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

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2answers
62 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
37 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
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2answers
147 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
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4answers
75 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?
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3answers
63 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
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1answer
77 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?
21
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1answer
205 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, ...
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3answers
62 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
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0answers
55 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 ...
3
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1answer
54 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 ...
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0answers
89 views

Factoring a polynomial over $\mathbb F_{2^8}$

How do you find the factors of $x^4+x+1$ in $GF(2^8)$ in terms of polynomials? Let me explain, We have primitive irreducible polynomial $p(x)=x^2+x+1$ in $GF(2^2)$ which has root $\alpha^2+\alpha$ in ...
3
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4answers
87 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
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2answers
44 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
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1answer
52 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
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1answer
93 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 ...
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2answers
100 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
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1answer
68 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 - ...
0
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2answers
89 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
42 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
36 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
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1answer
46 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 $a\in\Bbb Z$ ? ...
13
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0answers
207 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
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1answer
33 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
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0answers
27 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 + ...
1
vote
1answer
90 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
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0answers
36 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
58 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
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2answers
32 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
61 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
16 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
49 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
77 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
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1answer
68 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
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3answers
146 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: $$x + 1, x + 2, x^2 + x + 2, x^2 + 2x + 2$$ ...
3
votes
1answer
29 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
60 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
289 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
6 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
93 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
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1answer
29 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
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2answers
85 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
36 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
54 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
31 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
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1answer
60 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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1answer
92 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
66 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
30 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
101 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.
0
votes
1answer
39 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 $2$. ...