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

learn more… | top users | synonyms

1
vote
1answer
22 views

A non-UFD where prime=irreducible

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 ...
2
votes
1answer
46 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 ...
-2
votes
2answers
84 views

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

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
22 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 - ...
-5
votes
2answers
65 views

TAMS TOURNEMENT: exponential question (very hard) [on hold]

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
73 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
32 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 ...
1
vote
1answer
33 views

Finding the ideal

Determine all the ideals, prime ideals, and maximal ideals of $\mathbb{R}[x]/I$ where $I$ is the ideal generated by $(x^2+1)(x-2)^2$. I am currently doing some reading on ideals (see ...
0
votes
1answer
19 views

Factorization with a Primitive Factor of Polynomials

Question: Let $f,g\in\Bbb Q[x]$. Why is it that $\rm\color{#c00}{(1)}$ if $f$ is monic then $f=\frac{1}{a}f^*$ for some primitive polynomial $f^*\in\Bbb Z[x]$ and $a\in\Bbb Z$ ? ...
5
votes
0answers
35 views

Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
0
votes
1answer
18 views

Matrix Factorization of lower triangular and unit upper triangular.

So I have a matrix. $$A =\begin{bmatrix} 8 && -3 && 2 && -1\\ -3 && 8 && 0 && 2\\ 2 && 0 && 8 && -3\\ ...
0
votes
0answers
24 views

Unable to get matched answer using factorization

I have question to solve by factorization. the question is $$(a+b)x^2 + (a+2b+c)x + (b+c) = 0$$ the answer should be $$x = -a, -b.$$ i have done using it \begin{align} (a+b)x^2 + (a+b+b+c)x + ...
0
votes
1answer
54 views

How to fully factor a polynomial of 4th degree?

How to fully factor this polynomial? $$ 2x^4+3x^3-32x^2-48x$$ Can anyone describe the full steps to factor it? Thanks for the help.
1
vote
0answers
34 views

Numbers with special factorisation

We know that any natural number $n$ can be decomposed as $p_1^{k_1}p_2^{k_2}...p_n^{k_n}$. I am looking for numbers which have $k_1=k_2=k_3=....=k_n=1$ i.e. given a number n, identify if it has all ...
1
vote
3answers
52 views

Solving $3t^2-\frac{12}{3}t+\frac{4}{3}=0$

I need to to solve: $$3t^2-\frac{12}{3}t+\frac{4}{3}=0$$ The solution manual factorizes this to $\dfrac{1}{3}(3t-2)^2$. How can you do this easily?
1
vote
2answers
15 views

finding poles for a complex rational function

So in working out the details of a trig integration with complex integrals problem, I have ended up with an integrand of $$\frac{z}{z^4+6z^2+1}$$ I need to find the roots of $z^4+6z^2+1$ to use the ...
2
votes
2answers
54 views

Prove that $n^3 - n$ is divisible by 6 by factoring

I need to prove that $n^3 - n$ is divisible by $6$ by factoring it and by knowing that the product of each consecutive $3$ numbers is divisible by $2$ and $3$. I tried: $n(n^2 - 1)$ Factoring it ...
0
votes
3answers
15 views

Finding for which value of $a$ are two equations equal(need instructions for method)

I have the equations: $(a - 5x)^2$ and $25x^2 - 5x + a^2$ And I have a list of values for $a$ and for one of them, the two are equal. I just need to know what is the method for solving this - do I ...
0
votes
2answers
46 views

What is wrong with this factoring by completing the square?

This is the problem and my attempt at solution: $3x^2 + 2x - 1 = $ $3(x^2 + \frac{2}{3}x - \frac{1}{3}) = $ $3(x^2 + 2x + 1 - \frac{4}{3}x - \frac{4}{3}) = $ $3[(x + 1)^2 - \frac{4}{3}x - ...
2
votes
1answer
64 views

Average number of linear factors in a monic polynomial of degree $n$ over $\mathbb{F}_p$

Let $p$ be a prime and $P_n$ the set of all monic polynomials with coefficients in $\mathbb{F}_p.$ I am interested in the average number of linear factors of polynomials in $P_n.$ In an exercise in ...
1
vote
1answer
55 views

Factor of determinant with identical row

How the following fact applies to determinants (I came across it while solving problems): Consider A is a nxn matrix, the elements of which are real (or complex) polynomials in x. If r rows of the ...
1
vote
3answers
111 views

Irreducible factors of x^16 - 1 over GF(3)

Just want to double check my work. I'm trying to list the irreducible factors of $x^{16} − 1 $ over $GF (3)$ of degree $1$ and $2$ . Here's what I have: $$x + 1, x + 2, x^2 + x + 2, x^2 + 2x + 2$$ ...
2
votes
1answer
20 views

Basic question on Fermat's factorization method

Please excuse me if this is a basic question, or badly phrased, I'm very new to mathematics in general. In Fermat's factorization method - based on the fact that every odd number can be expressed as ...
2
votes
1answer
49 views

Factorize matrix determinant

When trying to diagonalize a matrix, say : $$\left(\begin{matrix} 0 & 2 & -1 \\ 3 & -2 & 0 \\ -2 & 2 & 1 \end{matrix}\right)$$ to find the eigenvalues, I have to find ...
9
votes
3answers
240 views

Intuitive understanding of the uniqueness of the Fundamental Theorem of Arithmetic.

Basically I am trying to understand why Fundamental Theorem of Arithmetic (FTA) exists, i.e why a natural number cannot be factored primely in two or more different ways. There are two proofs given ...
0
votes
0answers
3 views

Factoriaztion of quasi homogeneous function

Let $f(x,y) \in C[x,y]$ be a quasi-homogeneous polynomial, with $f(t^{w_1}x,t^{w_1}y)=t^df(x,y)$ Supposedly, after an analytic change of variables, we can always write it as: $f(x,y) = ...
2
votes
4answers
87 views

Why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$

I would like to understand why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$ I am particularly concerned with the term, $-4$.
0
votes
1answer
26 views

Factorizing Given Problem

I have searched through various site's and forums but couldn't find the answer to my problem, $$z^2-\frac{1}{2}z-\frac{1}{4}=0$$ How will you factorize this As I can't find $2$ numbers that give me ...
4
votes
2answers
73 views

Factoring in $\mathbb{Z}[\sqrt{2}]$

How would one factor a number, say $9+4\sqrt{2}$ in $\mathbb{Z}[\sqrt{2}]$? This is what I've attemped to do: $$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2}) $$ $$a_1a_2+a_1b_2\sqrt{2}+a_2b_1\sqrt{2}+2b_1b_2$$ ...
0
votes
1answer
24 views

Integer factorization complexity

Why isn't the problem of factoring an integer known to be in $P$? Isn't the naive algorithm of trying to divide a number by all the numbers up to its squre root polynomial?
2
votes
1answer
45 views

Possible values of $\gcd(a+b, a\times b)$

Main Question: Let $N \in \mathbb{N}$. What are the possible values of $\gcd(a+b, a\times b)$ given that $\gcd(a,b) = N$? Fact 0. If $\gcd(a,b) = N$, then $N \leq \gcd(a+b, a\times b) \leq ...
0
votes
2answers
28 views

How to extract factor when expression is with a power

$$f(x) = x^2(2x-3)^3$$ I tried to extract the 2 from the parenthesis. $$f(x) = 2x^2(x-\frac{3}{2})^3$$ But the graphic from this function is different. What should I consider when doing this kind ...
1
vote
1answer
40 views

Prime factorization difficulty

From Wikipedia: Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime ...
0
votes
0answers
25 views

Show a curve has no factor of degree 1 or 2

I have to show that $ h(x,y)=y^{2}(x^{2}+x+1)-x^{2} $ has no factors of degree 1 or 2. I know that h contains infinitely many points and is singular at the points (1,0,0), (0,1,0) and (0,0,1). I am ...
-4
votes
1answer
85 views

limit of function at $x \rightarrow 2$

ok, so this is a very basic question, i'm trying to find the limit of the following function at $x \rightarrow 2$: $|x^2 + 3x + 2| / (x^2 - 4)$ what i had previously done was simply plug in 2 for ...
3
votes
2answers
54 views

Limit of square root function at $x \to 6$

I'm trying to find the limit of the following function at $x \to 6$: $$\frac{x^2-36}{\sqrt{x^2-12x+36}}$$ i've simplified it so that it becomes $\dfrac{(x+6)(x-6)}{\sqrt{(x-6)^2}}$, which simplifies ...
2
votes
1answer
19 views

Factoring completely using complex cube of unity

How can you completely factor $a^2 + ab + b^2$ and $a^2 - ab + b^2$ completely using $\omega$, the complex root of unity? Is there some general rule for such complex factorisations? Any help would be ...
1
vote
4answers
87 views

Show that $(k!)^n$ divides $(kn)!$

Show that $(k!)^n$ divides $(kn)!$ I've tried it but without success. Any help would be great.
1
vote
1answer
36 views

Multiplying two fractions with complex numbers

I'm doing $$ \frac{6-7i}{1+i}\cdot\frac{1+i}{1+i}, $$ and I'm getting the correct value for the numerator (namely, $-1-13i$), but based on the problem answer, I need for the denominator to become $2$. ...
1
vote
2answers
41 views

Simplifying an inequality: $4x(x-2) \lt 2(2x-1)(x-3)$

I have: $$4x(x-2) \lt 2(2x-1)(x-3)$$ For the last part, do I multiply both things in $()$ by two then solve them like I normally would? If I solve them and then multiply will it work the same? Is that ...
2
votes
0answers
80 views

Humankind knows the prime factorization of the first how many consecutive integers?

I am only looking for an approximation. I'm guessing the answer must be somewhere between $10^{20}$ and $10^{50}$. . Edit: Okay so my first initial estimation was pretty poor... I should have ...
0
votes
3answers
59 views

Factoring Real and Complex polynomials.

Factor: a) $x^2 + 1 \in \mathbb{R}[x]$ b) $z^3 - i \in \mathbb{C}[x]$ Well I solved for $x^2$ and got $-i$ and $i$, but they aren't from Real. And I couldn't solve for Complex (part b).
0
votes
2answers
22 views

Basic complex factorisation

Let's say I want to find all the roots of $f(z)=z^8-256$. Factorising it, I find $f(z)=(z-2)(z+2)(z^2+4)(z^4+16)$. $z =\pm2,\,\pm2i$ is only 4 roots. Shouldn't there be another 4?
0
votes
4answers
46 views

Solving for x by completing the square in a problem where the solution doesn't seem to have symmetrical answers

So I've been given this problem: $-14x^2 + 45x + 14 = 0$ And I've tried it a number of times but can't seem to solve it. The answer is supposed to be found by completing the square, and the solution ...
0
votes
1answer
28 views

Please the box method for factoring trinomial of the form ax^2+bx=c

I was given this method for factoring trinomials of the form ax^2 + bx + c This is the method: find numbers p and q such as ac=pq and b=p+q With p and q (GCF(a,q)x + GCF(c,p))(GCF(a,p)x+GCF(c,q))= ...
2
votes
2answers
53 views

Factorize the given equation.

Factorize $$f(t) = t^3 - 11t^2 - 39t - 45$$ Assuming the above polynomial has a rational root, I tested the above equation using $+1$, $-1$, $+2$, $-2$. These did not work out. Then I tried $t =3$. ...
0
votes
2answers
20 views

Question about factoring/condensing equation rules

I have the equation $x^2 - 6x = 72$ and then $x^2 - 6x - 72 = 0$ that's supposed to turn into $(x-12)(x+6)$. 72/6 = 12. So could just do that with any equation that? Divide the end thing with the ...
0
votes
1answer
33 views

Calculate sum of all factors of expression

Expression: $$ \left(\frac{2x}2\right)^2 \left(\frac{3y}3\right)^3$$ Sum of all factors of above expression is $$2\cdot \left(\dfrac{2x}2\right) + 3\cdot\left(\dfrac {3y}3\right)$$ How ? Can ...
1
vote
1answer
22 views

Deflating (factoring) a 6th degree polynomial

What is the procedure to factor a 6th degree polynomial of a complex variable? $$P(z)=1+x^2+x^3+x^4+x^5+x^6$$ I do have the correct answer but no idea how to get to it. The answer is: ...
0
votes
2answers
45 views

Direct Proof even and odd

In trying to show that $n$ is even, is my final solution correct? First: If $n$ is even then $n^3+n$ is even. Since $n$ is even, then: $$n=2\cdot s$$ $$n^3+n = (2\cdot s)\cdot (2\cdot s)\cdot (2\cdot ...