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

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0
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3answers
78 views

How to find remain factor of this trigonometic equation?

The equation $$3\sin^2 x - 3\cos x -6\sin x + 2\sin 2x + 3=0$$ has a solution $x = 0$. That is mean it has a factor $\cos x - 1$. I tried write the given equation has the form $$(\cos x - 1)P(x)=0.$$ ...
0
votes
1answer
29 views

Derive the following expression:

Given the function $(x_1^2+x_2^2)^2-x_1^2+x_2^2=0$, where $r^4=x_1^2-x_2^2$ and $r^2=x_1^2+x_2^2$ for $-1\leq r \leq 1$ show that $x_1=\frac{1}{\sqrt{2}}r\sqrt{1+r^2}$ and $x_2= \pm ...
10
votes
5answers
778 views

Factor $(a^2+2a)^2-2(a^2+2a)-3$ completely

I have this question that asks to factor this expression completely: $$(a^2+2a)^2-2(a^2+2a)-3$$ My working out: $$a^4+4a^3+4a^2-2a^2-4a-3$$ $$=a^4+4a^3+2a^2-4a-3$$ $$=a^2(a^2+4a-2)-4a-3$$ I am ...
6
votes
4answers
264 views

Factorize $3m^4-6m^3+14m^2-6m+11$

I have this expression: $3m^4-6m^3+14m^2-6m+11=0$ and I want to factorize it in $(m^2+1)(3m^2-6m+11)$. How can I do it? Thanks for any help!
1
vote
1answer
339 views

Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$?

Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$? $Approach$: $N$=$11^2$.$13^4$.$17^6$ $N^2$=$11^4$.$13^8$.$17^{12}$ This ...
1
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0answers
51 views

Integer factorization using discrete logarithms

I'm reading up on RSA and attacks on it. At the end of one section of the notes, it asks (without giving an answer) whether or not integer factorization is easy given an oracle which computes discrete ...
7
votes
1answer
363 views

How prove that polynomial has only real root.

Let this polynomial $f(x)=\displaystyle\sum_{i=1}^{n}a_{i}x^i,\;\;a_{i}\in \mathbb{R} $ have only real roots. Prove: The polynomial $g(x)=\displaystyle\sum_{i}^{n}C_{n}^{i}a_{i}x^i$ has only real ...
0
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3answers
76 views

How do I factor this and simplify

I just found the derivative of a function and now I'm trying to factor and simplify the expression below. I have no idea how to factor terms with fractions as exponents especially when these fractions ...
1
vote
0answers
104 views

Prime factors of a random number

Let $r$ be a uniformly random integer between $1$ and $N$, for some large enough $N$ (i.e., I'm only interested in the asymptotics). What is the expected largest prime factor of $r$? Is there a good ...
3
votes
2answers
227 views

Write in polynomial in factored form in complex number

Write the following polynomial in factored form(in complex number): $$1+z+z^2+z^3+z^4+z^5+z^6$$ Also, is there general solution of factoring for $1+z+z^2...z^n$ types of polynomial?
1
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4answers
103 views

Help solving $\sqrt[3]{x^3+3x^2-6x-8}>x+1$

I'm working through a problem set of inequalities where we've been moving all terms to one side and factoring, so you end up with a product of factors compared to zero. Then by creating a sign diagram ...
0
votes
4answers
90 views

What algorithms are used to determine the difference of two squares?

I'm taking a discrete math class this semester. The professor said that by the end of the semester he wants us to have an understanding of RSA encryption. One thing that we have gone over in class is ...
2
votes
3answers
98 views

Let $F$ be a field, then the polynomial $x^n - 1$ has $n$ roots in $F$ if $F$ contains a multiplicative subgroup of order $n$.

Is this following true? If $F$ is a field, then the polynomial $x^n - 1$ has $n$ roots in $F$ whenever $F$ contains a multiplicative subgroup of order $n$.
1
vote
5answers
131 views

How to factor a strange trinominal.

I know how to factor normal trinomials, however I was stumped when I saw this on my homework, could anyone help me through this? The trinomial is as such: $-m^2 + 8m + 18$.
2
votes
3answers
141 views

irreducibility of polynomials with integer coefficients

Consider the polynomial $$p(x)=x^9+18x^8+132x^7+501x^6+1011x^5+933x^4+269x^3+906x^2+2529x+1733$$ Is there a way to prove irreducubility of $p(x)$ in $\mathbb{Q}[x]$ different from asking to PARI/GP?
0
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1answer
51 views

Factorising and limits

How do I factorize this expression? $$(2^n-3^n+n4^n)^{\frac{1}{n}}$$ so far I have: $$n4^n\left(\frac{1}{n} \left(\frac{1}{2}\right)^n-\frac{1}{n}\left(\frac{3}{4}\right)^n +1\right)^{\frac{1}{n}}$$ ...
4
votes
3answers
134 views

Factoring $x^8-x^4+1$ over $GF(7)$

Could anyone suggest any good way to do it? (The only way I can think of is by looking for roots (There are none), checking a factorization into the product of a 6 and a 2 polynomial (Many unknowns ...
7
votes
1answer
428 views

factorise, $x^3-13x^2+32x+20$

factorise, $x^3-13x^2+32x+20$ Let, $f(x)=x^3-13x^2+32x+20$ $f(x)=x(x^2-13x+30)+2x+20$ $f(x)=x(x-3)(x-10)+2x+20$ $f(-1)\lt 0$, $f(0)\gt 0$, which shows there is a root between $x=-1$ and $x=0$ ...
6
votes
2answers
179 views

Simple factoring in proof by induction

How would this: $$\frac{((n+1)+1)(2(n+1) + 1)(2(n+1) + 3)}{3}$$ Factor to this: $$(2(n+1)+1)^2$$ This is a part of an induction proof, which I would post an image if my reputation was higher... ...
8
votes
1answer
151 views

Factoring a couple $5$th degree polynomials

I'm reading an old (1895) textbook on algebra (doing a bit of review), and practicing factoring polynomials. The author started with polynomials where all terms share a common factor, like $4a^2 + 4a ...
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vote
3answers
115 views

Trouble with factorising a polynomial

I'm supposed to show that: $$y=\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$ and the required answers are: $$ P=5, Q=4, R=-4 $$ I tried to solve this with partial ...
1
vote
2answers
270 views

Factorization of three variables

Prove that : $(a+b+c)^3-(b+c)^3-(c+a)^3-(a+b)^3+a^3+b^3+c^3=6abc$ Since $(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a) $ Therefore the equation becomes : $2(a^3+b^3+c^3)+3(a+b)(b+c)(c+a) - [(c+a)^3 ...
1
vote
4answers
106 views

Factoring a large integer where the factors are prime

Suppose I know the value of $n$ (where $n = pq$), and I also know $k = p-q$. How can I efficiently factor $n$? Note that I don't know $p$ or $q$. EDIT: Thank you for your answers. I understand that ...
1
vote
2answers
63 views

Factoring polynomial with 4 terms

I cant figure this out. $$\frac{x-5}{x^3-3x^2+7x-5}$$ I tried by grouping and got $$x^2(x-3)+1(7x-5)$$ for the denominator. I need to use partial fractions on this so I cant use that yet.
4
votes
1answer
153 views

Does the difficulty of discrete logarithm depend on the difficulty of integer factorization?

The security of many (most? all?) public-key cryptography systems are based on the difficulty of the discrete logarithm or integer factorization. Are these two problems related at all? With the ...
0
votes
1answer
52 views

What is $q(x)$ and $r(x)$ when $(x^2-6x+9)q(x) + r(x) = x^3 -27$?

I just failed this question on a test, so I would please like to get some feedback on where my thinking was wrong. I need help with determining $q(x)$ and $r(x)$ when: $$(x^2-6x+9)q(x) + r(x) = x^3 ...
3
votes
1answer
86 views

Factorization of integers - why does it suffice to consider squarefree instances?

I sat a lecture where a proposition is proven that states the following: If computation of $(k!)_{k\in\mathbb{N}}$ is "easy", then integer numbers can be factored in non uniform polynomial time. ...
1
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0answers
572 views

Factorization of cyclic polynomial

Factorize $a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$ Since this is a cyclic polynomial therefore factors are also cyclic : $f(a) = a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$ $f(b) = b(b^2-c^2)+b(c^2-b^2)+c(b^2-b^2) ...
2
votes
1answer
139 views

Help with particular solution to solving $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$.

I asked a question about how to solve: $$z^4−2z^3+9z^2−14z+14 = 0$$ When all you know is that there is a root with the real part of 1. I was given great answers and you can find the question ...
7
votes
3answers
577 views

How do I completely solve the equation $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$ where there is a root with the real part of $1$.

I would please like some help with solving the following equation: $$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$ All I know about the equation is that there is a root with the real part of $1$. My approach ...
1
vote
3answers
222 views

finding residue with $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$

I am doing the integral $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$, and I am trying to find the residue at the pole $3i$;I am unsure how to do this. Could I factor $z^2 + 9$ further?
0
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1answer
136 views

Integer Factoring Algorithm Speeds

Given $N=pq$, would $\frac{p-1}{2}$ steps be fast compared with extant factoring methods?
3
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1answer
1k views

Using Parseval' s theorem to evaluate a sum..

On the function $f(x)$ = $x^3$ on $(-1,1)$ find Fourier coefficients for this function and then use Parseval's Theorem to evaluate: $$\sum_{n=1}^{\infty}\frac{1}{(n^6)}.$$ Current work: I have used ...
0
votes
1answer
90 views

Gaussian elimination with partial pivoting doubts

I have the following doubts about Gauss algorithm with partial pivoting: Say that I sum to the second row the first row multiplied by $k$. In the $L$ matrix, should I sum to the second row the first ...
0
votes
1answer
32 views

Distinct-degree factorization

I'm trying to understand distinct-degree factorization from Wikipedia. I'm trying the algorithm on paper with $q=9$ and $f(x) = (x+4)(x+5) = x^2+2 \in F_{q}$. We start with $i=1$. I calculate $g = ...
4
votes
5answers
131 views

Polynomial factoring $1-3x+4x^3$

I want to factorize (or factor ? can both verbs be used ?) $1-3x+4x^3$. I notice that $\frac{1}{2}$ and $-1$ are roots of the polynomial. My questions are : 1) how do you notice that $\frac{1}{2}$ ...
0
votes
1answer
21 views

What is the pair $(n,k)$ called where $n$ is an integer and $k$ is the ordered factorization index?

I’m developing a number class (as in Object-Oriented Programming) and am wondering what to call it. At its core, it represents an integer, but in a way in which not all integers are unique. What it ...
2
votes
3answers
218 views

Solve for $ x$, $-\frac{1}{2}x^2 + 2x + 5 = 0$

I'm having trouble solving this equation for $x$: $$-\frac{1}{2}x^2 + 2x + 5 = 0$$ What's the steps to take to solve it? Thanks.
2
votes
3answers
4k views

How many positive integers are factors of a given number?

I've been trying to find / generate a formula for the following problem: Given a number, how many positive integers are factors of this number. In practice, you could simply build a table as such ...
3
votes
1answer
101 views

Limit question factoring [duplicate]

Possible Duplicate: How do I find the delta analytically for $f(x)$ with a degree other than $1$ There is a question, prove that: $$\displaystyle \lim_{x\to3} x^{2} = 9$$ in linear ...
1
vote
1answer
98 views

Can the product of irreducible polynomials have non-constant factors other than those polynomials?

Can the product of irreducible polynomials over the reals, $P_1, P_2,...,P_n$, have non-constant polynomial factors other than those polynomials or products of them (eg. $P_1P_3$)? It seems that the ...
1
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1answer
81 views

legal value for prime factorization

My understanding of mathematics is poor, although I'm trying to improve it, so I hope you will forgive me for asking a rather basic question. I've written a computer code that prints the prime ...
0
votes
1answer
46 views

solving an equation by factoring

Need help solving an equation by factoring. PROBLEM: $3v^2-10v-12 = -28v + 36$ This is my solution, but it seems a little too much for school: $3v^2+18v-48 = 0$ $3(v^2+18v-48) = 0$ ...
3
votes
1answer
158 views

Matrix Processing in the Quadratic Sieve

I am working through the example in implementation of the quadratic sieve, and I have got stuck at the very last part: the matrix processing. In the example we must find the vector $S$ by left null ...
6
votes
1answer
1k views

Simply Explain the General Number Field Sieve

As a beginner to the world of integer factorization, my idea of factoring an integer is to generate a large list of prime numbers below this number and to repeatedly try to divide the integer by these ...
1
vote
3answers
92 views

Find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\dots+\frac{1}{d_k}$.

Let $d_1,d_2,\dots,d_k$ be all the factors of a positive integer $n$ including $1$ and $n$. Suppose $d_1+d_2+\dots+d_k=72$. Then the value of $$\frac{1}{d_1}+\frac{1}{d_2}+\dots+\frac{1}{d_k}$$ is ...
1
vote
1answer
65 views

probability of a number not having factors below n?

I want to know the probability that a number is not divisible by any prime smaller than or equal to n. ie, I find a big number, I trial division it up to 300000 and don't find any factors. I want to ...
2
votes
5answers
3k views

Factoring Polynomials with four terms and two variables

I've been working on this for hours and cannot figure it out. When I search, I find factorization techniques that I already know but don't seem to be able to apply here, or that are for polynomials ...
1
vote
2answers
183 views

How to break apart this sum?

I have a summation I need to break apart but I can't figure it out http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape10/PQDD_0027/MQ50799.pdf $p.15$, right after line $(3.8)$ Going from the ...
5
votes
3answers
97 views

Given $N$, find $ab = N$ with $a$ and $b$ as close as possible

Given a number $N$ I would like to factor it as $N=ab$ where $a$ and $b$ are as close as possible; say when $|b-a|$ is minimal. For certain $N$ this is trivial: when $N$ is prime, a product of two ...