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

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0
votes
2answers
70 views

Find monic grade 3 polynomial in $\mathbb Z_p[x]$ then factorize

Let $f = 15x^4+22x^3-x=0$ a polynomial in $\mathbb Z_p[x]$, find the first prime $p$ value that will make $f$ result in being grade 3 and monic. Then factorize $f$ in $\mathbb Z_3[x]$ as product of ...
1
vote
1answer
80 views

How to simplify [3a(b-c)+5][-3a(b-c)-5] by using special product?

In simplifying $$[3a(b-c)+5][-3a(b-c)-5],$$ I used $$4(au+bv)(cu+dv)=acu^2+(ad+bc)uv+bdv^2.$$ I failed to apply the formula to the equation because $a=3a$, $b=-3a$, $c=5$, $d=-5$, $u=(b-c)$, $v=?$ ...
0
votes
2answers
83 views

Find a prime number $p$ so that $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ is divided by $x-\overline{2}$ in $\mathbb Z_p$

Let $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ be defined in $\mathbb Z_p$. Find a prime number $p$ so that $f$ can be divided by $g = x-\overline{2}$, then factorize $f$ as ...
7
votes
3answers
110 views

How to see that the shift $x \mapsto (x-c)$ is an automorphism of $R[x]$?

In the process of studying irreducibility of polynomials, I encountered the criterion that $p(x)$ is irreducible if and only if $p(x-c)$ is irreducible. When trying to determine what properties of the ...
5
votes
1answer
672 views

Can any Polynomial be factored into the product of Linear expressions?

Specifically I am wondering if... Given a Polynomial of n degree in one variable with coefficients from the Reals. Will every Polynomial of this form be able to be factored into a product of n ...
0
votes
3answers
181 views

Determine monic and degree 3 polynomial in $\mathbb Z_p$

I stumbled upon this kind of problem and I really can't get the hang of it. Will anyone please outline the way to solve it? Determine for which of the first $p > 0$ values the polynomial $f = ...
1
vote
1answer
165 views

How to factor $x^5 - x + 1$

As I understand it $x^5 - x + 1$ is not solvable by radicals. But it splits over $\mathbb{C}$, so how does it factor into linear factors?
0
votes
0answers
148 views

Approximations for the number of divisors of an integer

Given an integer $n$, I want to know the asymptotic order of: a. the number of distinct prime factors b. the number of non-distinct prime factors c. the number of distinct divisors d. the number ...
2
votes
2answers
125 views

Polynomial-related manipulation

My question is: Factorize: $$x^{11} + x^{10} + x^9 + \cdots + x + 1$$ Any help to solve this question would be greatly appreciated.
1
vote
0answers
136 views

A special factorization

Suppose that monic polynomial $f(x)\in\Bbb Z[x]$ such that for all $m\in\Bbb Z$, $m>1$, there's no integers $\langle r,r_1,\ldots,r_m\rangle$ such that $f(r)=f(r_1)\cdots f(r_m)$. Is there any ...
2
votes
3answers
298 views

Factoring by grouping: $x^4 - y^4 -4x^2 + 4$

Please help me factor $x^4 - y^4 -4x^2 + 4$ by grouping terms. Thank you.
1
vote
2answers
138 views

Factoring $x^4z-2z^2-4x^6+x^2z$

We want to factor $8x^4y^4-2y^8-4x^6+x^2y^4 = -2y^8 + (8x^4+x^2)y^4 -4x^6$. We substitute $x^4$ with $z$: Now we want to compute this $8x^4z-2z^2-4x^6+x^2z = -(x^2-2z)(4x^4-z)$ by hand. Therefore we ...
1
vote
2answers
481 views

Factoring multivariate polynomial

I'm trying to factor $$x^3+x^2y-x^2+2xy+y^2-2x-2y \in \mathbb{Q}[x,y].$$ The hint for the exercise is to use the recursive multivariate polynomial form. So I'm using $\mathbb{Q}[x][y]$: $$ x^3 + ...
2
votes
6answers
578 views

How do I factor trinomials of the format $ax^2 + bxy + cy^2$?

Take, for example, the polynomial $15x^2 + 5xy - 12y^2$.
3
votes
1answer
166 views

Dixon's random squares algorithm: a step in the proof of its subexp. running time

I am currently working to understand Dixon's running time proof of his subexp integer factorization algorithm (random squares). But unfortunately I can not follow him at a certain point in his work. ...
2
votes
4answers
70 views

Probably simple factoring problem

I came across this in a friend's 12th grade math homework and couldn't solve it. I want to factor the following trinomial: $$3x^2 -8x + 1.$$ How to solve this is far from immediately clear to me, ...
0
votes
2answers
102 views

Help with $1 + a + a(a-1) + a(a-1) (a-2) +\cdots+a(a-1)\cdots(a-(n-1))$

I want to rewrite the series $$1 + a + a(a-1) + a(a-1) (a-2) +\cdots+a(a-1)\cdots(a-(n-1))$$ as $(a^n-1)Y$ or $(a^{n-1}-1)Y$ Short-form: $$\{1+\sum_{i=1}^{n} \prod_{j=0}^{i-1}(a-j)\}$$ as $(a^n-1)Y$ ...
3
votes
2answers
206 views

Find $X$, $a$ : ALL prime factors of $(X^a - 1)/(X - 1) < X$

where $X$ is an odd prime, and $a$ is an odd integer. For example, let $X = 37$, $a = 3$, we get $$\frac{37^3-1}{36} = 3 \times 7 \times 67.$$ When factoring numbers such as this, I've noticed that ...
3
votes
3answers
201 views

Factorization of polynomials over $\mathbb{C}$

I'm stuck I don't know how to write this complex number equation as two factors although I know one of those factors is $z - 3$. Any ideas/advice appreciated. $$ f(z) = z^3 + (-6+2j)z^2 + ...
1
vote
4answers
1k views

How to factor the quadratic polynomial $2x^2-5xy-y^2$?

How do I factor this polynomial: $2x^2-5xy-y^2$ ?
1
vote
1answer
145 views

Unique decomposition of a mapping by an equivalence relation

I have a math question from computer science. The following should be a fundamental fact from mathematics. Can you the mathematicins tell me how you would say it in a more elegant way? Given a ...
4
votes
1answer
72 views

Factoring of a fraction, possibly made a mistake

Here is the initial expression and the steps I've made so far, but from the final line I can't go on. Have I made a mistake somewhere? ...
2
votes
2answers
186 views

Splitting polynomials

I have a polynomial ${\frac{{{{({z^2})}^p} \pm {p^p}}}{{{z^2} \pm p}}}$ where $p$ is an odd prime number, and I know it splits into two factors $$ \sum_{i = 0}^{p - 1} a_i z^i \text{ and } \sum_{i = ...
6
votes
1answer
461 views

Smallest number with a given number of factors

From my rather rudimentary explorations of this fascinating problem, I believe it to be a layered and rewarding subject for investigation. My question, essentially, is: How do you find the smallest ...
2
votes
1answer
413 views

Factoring a number $p^a q^b$ knowing its totient

We are given: $n=p^aq^b$ and $\phi(n)$, where $p,q$ are prime numbers. I have to calculate the $a,b,p,q$, possibly using computer for some calculations, but the method is supposed to be symbolically ...
2
votes
1answer
235 views

How to find the factors of numbers around 1e7?

I don't have a maths background but I'm solving problems on the awesome Project Euler .net in JavaScript as programming practice. I don't want to link directly to the question or post it verbatim ...
3
votes
2answers
72 views

Factoring a number to get an encoded string

Say we encode the string $BCA$ in the following way: \begin{align*}BCA &= 2 \times 26^2 + 3 \times 26^1 + 1 \times 26^0 \\ &= 1456 \end{align*} Is it possible to get back to the string $BCA$, ...
1
vote
1answer
108 views

Simplifying a fraction through factoring

I have the following fraction: $\frac{a^3-8}{a^2+2a+4}$ Because the numerator is the difference of two cubes, I've factored it like this: $(a-2)(a^2+8a+64)$. The denumerator does not have natural ...
3
votes
4answers
371 views

Factoring $ac$ to factor $ax^2+bx+c$

I was watching a first-year high-school-algebra student struggle with factoring quadratics last night. Given a quadratic $ax^2+bx+c$ (I'll give you the exact example in a moment), her method — ...
5
votes
1answer
195 views

Is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$?

If $p\in\mathbb{N}$ is a prime, is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$? I've proved that any non-unit factor in $\mathbb{Z}[x]$ must have degree at least 2. Eisenstein's criterion doesn't ...
1
vote
2answers
112 views

Basic factoring problem.

I trying to work through a problem and have become stuck at the following equality: $ \sum_{n=1}^{100}{ n^2+n - 1 - (n-1)^2} = \sum_{n=1}^{100}{(3n - 2)}$ I can't quite get my head around the ...
3
votes
1answer
111 views

Factoring $y$ out of $y^2 + e^y$ possible?

I'm working on solving a separable first order differential equation where I get down to: $$.5y^2 + e^y = .5x^2 + e^{-x} + c$$ Is it possible to solve for $y$ here? I can't think of a way to factor ...
1
vote
1answer
119 views

Quick way to find the highest multiplicity of a divisor of a number?

Not sure if worded properly. For instance, the highest multiplicity of 2 in 60 is 2 because the prime factorization of 60 is 2^2*3*5. For 16, the highest multiplicity of 2 would be 4, etc. Is there ...
4
votes
2answers
712 views

Factorize polynomial over $GF(3)$

I want to factorize $x^{11}-1$ over $GF(3)$ but I'm stuck at $(x-1)(x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1).$ I have tried to do it trial and error but failed. Is $$ ...
1
vote
2answers
1k views

Notation : What is the meaning of the (mod n) in factoring algorithms?

Pretty much every thing is in the title, really! I'm trying to come up with an efficient algorithm to factorize large integer as an homework for a parallel programming course. I've seen a few pages ...
2
votes
0answers
88 views

Factors of a polynomial in several variables

Fix an embedding of $\overline{\mathbb{Q}}$ into $\mathbb{C}$. Suppose you have a polynomial in several variables, with algebraic coefficients: $P\in \overline{\mathbb{Q}}[z_1, \ldots, z_n]$. Also ...
4
votes
1answer
184 views

Help with basic high school math. What happens to $j$?

I know my math is very rusty, actually, its always been that way. but I need help with this. The question below has me stumped. I've tried to show the steps I went through to get the answer. Please ...
0
votes
2answers
127 views

What are the specifics and the possible outputs of Pollard's Rho algorithm?

I'm trying to implement a simple prime factorization program (for Project Euler), and want to be able to use Pollard's Rho algorithm. I read the Wikipedia, wolfram|alpha, and planet math explanations ...
31
votes
4answers
1k views

Could G. H. Hardy make a product of two primes so big he couldn't find out which?

This question of course began as a slightly irreverent play on the riddle, "Can God make a stone so big He can't lift it?" Then I wondered about the answer. Is it possible to exhibit a number that is ...
0
votes
1answer
87 views

Beginners question: Factorised expression, not sure how book got the answer

I have an expression: $2a^3 / 3a^2 * 6a^5$ and my book gives the answer: $4a^6$ I get where the $a^6$ is from but how is the $4$ worked out?
2
votes
3answers
127 views

How to express $\frac{x^3+4x^2-1}{(x^2+1)^2}$ as a polynomial plus a proper fraction, using long division?

I'm trying to express $$\dfrac{x^3+4x^2-1}{(x^2+1)^2}$$ as a polynomial plus a proper fraction, using long division but I don't know how to do that. It'd be cool if you can solve this. Thanks.
1
vote
1answer
162 views

What are the steps for factorizing $1 - ab - cd + abcd$?

I can see that $1 - ab - cd + abcd$ factors to $(1-cd)(1-ab)$ but only because I tried a lot of different factors in a trial and error method, so it took me a while. I was wondering what the pattern ...
2
votes
2answers
144 views

For $x_1,x_2,x_3\in\mathbb R$ that $x_1+x_2+x_3=0$ show that $\sum_{i=1}^{3}\frac{1}{x^2_i} =({\sum_{i=1}^{3}\frac{1}{x_i}})^2$

Show that if $ x_1,x_2,x_3 \in \mathbb{R}$ , and $x_1+x_2+x_3=0$ , we can say that: $$\sum_{i=1}^{3}\frac{1}{x^2_i} = \left({\sum_{i=1}^{3}\frac{1}{x_i}}\right)^2.$$
1
vote
2answers
499 views

equations with lcm and gcf

If $p$, $q$, and $r$ are three different odd prime numbers what are the $\operatorname{lcm} (2pq, 2pr, 2qr)$ and $\operatorname{gcf} (2pq, 2pr, 2qr)$. Anyone have any suggestions on how to even start ...
4
votes
4answers
1k views

How can we prove that among positive integers any number can have only one prime factorization?

I have read right from school that prime factorization is unique, but have never found proof for this. Can someone show me the proof?
0
votes
1answer
85 views

How to solve for x

$W(4d^2 (1-x^2)^2) = abc^3x \sqrt{(\pi^2 (i-x^2)^2 + 16 x^2) }$ I have to find x ,i have the values of all other constants , I tried to separate it using partial fraction but I am stuck. a=3 b=4 c=7 ...
3
votes
5answers
606 views

Number of Solutions of $3\cos^2(x)+\cos(x)-2=0$

I'm trying to figure out how many solutions there are for $$3\cos^2(x)+\cos(x)-2=0.$$ I can come up with at least two solutions I believe are correct, but I'm not sure if there is a third.
1
vote
1answer
147 views

Interesting prime factorization function divisibility problem [duplicate]

Possible Duplicate: Is the set of all numbers which divide a specific function of their prime factors, infinite? Let the function $f(n) =(p_1^{a+1}-1)(p_2^{b+1}-1)... $ where $n$ is an ...
5
votes
2answers
629 views

Why is integer factorization considered to be in NP if a quantum computer can compute a factorization in polynomial time?

Sorry if this seems off topic, the cstheory guys told me it was off topic over there, and sent me here. Shor's algorithm on a quantum computer can solve an integer factorization problem in polynomial ...
4
votes
3answers
987 views

Factors of a number?

I'm studying for the SAT on collegeboard.org, and I came across the problem: What is the least positive integer that has the same number of positive factors as 175? ...