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

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0answers
17 views

Notation for separating out factors of a number

I have an integer (let's call it $n$), and I want to define it as the product of two values: one that's a pure power of two, and another that is odd. Obviously, these two values are unique for a ...
2
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3answers
46 views

How to factor intricate polynomial.

I would like to know how to factor the following polynomial. \begin{equation} ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c \end{equation} What is the method i should use to factor it? If anyone could help.. ...
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1answer
39 views

polynomial factorization when exponent is not given

How can I factorise this equation, given i already know some of its factors which are: $(a-b)(b-c)(c-a).$ Equation is : $$a^nb^{n-1} + a^{n-1}c^n - a^nc^{n-1} - a^{n-1}b^n - b^{n-1}c^n + b^nc^{n-1}$$ ...
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3answers
23 views

simplification of multiplying rational expressions

I cant' get the right answer to simplifying this expression: $$\frac{x^2+x-6}{x} \cdot \frac{x^2-3x}{x^2-9}$$ The answer I have is ${x - 2}.$ I can simplify it to $$\frac{x^2-6}{x} \cdot ...
0
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1answer
48 views

In Maple, how can I partially factor a lengthy symbolic expression (23 terms in 6 variables)?

I need to show that the following expression, $$a^3b-a^3c+a^3z+a^3x+a^3y-a^2bx+a^2by+a^2cx-a^2cy-a^2zx+a^2zy-a^2x^2+a^2y^2-abcz-abcx-aczx-acx^2+b^2c^2+2bc^2x+c^2x^2-b^2c-2bcx-cx^2,$$ is positive ...
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1answer
30 views

Factoring trinomials. [closed]

A student factored $m^2 + 12mn + 144n^2$ as shown. I know that since $m^2$ squares = $m^4$ and $144n^2$ squared = $12n$, the first and third terms of the trinomial are perfect squares. This means ...
2
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5answers
83 views

The expression $(1+q)(1+q^2)(1+q^4)(1+q^8)(1+q^{16})(1+q^{32})(1+q^{64})$ where $q\ne 1$, equals

The expression $(1+q)(1+q^2)(1+q^4)(1+q^8)(1+q^{16})(1+q^{32})(1+q^{64})$ where $q\ne 1$, equals (A) $\frac{1-q^{128}}{1-q}$ (B) $\frac{1-q^{64}}{1-q}$ (C) $\frac{1-q^{2^{1+2+\dots +6}}}{1-q}$ ...
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1answer
80 views

Factorizing a cubic polynomial

This is the result of determinant evaluation: $$p(x) = (x-3)((x-1)(x-2)-1)+1$$ How can I factor this polynomial?
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4answers
89 views

Factorizing Faster

As one of the steps in a calculation, I had to factorize the following quadratic polynomial: $$12x^2 - 11x - 15$$ Using guess and check, and after roughly 10 minutes, I came to the following ...
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2answers
26 views

Methods for verifying correct factorisation of polynomials

In an attempt to factor using a GCF, Mia wrote $8x^2 + 4x = 4x(2x – 0)$, which is not correct. a. Explain how Mia could check her work. b. What error did Mia make? She didn't factor using the GFC ...
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2answers
16 views

Steps to Simplify

I am struggling to see how the following problem is simplified. Can someone include any steps that may have been skipped? Original Equation= $\frac{T(p-b)}{(p-b+q-a)}$ Simplified Equation= ...
3
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3answers
39 views

Limit and factorization

I have the following very interesting homework exercise: Let $$f(x)=\frac{x^2-4}{2-x\cdot \sqrt {x+2}+\sqrt{x+2}}$$ Find the following limit, if it exists: $$\lim_{x\to 2}f(x)$$ I understand ...
2
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1answer
44 views

Number of digits in the number $N=(1.6 \times 10^{32})!$

I am trying to find the number of digits in $$N=(1.6 \times 10^{32})!$$ where ! denotes Factorial. I have no idea how to proceed, please help me.
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1answer
43 views

Sum of factors of multiplication of different numbers

Given $N$ numbers $n_i$ such that $\forall i \le N, n_i$ $\le10^9$, is there a method to calculate the sum of divisors of their product? For example, given $\{11,15,17\}$ their product would is ...
1
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1answer
12 views

Factorization Process in a polynomial ring

Reading the book "Field Theory" by S. Roman, in chapter $0$ I found the following problem: Let $F$ be a field and consider the polynomial ring $F[x_1,x_2,\ldots]$ where $x_i^2 = x_{i-1}$. Show that ...
0
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3answers
45 views

Factoring Trick - Adding Up Coefficients

My professor told me this for factoring polynomials: Add up the coefficients and if they equal 0 then the polynomial has root of 1. Add up, but switch the signs of the coefficients with odd ...
2
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2answers
31 views

How to use the factor theorem on $a(b^2-c^2) + b(c^2-a^2) + c(a^2-b^2)$?

I know the factor theorem i.e, Let $P(x)$ be a polynomial of degree greater than or equal to $1$ and $a$ be a real number such that $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$. I have an question ...
0
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1answer
43 views

Time complexity of a simple factoring algorithm?

This has puzzled me for a little. I start off with a list of primes that is sufficiently large. For my number $n$, I do trial division of primes in ascending order until I reach a prime that divides ...
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2answers
38 views

Prove (or derive) the de Polignac formula for the prime decomposition of $n!$

I can't seem to find any papers published dedicated to show that the de Polignac formula has a rigorous derivation. From Wikipedia's entry for the formula: Let $n \geq 1$ be an integer. The prime ...
0
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1answer
35 views

Ratios between Factorial numbers and the sum of their factors

Let a factorial number be called $f!$. Let the sum of its factors be called $S(f!)$. Let the ratio between the two be “r”, such that $r=\frac{S}{f!}$. It is conjectured that: ...
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3answers
52 views

Factor a cubic polynomial

Is there a simple way to find out that, for example, $u^3 - 54u + 108$ is $(u - 6)(u^2 + 6u - 18)$?
0
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1answer
45 views

Newton-Raphson For Integer Factorization

Per my earlier question on Naive Grouping for factorization here, below is the modified Newton-Raphson method (integers only) for the polynomial $N -x^2 - yx - x = 0$. ...
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0answers
33 views

Notation for indexing the factorizations of a number?

Background Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ ...
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3answers
46 views

Finding the Characteristic Equation

For the following matrix I need to find $$\begin{bmatrix}-3 & 2 &1 \\3 & -4 & -3 \\-8 & 8 & 6 \end{bmatrix}$$ a. Characteristic Polynomial of $A$ b. Eigen Values c. Eigen ...
3
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0answers
18 views

For any field $k$ and $n>0$: $X^n-a\text{ reducible }\ \Leftrightarrow\ a=b^p\ \vee\ a=-4b^4$.

I'm stuck on the following exercise: Let $k$ a field and $n$ a positive integer. Prove that $X^n-a\in k[X]$ is reducible if and only if there exist a prime $p$ dividing $n$ and a $b\in k$ such ...
0
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3answers
63 views

Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
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1answer
38 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 ...
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2answers
33 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 : ...
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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
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1answer
33 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
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1answer
66 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 ...
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1answer
39 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
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0answers
44 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
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6answers
165 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
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1answer
59 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 $\det(A - \lambda I)$, ...
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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 ...
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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 ...
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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?
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0answers
81 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 ...
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11answers
141 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?
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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 ...
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2answers
30 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 $3$ ...
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2answers
45 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?
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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: $(x+\surd 741 -2a)(x-\surd 741 -2a)$.
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1answer
34 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. ...
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0answers
34 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
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0answers
23 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
54 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
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3answers
43 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
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1answer
35 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 ...