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

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1answer
40 views

Negative factors vs positive factors

I'm learning about factoring and the lecturer show this example: $$-3x^2+12x-18$$ For start he factor this polynomial like: $$3(-x^2+4x-6)$$ So far so good but now he said: In some ...
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2answers
59 views

Factorization of $x^8-x$ over $F_2$ and $F_4$

How can I factorize $x^8-x$ over the fields $F_2$ and $F_4$?
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1answer
28 views

Expansion of polynomial

Expand the following: $-4(5x - 3) ^2$ As for this one, factorise : $5(y^2 - 45) $ Can't it just be $5(y^2 - 9)$? Why is it $5 (y+3) (y-3)$
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2answers
53 views

When factoring polynomials over the {reals,rationals,integers}, can one get stuck with an incorrect partial factorization?

Suppose I have a polynomial $A$, which I factorize as $A=BC$ (where $B$ and $C$ are polynomials with integer, rational or real coefficients). When factoring $B$ as $B=B'$ and $C$ as $C=C'$ (to ...
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2answers
47 views

factoring polynomials with 3rd degree or higher

I was searching for a method that would allow me to factor polynomials like this one $x^3 - 13x^2 +(14+4y)x + 8y=0$ I failed, I've only found how to factor by grouping or long division with already ...
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2answers
89 views

Factorizing $(x+a)(x+b)(x+c)$

I was solving questions related to polynomial factorization. I have learnt the remainder and factor theorems, and some basic identities. There was a question like this one: $$p(x)=x^3+8x^2+19x+12$$ ...
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1answer
42 views

Does there exist a natural number $a$ such that $a^2+1$ is divisble by $9$?

Can the above question be solved? Or can it be proved that it can not be solved? What is the best approach to solving such questions?
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2answers
16 views

How do you find the common factor of these expressions?

I have the answer in my answer book but I don't know how to work it out. $2a^2 - a - 3$ ----- $(2a - 3)^2$ ------- $4a^2 - 9$ $a^2b^2 - b^4$-------- $ab^2 + b^3$---------- $ab - b^2$ (I used ...
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1answer
42 views

If I'm factoring $2p^2+p-10$ would the answer be $p(2p+5) -2(2p+5)$?

If I'm factoring $2p^2+p-10$ would the answer be $p(2p+5) -2(2p+5)$? And to check would I just distribute and see if it matches up to the original problem?
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1answer
26 views

Factors of a polynomial over $\mathbb{C}$

For one of the problems on my algebra homework I am asked to find the zeros of $p(t)=t^5+t+1$ over $\mathbb{C}$. I have factored it into $$ p(t) = (t^2+t+1)(t^3-t^2+1).$$ We can compute the zeros of ...
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1answer
92 views

$n$ is a divider of $c$ if and only if $n = 2(c \mod (n-1)) - (c \mod(n-2)) + 2$

While working on Integer factorization problem I came to this conclusion: If and only if $n$ is a divider of $c$ $$c\mod n = 0$$ Than $$n = 2(c \mod (n-1)) - (c \mod(n-2)) + 2$$ c,n are positive ...
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1answer
38 views

How to factorize this.

We just started calculus and busy with limits. we were told that use a limit as long as it does not make the equation undefined. So the question is: $\displaystyle \lim_{x\to 0} \dfrac{2x}{x^2+x}$ ...
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2answers
59 views

Irreducible polynomials of the form $x^n - q$

Is there any easy way to see the following? Let $n\in\mathbb{N}$. Let $a,b\in\mathbb{R}$ s.t. $a < b$. Then, there exists $q\in\mathbb{Q}$ s.t. $a<q<b$ and $p(x) = x^n - q$ is irreducible ...
2
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2answers
148 views

Integer factorization: What is the meaning of $d^2 - kc = e^2$

I found an interesting behavior when placing the integer factorization problem in to geometry, I call it pyramid factoring. Lets assume we have $c$ boxes and we want to order them in to rectangle. ...
1
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1answer
45 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? $$a^2-c=b^2$$
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0answers
67 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 ...
1
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1answer
63 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 ...
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1answer
52 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 ...
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0answers
26 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 $e$ ...
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2answers
147 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)} = ...
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3answers
33 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
48 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
37 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 ...
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3answers
35 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
35 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 ...
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2answers
62 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 ...
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2answers
46 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
309 views

Integer Factorization: Possible progress [closed]

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 ...
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0answers
23 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 ...
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2answers
23 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
162 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 ...
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0answers
40 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
28 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 ...
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2answers
24 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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1answer
26 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 ...
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1answer
50 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 ...
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1answer
145 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
65 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 ...
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2answers
25 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
53 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 ...
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3answers
82 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 $ ...
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2answers
72 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
62 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
78 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
37 views

How to solve this rational equation?

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
54 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
70 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
121 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 ...
2
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0answers
53 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
53 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 ...