numbers? And what might be the algorithm? ...
3
votes
1answer
28 views

Degree Of Polynomial Factored Function $g(x) = 0.5x (x+4)^2(2x-3)$

I'm confused about the process of how to find the degree of a polynomial factored function. I'm not sure that in this specific question if there is only two zeros/factors or if the 0.5x also plays ...
3
votes
1answer
61 views

Does $a^2(x) = b^2(x) (1 - x^2)$ imply $a = b = 0$?

I have been working through an exercise and I have found out that $a^2(x) = b^2(x) (1 - x^2)$, where $a(x), b(x) \in \mathbb{R}[x]$. Is this enough to deduce $a = b = 0$? I think so, because ...
0
votes
1answer
36 views

not easily factored quadratic expression how to find its roots [closed]

Could you please help me and explain this issue: If a quadratic equation is not easily factored then its roots can be found using quadratic formula: If $ax^2+bx+c=0$ ($a\ne0$), then the roots are ...
0
votes
0answers
43 views

About Bâle's problem Euler's proof [duplicate]

Can someone explain me with the more details as possible the factorisation in the Euler's proof of $$\sum_{k=1}^{\infty} \frac{1}{k^2}=\frac{\pi^2}{6}$$ In wikipedia they only say that ...
2
votes
6answers
163 views

Given $x^2 + 4x + 6$ as factor of $x^4 + ax^2 + b$, then $a + b$ is [closed]

I got this task two days ago, quite difficult for me, since I have not done applications of Vieta's formulas and Bezout's Theorem for a while. If can someone solve this and add exactly how I am ...
0
votes
1answer
55 views

How can I find the eigenvalues of a non-triangular/non-diagonal matrix?

I'm trying to solve for the eigenvalues of a matrix $$ A = \begin{pmatrix} 1 & 2 & 0 \\ -1 & -1 & 1 \\ 0 & 1 & 1 \\ \end{pmatrix} $$ I tried expanding $\det(A - \lambda I)$, ...
-1
votes
0answers
26 views

Help factorizing an expression with different powers

I was asked to find the length of the curve $y = \frac{x^2}{8} + \frac{1}{4x^2}$ from $x=1$ to $x=2$. First I have to differentiate and then add $1$. $$\frac{dy}{dx} = \frac{x}{4}-\frac{1}{2x^3}$$ ...
0
votes
2answers
18 views

Explanation of a factoring process

I have found in a solved exercise the following: $$x^{-2}-\sqrt{2}x^{-1}+1=(1-e^{j\frac{\pi}{4}}\cdot x^{-1})(1-e^{-j\frac{\pi}{4}}\cdot x^{-1})$$ Can someone explain how did the exponentials appear ...
-4
votes
3answers
31 views

Polynomial Factorizations [closed]

Find the zeros of this function: $f(x) = x^2 - 3x - 40.$ My homework tells me to use a graphing calculator but I don't have one. How can I solve this w/o one? Thank you everyone for your help. I ...
0
votes
0answers
13 views

Factorization of polynomials over the integers without finite fields

Is there any algorithm for factoring polynomials over the integers without "using" finite fields other than the inefficient Kronecker's algorithm?
0
votes
0answers
54 views

Naive Grouping for Factorization

I have a naive grouping method for factorization. I am curious as to its novelty and aspects of the code below that will increase its efficiency. The method is best described with an example: For n ...
6
votes
11answers
140 views

Why is $2^{32}-1=(2+1)(2^2+1)(2^4+1)(2^8+1)(2^{16}+1)?$

Is there an easy explanation for this curious pattern?
1
vote
3answers
60 views

What is the name of this algebraic property?

So this is the formula I'm working with: $$ E^2 = m^2c^4+p^2c^2 $$ From that we can get this: $$ E = \sqrt{m^2c^4+p^2c^2} $$ But I'm wondering what the process between ^ this equation and the one ...
1
vote
2answers
28 views

Finding the upper bound for a number's factors length

Okay, so the title is a bit misleading but I had to keep it short.. Anyhow, if I have a number X what will the length of it's longest two factors be? For example: $X = 10000$ I want $3$ and $3$ ...
-1
votes
2answers
43 views

Factoring $a^{k} - b^{k}$ [duplicate]

I am a bit lost how to factor $a^{k} - b^{k}$. I know it links to the binomial theorem but I can't remember how to do it. Could anyone explain?
0
votes
2answers
47 views

How to factorise an expression which doesn't factorise over the integers?

How can I factorise this expression: $$x^2+58x+100$$ I got an answer and I don't know if it's correct: $(x+\surd 741 -2a)(x-\surd 741 -2a)$.
2
votes
1answer
33 views

Irreducibles and factorization in $\mathbb{Z}[\sqrt{5}i]$

Consider the ring $\mathbb{Z}[\sqrt{5}i]=\{m+n\sqrt{5}i:m,n\in\mathbb{Z}\}$. Show that $21$ has two distinct factorisations into irreducibles in $\mathbb{Z}[\sqrt{5}i]$, which is thus not a UFD. ...
1
vote
0answers
30 views

Factorizing a Polynomial over the Integers

What are the most efficient algorithms to factorize a polynomial over integers, knowing that it has only integer roots? I googled around a lot, but most of the work seems to be around Finite fields. ...
2
votes
0answers
19 views

Factorization by multiplying and representation as difference of two squares

Definition 1.$$R: \mathbb N \to \mathbb N: \ R(n) = \lceil\sqrt{n}\rceil^2-n.$$ This is the distance from $n$ to the smallest square greater or equal to $n$. Definition 2. Let $a$ be as positive ...
2
votes
1answer
52 views

Irreducibility of polynomials $x^{2^{n}}+1$

I would like to if the polynomials of the form $x^{2^{n}}+1$ are irreducible over $\mathbb{Q}$ and in that case if there is some "easy" proof for that (where easy means not using a big theory like ...
1
vote
3answers
40 views

How to prove that a number cannot have factors that are large than the number itself?

For instance, how does the proof for 7 being prime work? We can start from 1 and work up to to 7 and show that 7 has exactly two factors, namely 1 and 7. But, how do we rigorously establish that no ...
2
votes
1answer
33 views

Terminology: name for integer “factor” of a rectangle?

Basic terminology question from a non-mathematician. I started trying to express this with mathematical terms, but decided any potential errors might be more frustrating than the imprecision of ...
1
vote
0answers
24 views

Factoring quaternion into three parts

I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors. What I would like to know is angles ...
4
votes
3answers
218 views

Factors in a cubic equation

I have no idea how to go about this. Any Hint? Suppose that $(x-3)$ is a factor of $$kx^3 - 6x^2 + 2kx - 12.$$ Solve for $k$.
2
votes
2answers
19 views

Stuck in expressing factor into a sum of three perfect squares.

Two part question. (i) Consider the function $f(x)=x^3-6kx+k^3+8$. Show that we can write $f(x)$ as $(x+k+2)P(x)$ where $P(x)$ is a quadratic function. (ii) Show that $2P(x)$ can be written as the ...
2
votes
5answers
69 views

Remainder of $(1+x)^{2015}$ after division with $x^2+x+1$

Remainder of $(1+x)^{2015}$ after division with $x^2+x+1$ Is it correct if I consider the polynomial modulo $5$ $$(1+x)^{2015}=\sum\binom{2015}{n}x^n=1+2015x+2015\cdot1007x^2+\cdots+x^{2015}$$ ...
5
votes
3answers
120 views

Showing that $x^4 -2x^2 +8 x+1$ is irreducible over $\Bbb Q$

I want to show that the polynomial $$f(x)= x^4 -2x^2 +8 x+1$$ is irreducible over $\Bbb Q$. I've proved it by a long method, but I need an easy and short method. I've try to put $x=t+1$, but this ...
3
votes
2answers
28 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 $P_f$: ...
0
votes
3answers
52 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 ...
0
votes
2answers
26 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
-2
votes
4answers
73 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
47 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
0answers
64 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 ...
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 ...
0
votes
1answer
41 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 ...
0
votes
0answers
21 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 = ...
6
votes
1answer
53 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
3answers
96 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?
1
vote
1answer
96 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 ...
0
votes
1answer
34 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 = ...
6
votes
1answer
69 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, ...
1
vote
2answers
63 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
49 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
50 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 ...
1
vote
1answer
42 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 $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...