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

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0answers
70 views

The annihilator of finitely generated modules over PID

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Suppose there exists prime ideal $p$ and integer $i$ such that $p^i=\operatorname{Ann}(M)$. Then prove: (1) there ...
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1answer
45 views

Question about factor of a function

can you explain this question for me? I don't quite understand it. Thank you in advance. $x^2+1$ is a factor of $f(x)$. Which of the following is TRUE? $\text{a)}\qquad f(-1)=0 ...
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2answers
56 views

How to simplify this fraction

It's embarrassing, but I need help solving this one... Need some refresher course for algebra. $$ \frac y{y+\sqrt y} $$
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1answer
30 views

If $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$

Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$. Show that if $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$. I showed that if $a$ is ...
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3answers
109 views

Is there a branch of mathematics that studies the factors of rational numbers?

Is there a branch of mathematics that studies the factors of rational numbers? I am imagining that defining this would work pretty much the same way as defining the factors x of an integer n: $\{x ...
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2answers
33 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
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1answer
34 views

Factors of $x^n+1$ over $\mathbb{Z}[x]$

Is there any equivalent to $x^n-1 = \prod\limits_{d|n} \phi_d$ where $\phi_d$ is the $d$th cyclotomic polynomial but for $x^n+1$? Even better, can we generalize any further?
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4answers
221 views

Factoring a cubic polynomial?

So I have a matrix $$A = \begin{pmatrix} -5 & -6 & 3 \\ 3 & 4 & -3 \\ 0 & 0 & -2 \end{pmatrix} $$ I'm to find the characteristic polynomial and all the eigenvalues of ...
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2answers
75 views

How to factor $x^3-x^2+x+3$ [closed]

I have a partial fractions integral with $x^3-x^2+x+3$ in the denominator. How do I factor this?
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1answer
39 views

Inversion in factor rings

I have this polynomials: $f = x^{4} + 3x^{3} + x^2 + 3 \in \mathbb{Z}_{5}[x]$, $g = x + 2 \in \mathbb{Z}_{5}[x]$ Does g + (f) have inversion in ring $(\mathbb{Z}_{5}[x]/(f),+,.)$ ? I should found ...
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2answers
212 views

Use a given zero to write P(x) as a product of linear and irreducible quadratic factors

The polynomial in question is: $x^4 - 8x^3 - 19x^2 + 288x - 612$ and the zero is $4 - i$. What I don't understand is how to go from the given zero to factorizing, especially as it's imaginary. ...
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2answers
43 views

*Step* in proving that there are infinitely many primes that suffice…

Let $k,n\in \mathbb Z$ with $n=k^2+1$ and let $p$ be an odd prime with $p\mid n$. Prove that $p\equiv1\text{ mod }4$. I found out that $\bar{n}\in\left\{ \bar{1},\bar{2}\right\} $ (denoting ...
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2answers
90 views

Intermediate Problem Solving Patterns involving Prime Factoring

a and b are positive integers such that $a\times b= 500000000,$ where neither a nor b contain any zeros. Find a and b where $a<b.$
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1answer
57 views

If n > 3 and (n + 1) is a square, is there any n that is a prime?

I am looking at properties of squares and came about this property. I am investigating the difference of squares in relation to primes.
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3answers
55 views

Can someone please explain how this was factored?

How was $x^2(x+1)-4(x+1)$ factored into $$(x^2-4)(x+1)?$$ I know this seems very basic but can someone please explain this?
2
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1answer
57 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
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2answers
59 views

Common divisor of natural number sequence [closed]

For all natural numbers $n$, the products $(n+2)(n+3)(2n+5)$ will always share what common divisor ?
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3answers
52 views

Trouble with factoring polonomial to the 3rd degree

I am having trouble factoring this problem: $\displaystyle{-x^{3} + 6x^{2} - 11x + 6}$ I know the answer but i can't figure out how it is done with this. I have tried by grouping and is doesn't seem ...
8
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6answers
319 views

Find $x$ and $y$ in $2^{x-y} + 1 = 2^x,$ where $x,y$ are integers

I have no idea what to do now. Is there any way to find the integers $x$ and $y$ by factoring? Thank you.
2
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1answer
71 views

How many integral solutions of $a,\ b,\ c$ are there such that $2^a \cdot 3^b + 9 = c^2 $

How many integral solutions of $a,\ b,\ c$ are there such that $$2^a \cdot 3^b + 9 = c^2.$$ we can get that $$2^a \cdot3^b = (c-3)(c+3) $$ we can make cases if $b \ge 2$ then $c=3k$ then ...
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1answer
61 views

Please help me with factorisation

Is it possible to write $$64x^6-112x^4+56x^2-7$$ in linear factors? If so, what are they? (Finding it really difficult to ask this question!!)
2
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1answer
175 views

Factorizing $(x-1)(x-3)(x-5)(x-7)-64$

We need to factorize: $$(x-1)(x-3)(x-5)(x-7)-64$$ We can, by the rational root theorem, see that there are no roots of this polynomial.Next observation is that $64=(8)^2$. So this means that if the ...
0
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1answer
95 views

How long is an arrow in the air?

The height $h$ of an arrow in feet is modeled by $h(t) = -16t^2 + 63t + 4$, where $t$ is the time in seconds since the arrow was shot. How long is the arrow in the air? Could someone explain where to ...
0
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1answer
17 views

How do I identify repeated irreducible factors?

I thought my solution was correct - but it seems like that's not the case. Can anyone possibly explain to me why I'm wrong?
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2answers
219 views

Spivak Calculus chapter 1, problem 1v

this is my first question here. I am self-studying Spivak's Calculus Fourth Edition and am stuck on Chapter 1 Problem 1v. The question is to prove that: $x^n - y^n = ...
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0answers
234 views

Find the set of cyclotomic cosets of q modulo n

Calculating finite field and factoring $x^n - 1$ over $GF(q)$ first step is to calculate cyclotomic cosets. For example : For $n=9,q=2$ $C_1=\{1,2,4,8,7,5\} = C_4 = C_8 = C_7 = C_5$ $C_3=\{3,6\} ...
0
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1answer
110 views

What is the mathematical proof behind the shortcut used in this video, Factoring Trinomials with Leading Coefficient not 1 (fast way)?

My teacher found this cool shortcut for factoring. I would like to use, for it saves time, but I feel hesitant using it without knowing the mathematical proof. Can anyone watch the video and explain ...
0
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1answer
55 views

Frobenius matrix norm vs. 2-norm

From this article about the singular value decomposition: Let $A$ be an $n \times d$ matrix and think of the rows of $A$ as $n$ points in $d$-dimensional space. The Frobenius norm of $A$ is the ...
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1answer
34 views

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $$\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$$ I have a theorem (shown in my text) that says $$\fbox{If $a_1,...,a_m\in\mathbb{N}^*$ then ...
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1answer
36 views

Find the smallest value of $n$ so that the greater potency of $5$ which divides $n!$ is $5^{84}$. What are the other numbers that enjoy this property?

Find the smallest value of $n$ so that the greater potency of $5$ which divides $n!$ is $5^{84}$. What are the other numbers that enjoy this property? I thought I would put together an equation ...
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1answer
30 views

Computations question

a) Determine the prime factorizations of 3850 and 4125 b) Find the value of d = gcd(3850,4125) c) List all the positive divisors of d This is what I have so far. a) 3850: 11, 5, 5, 7, 2 4125: ...
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2answers
129 views

How useful is factoring large numbers - Not for cryptography!

This is a question for my curiosity. Apart from its implications to cryptography. Is factoring large numbers really useful? Are there any examples of where the ability to factor huge numbers, 1K ...
2
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1answer
83 views

Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number.

Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number. Actually, I know a way to solve this, but even if it is very large and ...
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2answers
203 views

Convert Circuit SAT to 3-SAT

I am trying to convert Integer Factorization to $3-SAT$. So far I managed to convert it to Circuit SAT, but I don't know how to make the final step. This is how it look for 3*3 multiplication: ...
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0answers
36 views

Polynomial factorization over an infinite field - is there an algorithm

In my previous questioned I asked how do I factor a polynomial, and I gave an easy example of a polynomial of degree 2. But now I have another question I need to solve. I need to factor ...
3
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1answer
74 views

Polynomial factorization to irreducible factors with respect to field

I have a question, I think I don't understand this material very well and could use an explanation / some help. Basically we are asked to decompose $x^5-x$ to irreducible factors over $R,F2,F5,C$ ...
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2answers
31 views

If $N$ is not a power of a prime, why does 1 have -1 as a root modulo $N$?

If integer $N$ is not a power of a prime, it is the product of two coprime integer numbers greater than 1. As a consequence of the Chinese remainder theorem, the number 1 has at least four ...
3
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1answer
76 views

Probability of an ECM factor

Suppose I have a composite number $N$ divisible by some prime $p\le x.$ What is the probability that one iteration of ECM finds $p$, given parameters B1 and B2? Usually people look for factors in ...
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1answer
35 views

$\frac{x^4 - x^3 + ax^2 + bx + c}{x^3 + 2x^2 - 3x + 1}$, remainder $3x^2 - 2x + 1$. Find $(a + b)c$.

Given the polynomials $P(x) = x^4 - x^3 + ax^2 + bx + c\\ Q(x) = x^3 + 2x^2 - 3x + 1\\ R(x) = 3x^2 - 2x + 1$ such that $P(x) = D(x)Q(x) + R(x)$, find $(a + b)c$. I would normally apply little ...
3
votes
1answer
66 views

Finding $a_n$ such that $x^n+a_1x^{n-1}+\cdots+a_{n-1}+a_n$ cannot be factored when $a_1,\cdots,a_{n-1}$ given

Let $n\ge 4\in\mathbb N$. Suppose that $a_1,a_2,\cdots,a_{n-1}$ are given integers. Then, here is my question. Question : Is the following true for any $(a_1,a_2,\cdots,a_{n-1})$ ? There ...
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2answers
116 views

Defining irreducible polynomials recursively: how far can we go?

Fix $n\in\mathbb N$ and a starting polynomial $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_n\ne0$. Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+a_rx^r$ such ...
3
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1answer
79 views

Cholesky decomposition: any theoretical value?

Just read the Wikipedia article on Cholesky decomposition. All the applications listed there were numerical. Are there theoretical arguments to which it is important? For instance, here there is an ...
2
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3answers
136 views

Factoring out an exponential?

I have the following expression $$\frac{2^{k+1}(k+1)!}{(k+1)^{k+1}}\cdot\frac{k^k}{2^k k!}$$ I get $$\frac{2(k+1)(k^k)}{(k+1)^{k+1}}$$ But how do I factor out the ${(k+1)}^{k+1}$
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3answers
69 views

How do you factor $(10x+24)^2-x^4$?

I tried expanding then decomposition but couldn't find a common factor between two terms
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2answers
81 views

Computing p and q from private key

We are given n (public modulus) where $n=pq$ and $e$ (encryption exponent) using RSA. Then I was able to crack the private key $d$, using Wieners attack. So now, I have $(n,e,d)$. My question: is ...
2
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1answer
89 views

Number of divisors of a number

Is there any trick to find the number of divisors of any number? For e.g., a quick way to tell the number of divisors of 987655432 (chosen randomly)? EDIT: And it has to be done without prime ...
0
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1answer
45 views

How to approach factoring problems?

Generally speaking, how should I approach a problem involving factoring? I usually don't have a problem with the more typical forms, but sometimes I just don't know what to do. My calc2 question is ...
3
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1answer
37 views

How to factor these monomials?

This is the original problem: $x^3+x^2y+xy^2+y^3$ Answer: $(x+y)(x^2+y^2)$ I understand that the answer is correct, but I can't figure out how to get to it.
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7answers
510 views

If $a^3 + b^3 +3ab = 1$, find $a+b$

Given that the real numbers $a,b$ satisfy $a^3 + b^3 +3ab = 1$, find $a+b$. I tried to factorize it but unable to do it.
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2answers
99 views

How to find the factors whose sum is minimum

Lets take a number 108. How to find natural numbers a and b such that ab=108 but there sum should be minimum. Please show the solution for number 108.