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

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1answer
33 views

Is it easier to find $a^2-8c=b^2$ than $a^2-c=b^2$

I found a way to factor numbers if I find: $$a^2-8c=b^2$$ Where $c$ is the number I want to factor Is it easier than searching for the next equation? $$a^2-c=b^2$$
1
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0answers
57 views

How to find integer solutions for $ax^2 + bx + c$

I am working on Integer factorization problem and I came to this: $$a = \frac{1-2b+\sqrt{4b^2 + 4b + 8c + 1}}{4}$$ c is the number that I want to factor $2a -1$, $a+b$ are factors of $c$ How to ...
1
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0answers
24 views

Does this approach for factorizing RSA numbers help in any way?

I was thinking about why factorizing RSA numbers is so hard. When humans perform any kind of maths manually, they often employ various 'tricks' that get them closer to the answer. Some are based on ...
-2
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0answers
39 views

If Integer Factorization is in $P$ — what are the implications?

If Integer Factorization is in $P$ Beside the bad news of the wide use, RSA cryptosystem, becoming insecure, is there any good news for such an algorithm to be found? For the seek of this question, ...
0
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1answer
32 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
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2answers
103 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 ...
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3answers
37 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
34 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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3answers
17 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 ...
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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
24 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 ...
4
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0answers
196 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
122 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 ...
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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
57 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 $2(1-2x)(x+2)(x^2+2)^{-5/2}$?
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3answers
79 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
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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 ...
0
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0answers
62 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
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1answer
61 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 $\mathbb{R}$ ...
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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
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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
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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
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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
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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
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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 ...
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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
27 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
vote
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
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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?