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

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0
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2answers
17 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 ...
1
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0answers
54 views

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. It can factor RSA1024 within ...
1
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0answers
15 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
20 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
votes
2answers
120 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
votes
0answers
30 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
votes
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?
1
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2answers
51 views

Factorise the following polynomial [on hold]

$$x^6+3x^4+4x^2+2$$ 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
18 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
75 views

factorizing without removing brackets [on hold]

\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
21 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
vote
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 $\mathbb{Z}[11]$ [on hold]

Prove that 2 is reducible over $\mathbb{Z}[11]$.
0
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0answers
42 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
58 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
votes
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
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3answers
62 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 advace that I have done is factor $ ...
0
votes
2answers
70 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
votes
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
72 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
45 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
39 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
59 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
votes
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
101 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
21 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
47 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
115 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
56 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
159 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
47 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 ...
1
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0answers
18 views

Computer program for factorization into irreducible polynomials over $\mathbb{Z}_p^k$

Hensel's Lemma allows us to factor a polynomial uniquely into basic irreducible factors over $\mathbb{Z}_p^k$. Is there a SAGE or Magma command that gives this factorization? Or can anyone help in ...
3
votes
4answers
82 views

AlgebraII factoring polynomials

equation: $2x^2 - 11x - 6$ Using the quadratic formula, I have found the zeros: $x_1 = 6, x_2 = -\frac{1}{2}$ Plug the zeros in: $2x^2 + \frac{1}{2}x - 6x - 6$ This is where I get lost. I factor ...
0
votes
2answers
35 views

What's the relation between factors of a number and its square root?

For instance, if the square root of a number $N$ is an integer, $N$ is a square number. But for instance $\sqrt{80} = 8.944...$, the fractional part is close to an integer, and indeed $81$ is a square ...
2
votes
1answer
42 views

A non-UFD where prime=irreducible [duplicate]

It is easy to see that in an atomic domain (where every element factors into irreducibles), we have that all irreducibles are prime iff the domain in question is an UFD. I think it is not true for a ...
3
votes
1answer
88 views

In $\triangle ABC$ , find the value of $\cos A+\cos B$

The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{(2b)!}$, Then prove that the value of ...
-3
votes
2answers
99 views

If $\sqrt{n}+ 8= n+1$, what is $n$? [closed]

If $\sqrt{n}+ 8= n+1$, what is $n$? Please show as many steps as possible so I can understand the process.
0
votes
1answer
30 views

Determining how many roots a cubic equation has.

I am working through some of the quizes on brilliant.org I came across this question. Suppose that the following cubic polynomial has one rational root and two non-real complex roots: $$ x^3 - ...
-4
votes
2answers
91 views

TAMS TOURNEMENT: exponential question (very hard) [closed]

What is the sum of the roots of $(2−x)^{2012} −x^{2012} = 0$ Any tips or solutions to this question would be greatly appreciated!
0
votes
2answers
78 views

How to factor polynomials in $\mathbb{Z}_n$?

How to factor a certain polynomial over $Zn$. for example factor the following polynomial into irreducible polynomials in $Z5$: $X^3+X^2+X-1$ or factor the following polynomial into irreducible ...
0
votes
1answer
33 views

How to take apart a characteristic polynomial

Suppose I have a polynomial: $x^3-8x^2+17x-4$. How do I know it will always be $(x-4)(x^2-4x+1)$ by solving it? I'm struggling to figure out what to look for in the polynomial to give me a hint or ...