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

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1answer
28 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 ...
2
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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
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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.
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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 - ...
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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!
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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
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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
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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 ...
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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 $a\in\Bbb Z$ ? ...
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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 ...
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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\\ ...
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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
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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.
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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
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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?
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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
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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
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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
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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
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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 ...
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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 ...
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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
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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
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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 ...
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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, $-4$.
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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 ...
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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
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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
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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
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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 ...
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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 ...
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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 ...
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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
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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
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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
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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
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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 $2$. ...
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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
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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
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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).
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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?
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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
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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
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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 $t =3$. ...
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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
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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
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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 ...