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

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0
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1answer
26 views

Integer factorization simplification

I found a small improvement to the brute force algorithm for the Integer Factorization. Please tell me if there is a point to investigate it more or there are better similar ideas. I found that if ...
0
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1answer
35 views

Factoring polynomial $x^3−2x^2−4x−8$ that fails Bezout's identity test

I usually factor 3rd degree polynomial in two steps. First, I find all the divisors of the last, coefficient-free part of the polynomial (in this case that's 8) and try (applying Bezout's identity) to ...
0
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0answers
25 views

When $\sqrt{(bce + ae + 1)^2 - 4bed} $ is integer?

While been working on my factoring algorithm I came to this: $$\sqrt{(bce + ae + 1)^2 - 4bed} = x$$ $a,b,c,d$ are known positive integers $e$ positive prime integer How can I find all the $e$ ...
4
votes
2answers
101 views

How do I simplify this expression about factorization?

I am trying to simplify this $$\frac{9x^2 - x^4} {x^2 - 6x +9}$$ The solution is $$\frac{-x^2(x +3)}{x-3} = \frac{-x^3 - 3x^2}{x-3} $$ I have done $$\frac{x^2(9-x^2)}{(x-3)(x-3)} = ...
1
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3answers
28 views

What is $c$ in $\left\lfloor\frac{a}{bc}\right\rfloor=d$

As part of my attempts to solve integer factorization problem. I came to this equation: $$\left\lfloor\frac{a}{bc}\right\rfloor=d$$ $a,b,c,d$ are positive integer values $\frac{a}{b}$ is an integer ...
1
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3answers
36 views

Can every polynomial be factored into constant and linear complex factors?

That is, can any polynomial, $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1+a_0$, be expressed $b_0\left(x + b_1\right)\left(x + b_2\right)\ldots \left(x + b_n\right)$ where $b_i \in \mathbb{C}$?
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0answers
33 views

When $\frac{-1436+y\pm \sqrt{(1436-y)^2 - 4\cdot480547}}{-2}$ is integer

While working on my algorithm I came to this problem: $$x_{1,2} = \frac{-b-a+y \pm \sqrt{(b + a-y)^2 - 4c}}{-2}$$ $$x_1 = \frac{c}{x_2}$$ $a,b,c$ are positive known integers $y$ is positive integer ...
0
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2answers
16 views

Factors of polynomial not passing the Bezout's identity test

When factoring $x^3 - 2x^2 - 4x - 8$ the result you get is $(x-2)(x^2 - 4)$ or $(x-2)^2 (x+2)$ , meaning that the mentioned polynomial is divisible by each of these factors. When using the Bezout's ...
0
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2answers
32 views

Why $(10^ab+c)^{4d+1}-c \mid 10$?

I came across the following equation: $$x=(10^ab+c)^{4d+1}-c$$ Why is $x$ a multiple of $10$ for any natural number values for $a$, $b$, $c$ and $d$? The only progress I made was that $a$ could be ...
0
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2answers
43 views

Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$

I have to show that if $d > 2$ is a prime number, then $2$ is not prime in $\mathbb Z[\sqrt{-d}]$. The case when $d$ is of the form $4k+1$ for integer $k$ is quite easy: $2\mid ...
0
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2answers
19 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
5
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0answers
165 views
+50

Integer Factorization: Possible progress

I build an algorithm for solving Integer Factorization problem when the number to factor is selected by multiplication of two prime numbers with the same bit count. So far I only been able to factor ...
1
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0answers
17 views

Should this be rephrased into saying no common factors but 1?

This is from Discrete Mathematics and its Applications For the phrase "a and b have no common factors" , does that actually mean a and b have no common factors other than 1? I feel like this would ...
1
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2answers
21 views

Using the distributive property to factor $(5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x)$

I can't seem to understand the distributive property. Take this: $$ 5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x$$ becoming this: $$ 5^x\left(\frac 15 - 1 - 5 +25\right) $$ Help? :D
2
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2answers
121 views

If GCD of a list of numbers is 1, is it a necessary condition that GCD of at least one pair of numbers from the list should be 1?

Suppose our numbers are {2, 6, 3}. GCD (2, 6, 3) = 1, GCD (2, 6) = 2, GCD (6, 3) = 3, but GCD(2, 3) = 1 If GCD(a,b,c) = p, GCD(a,b) = q, GCD(b,c) = r, GCD(c,a) = s, is it possible that p = 1 and (q ...
0
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0answers
32 views

Integer factorization: does $az + by$ helps?

I am trying to find solutions for integer factorization problem. And particularly I am curries in RSA cracking. I came to the next equation: $$az + by = \frac{c}{x}$$ $c$ is the number that I am ...
-2
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0answers
18 views

Factorise the following polynomial

file://localhost/var/folders/0p/frxrkc9d4_z99dy684t4_9100000gn/T/LaTeXiT-2.6.1/latexit-drag.pdf How do you factorise the above?
0
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2answers
56 views

Factorise the following polynomial [closed]

$$x^6+3x^4+4x^2+2$$ How do you factor this polynomial if it has no real solutions?
0
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0answers
17 views

length plus width equals price, factoring?

im trying to grasp what this means as I usually work with areas (L x W = A) or perimeters (2L + 2W = P)... This problems was presented to me by a colleague and i'm just trying to wrap my head around ...
0
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2answers
20 views

How to factor this expression

Could someone explain to me the process of factoring this $-6x(x+1)(x^2+2)^{-5/2} + 2(x^2+2)^{-3/2}$ into this $2(1-2x)(x+2)(x^2+2)^{-5/2}$?
1
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3answers
78 views

factorizing without removing brackets [closed]

\begin{equation} 2y(y+z)-(x+y)(x+z) \end{equation} Factorize this without removing bracket at any stage I dont know what method to use here Should I factor the brackets separately
-1
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1answer
23 views

. Find the partial fraction decomposition of the following rational function.

Find the PFD of the folowing: $$\frac{x^6-x^5+48x^3-53x^2+99x+48}{x^5-2x^4+2x^3-4x^2+x-2}$$ Initially I used long division and got $$x+1+\frac{50(x^3-x^2+2x+1)}{x^5-2x^4+2x^3-4x^2+x-2}$$ I then used ...
1
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1answer
39 views

proof factoring of $x^n-1$

I have a question about the proof of the theorem stating that $$x^n-1 = \prod_{i \in I} m^{(i)}(x)$$ is a factorisation in irreducible factors of $x^n-1$ over a field $\mathbb{F}_q$. Here $m^{(i)}$ is ...
-1
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0answers
26 views

Prove that 2 is reducible over $\mathbb{Z}[11]$ [closed]

Prove that 2 is reducible over $\mathbb{Z}[11]$.
0
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0answers
57 views

Is reducing factoring of integers to finding a polynomial which takes a perfect square value useful?

Below we only consider numbers $N=pq$, where $p$,$q$ are primes of the form $ (6j+1)$. It's easy to show that $N + 9n^2 = d^2$. $9k^2$ is related to $(p-q)$ and $d^2$ is related to $(p+q)$. the $n$ in ...
1
vote
1answer
60 views

Integer factorization: Single solution for integer equation

While working on my integer factorization project, I came to this: $(A + CX)(B + CY) = D$ $X,Y,A,B,C,D$ Are integer numbers $A,B,C,D > 0$ $X,Y >= 0$ $A,B,X,Y < C < D$ If $X=Y$ than $Y ...
0
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2answers
24 views

Why can we write a uncontinual function continual?

Let's consider $$f(x) = \frac{(x-1)(x-2)(x-3)}{x^2-3x+2}$$ with definition $D_{f} = \mathbb{R} \setminus \lbrace 1, 2 \rbrace$. This means we are allowed to set $x$ to every value of $\mathbb{R}$ ...
2
votes
2answers
34 views

How to factorize polynomial in GF(2)?

I want to know how to factorize polynomials in $\ GF(2) $ without a calculator in a product of irreducible factors. For example, in my exercises, I must factorize $\ p(x) = (x^7 - 1)$ The response is ...
1
vote
3answers
75 views

How do I solve this, first I have to factor $ 2x\over x-1$ + $ 3x +1\over x-1$ - $ 1 + 9x + 2x^2\over x^2-1$?

I am doing calculus exercises but I'm in trouble with this $$\frac{ 2x}{x-1} + \frac{3x +1}{ x-1} - \frac{1 + 9x + 2x^2}{x^2-1}$$ the solution is The only advance that I have done is factor $ ...
0
votes
2answers
71 views

Is it possible to factor $4x^2-3$?

Is it possible to factor $4x^2-3?$ I honestly can't thing of any way to factor this, but I wanted to be sure it was, in fact, impossible to factor. EDIT: Thanks for the help in the comments. ...
0
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2answers
56 views

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomials over Q for any natural number n.

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomial over $\mathbb{Q}$ for any natural number n. This is an extended version of my previous question here. I know if $n=4$, ...
2
votes
0answers
73 views

Show that $x^4+n(2-n)x^2+n^2$ reducible over Q for any natural number n.

Show that $$x^4+n(2-n)x^2+n^2$$ reducible over Q for any natural number n. Here is what I did $$x^4+n(2-n)x^2+n^2=x^4+2nx^2-n^2x^2+n^2=x^4+2nx^2+n^2-n^2x^2$$ $$=(x^2+n)^2-(nx)^2$$ ...
2
votes
3answers
32 views

How do I solve this rational expression?

I'm stuck on this rational expression. I factored and simplified, by what do I do next? Should I divide x/2x and 8/4? I posted my work below. Thank you!
2
votes
1answer
46 views

Factoring algebraic polynomials that are neither cyclic nor symmetric, and don't have obvious zeros

In a set of $40$ problems, I was not able to factor these three polynomials. (The polynomials are neither cyclic nor symmetric, and don't have obvious zeros.) Any help is appreciated: 1) $x^3+2 ...
1
vote
1answer
40 views

Factoring a fourth degree polynomial with missing degrees

Can someone explain how to factor this polynomial: $$x^4 - 4x^2 + 9x + 4 = 0.$$ The answer should be this: $$(x^2 - 3x + 4)(x^2 + 3x + 1) = 0,$$ but I can't find a way to figure it out on my own. ...
4
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0answers
60 views

Reducing multivariate rational fractions to lowest terms

I wish to simplify multivariate rational fractions to a canonical form. Thanks to some very helpful mathematically inclined people who verified that my understanding of Wikipedia was correct, I'm now ...
1
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0answers
39 views

factor theorem for multivariables

My understanding of the remainder theorem for one variable is that for $$f(x)=(x-a)q(x)+r(x)$$$\qquad$ if $x=a\implies f(a)=0\times q(a)+r(a)$ so $f(a)=r(a)$ Is this correct for a multivariate ...
-1
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2answers
47 views

finding the limit right answer wrong sign

I have the following equation Given $$\lim_{x\to 2}\frac{2-x}{x^2-4}$$ using substitution we know that both the top and the bottom solve to $\frac{0}{0}$ this means that (per my text book and this ...
2
votes
3answers
102 views

Better way of factorising $x^2-a^2+x+a$

I am currently at the subject factorisation and I have the following problem: Fully factorize: $$ {x^2}-{a^2}+x+a $$ What I did was the following: Create a common factor: $$ x({1^2}+1)-a(1^2-1) $$ ...
0
votes
1answer
25 views

Complexity of factoring integers by trial division

Ok, I have a real problem with understand the complexity of this algorithm: set k=n; while k!=1{ while True{ d=k/i; if type(d)=integer{ i is a factor; break; } } } So we go through the internal while ...
1
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2answers
50 views

Factorise a matrix using the factor theorem

Can someone check this please? $$ \begin{vmatrix} x&y&z\\ x^2&y^2&z^2\\ x^3&y^3&z^3\\ \end{vmatrix}$$ $$C_2=C_2-C_1\implies\quad \begin{vmatrix} x&y-x&z\\ ...
1
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2answers
60 views

Factoring the polynomial $ P(X)= X^6-X^5-X+1 $ over real and complex numbers

I'm trying to solve this exercise, I'm starting with polynomials and I'm wondering how to answer the 2 and 3 with the help of 1). We have the polynomial $$ P(X)= X^6-X^5-X+1 $$ Prove that 1 is a ...
0
votes
1answer
31 views

Factorise expressions of the form aⁿ ± a⁻ⁿ

In order to simplify the expression $\frac{a^{3x}-a^{-3x}}{a^{x}-a^{-x}}$, the numerator can be factorised into $\left(a^{x}-a^{-x}\right)\left(a^{2x}+1+a^{-2x}\right)$. Similarly, ...
2
votes
2answers
123 views

prime factors of number with a particular form

I try to factorize this huge number $2^{(3^{(5^7)})} +7^{(5^{(3^2)})}$ .but i have no idea,the only thing i know is that it's not divisible by 7 and 11. can you help me find some prime factors of ...
1
vote
4answers
70 views

how to factor this cubic polynomial

Let $f(t)=36t^3-19t+5$ be a cubic polynomial. How we can factor $f$ to its roots? Mathematica says that $f(t)=(-1+2 t) (-1+3 t) (5+6 t)$. How?
5
votes
3answers
57 views

Find a and b of $x^3+ax^2+bx−26=0$

I am doing practise papers and one of the questions is: The cubic equation $x^3+ax^2+bx−26=0$ has $3$ positive, distinct, integer roots. Find the values of $a$ and $b$ The mark scheme ...
2
votes
1answer
69 views

Factor $2^{15}-1=32767$ into a product of two smaller positive integers. Is there a method?

I can't think of anything short of dividing it until I find a factor. What could be a practical way of doing it?
20
votes
1answer
162 views

Solve $x^4+3x^3+6x+4=0$… easier way?

So I was playing around with solving polynomials last night and realized that I had no idea how to solve a polynomial with no rational roots, such as $$x^4+3x^3+6x+4=0$$ Using the rational roots test, ...
1
vote
3answers
49 views

Factorizing a polynomial of degree 4 that has complex roots

While working on differential equations with constant coefficients I came across the following auxiliary equation: $r^4 - 4r^3 + 9r^2 - 10r + 6 = 0$. Initially I tried the hit and trial method for ...
1
vote
0answers
53 views

Findind 3 factors for a integer number

As a background I'll explain what I'm trying to achieve and where's from. In Person of Interest, a TV series, one of the characters gets a phone number in the form of area code and phone number, like ...