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

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2
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3answers
115 views

Algebra (not so simple) Factoring

I got stuck on this problem from my Math Challenge II Algebra Class: Factorize the following: $$(x^2+xy+y^2)^2-4xy(x^2+y^2)$$ Hint: Let $u=x+y$ and $v=xy$. Here's what I did: ...
1
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1answer
39 views

On the existence of polynomial roots

Assume $F$ is a field, and $f\in F[x]$ is polynomial. To see that $f$ has a root in some extension of $F$, without loss of generality we can assume $f$ is irreducible. Indeed any polynomial $f$ is ...
0
votes
1answer
54 views

Finding greatest common divisor between two polynomials.

I have the following past exam question: Calculate $\operatorname{gcd}(x^3 + 2x^2 + 2,2x^2 + 1)$ in $\mathbb{F}_3$ Now I haven't encountered this sort of gcd before(usually I am trying to solve ...
1
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1answer
69 views

Factorial simplification

How can I work with this? $$\frac{(3n)!}{(3(n+1))!}$$ I really don't know how to open this fatorial and then, simplify it. Actually, I have to calculate the limit when $n\to\infty$. Thanks :)
1
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0answers
24 views

Finding coefficients of the min polynomial of an $n\times n$

Given an $n\times n$ matrix, for ease assume this matrix is over the $F_m$. What we know about min poly is the the non-zero components of the min polynomial for this case, ie if there is $x^2$, or ...
1
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2answers
42 views

Factorising after adding a square

I have been thinking about it for quite some time but am unable to find an answer. Let $a,b,c,d,e$ be any distinct natural numbers. Will the relation : $(x-a)(x-b)+c^2=(x-d)(x-e)$ ever hold? I am ...
-1
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6answers
55 views

Polynomial factors involving inequalities

How to factorise the polynomial $p(x) = x^4-2x^3 + 2x - 1$. Hence, solve the inequality $p(x) \gt 0$ ?
3
votes
2answers
78 views

Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
1
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3answers
104 views

How would I factor $a^3+b^3+c^3-6abc$

How would I factor the polynomial $a^3+b^3+c^3-6abc$? The values are homogenous, so so must be the factors. I don't know where to go from there.
2
votes
1answer
53 views

Polynomial factorisation on integers modulo n

Is there a known (efficient) algorithm to compute the list of factors of a polynomial modulo $n$ (for any integer $n$)? For example in $\mathbb Z_8$, $X^2+2X$ has a list of 4 factors (multiplicity 1 ...
1
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1answer
59 views

Any simpler way to do Pollard's p-1 method?

I found calculating factorization by Pollard's p-1 method is almost impossible if use a conventional scientific calculator. For example, I am trying to factor ...
2
votes
1answer
98 views

Expanding Square Roots, Why No Negative?

I haven't thought through algebra in a while and the last explanation I received of this seemed arbitrary. I hope I can get some clarification here. I understand that $\sqrt{+a} = \pm b$. Here's ...
0
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1answer
41 views

Factoring Polynomial Questions

How do you decide whether to use synthetic division or the factor theorem to help you factor a polynomial? Please help me answer.
1
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3answers
486 views

Can any factor pairs (where pair is unique) have the same sum?

I have used Stack Overflow but I'm new to this site so I apologize if this is a trivial question. I am creating mathematics software using javascript. I am using a for loop to find all factors for a ...
-1
votes
5answers
71 views

Help in factoring $x(y^2+z^2) + y(z^2+x^2) + x(y^2+z^2) + 3xyz$ [closed]

Some hints on factoring the above expression please
5
votes
5answers
315 views

Factoring in the derivative of a rational function

Given that $$ f(x) = \frac{x}{1+x^2} $$ I have to find $$\frac{f(x) - f(a)}{x-a}$$ So some progressing shows that: $$ \frac{\left(\frac{x}{1+x^2}\right) - \left(\frac{a}{1+a^2}\right)}{x-a} = ...
5
votes
1answer
126 views

Is 292229292292 the longest 29-smooth number made of 2's and 9's?

Is 292229292292 the longest 29-smooth number made of 2's and 9's? The factorization is $2^2 7^8*19*23*29$. Is there a general way to find other numbers of this sort without resorting to brute force ...
1
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1answer
29 views

Help with a technique in factoring a polynomial of four terms and two variables

I could not simplify this expression into factors despite the time I put into it: $x^3-64y^3-24xy-8$ I really want to learn how to do this well. Thank you very much for insights/techniques which you ...
0
votes
1answer
24 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
6
votes
1answer
169 views

Factorise $x^4 + 3x^2+ 6x+ 10$

I need to factorise $x^4 + 3x^2 + 6x + 10$ completely over $\mathbb{Q}$. I am not sure how to do this. I can't find any roots of this equation in $\mathbb{Z}$.
1
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0answers
26 views

Factorisation algorithm for polynomials in several variables over $\mathbb{Z}$.

What algorithm is used by a CAS to decide how to factor a polynomial in several variables over $\mathbb Z$?
0
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1answer
50 views

Is there a name for numbers that have 2 as their greatest common divisor?

Is there a name for numbers that have two as their greatest common divisor? Such as 8 and 130.
12
votes
0answers
397 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
0
votes
1answer
27 views

How do I work out the last sentence in this section of a proof of the Unique Factorization Theorem?

The last sentence states that the number of possibilities is $2\log_2 n$ (see the below image to follow the proof). I don't understand how to get $2\log_2 n$ but I understand everything that comes ...
1
vote
1answer
52 views

Power Factoring Contest Question

The question was as follows: Compute the smallest positive integer $n$ such that $n^n$ has at least $1,000,000$ positive divisors. I did some work, finding that if $n=2^a*3^b*5^c*7^d$ then the $n^n= ...
0
votes
0answers
55 views

Calculating all possible combinations to reach a specified integer by multiplying three other integers together

I have an integer $n$ and I need to find a way to calculate all the possible ways that $n$ can be reached by multiplying three other integers ($a, b, c$) together to reach $abc=n$ What would be the ...
11
votes
3answers
231 views

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

If $\begin{cases}a^4+b^4\in\mathbb Q\\ a^3+b^3\in\mathbb Q\\ a^2+b^2\in\mathbb Q\end{cases}$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$. It is given that $a,b\in\mathbb R$. The proof of ...
2
votes
1answer
96 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
0
votes
1answer
32 views

Find subset of rows whose entries sum to an even number in each column

I am trying to implement Fermat factorization with factor bases. The textbook suggests using row-reduction to find a linearly dependent set of rows. How does one go about finding such a linearly ...
14
votes
1answer
886 views

Finding smallest and largest prime factor of $\frac{200!}{180!}$

I originally started off by listing all the primes: $p<200$ then trying to calculate the prime factorisation of each (which I realise is a silly thing to do) I believe there must be a simpler way ...
1
vote
1answer
17 views

Using information from a congruence to factor a number

I am being asked to factor $15347$ given that $7331^2 \equiv 1460^2 \pmod{15347}$. I've tried playing around with each of the numbers -- prime factorization, gcd, lcm, etc., but I can't find a ...
3
votes
2answers
54 views

$a,b,c$ are three distinct natural numbers. Then how many ordered triplets $(a,b,c)$ will exist such that L.C.M (a,b,c) = 144.

Let $a,b,c$ be three distinct natural numbers. Then how many ordered triplets $(a,b,c)$ will exist such that L.C.M (a,b,c) = 144. Here's how I proceeded, 144=$(2^4)(3^2)$, so 144 has 15 factors(1 ...
0
votes
2answers
60 views

Cubic factoring question

I'm trying to figure out how a colleague factored an expression. I don't get how: $$a^3+a^2b-(b+1)=(a-1)[a^2+a(b+1)+(b+1)]$$ Multiplying the result I see it's true, but not sure how he got there..is ...
2
votes
2answers
115 views

Can fractions be relatively prime?

Two numbers are relatively prime if they do not share any factors, other than 1. Is it possible for fractions to be relatively prime? To reword this, do fractions even have factors?
2
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3answers
142 views

Using telescoping property to prove difference of powers

Ok so I have started working through Apostol calculus and as you can see I am stuck. The problem is that I can not see the telescoping pattern anywhere for following problem. Prove that $$a^n - b^n ...
1
vote
3answers
108 views

How to find sum of factors of $2^{2012}$?

This question really is confusing me and I was wondering if there was a simple way this could be achieved. I've come up with this so far after skimming through a few articles on the net. I assumed ...
1
vote
1answer
85 views

Efficient factorization of numbers with unique prime factors

I need a factorization algorithm for numbers of the form $n = p_{1}p_{2}\cdots p_{k}$ with $p_i \neq p_j$ for $i \neq j$ and $p_j \in \{p : p \mbox{ is a prime and } p \leq P_s\}$, where $P_s$ is the ...
0
votes
1answer
83 views

Given a polynomial of degree 5, get minimum and maximum without using derivatives

Given a quintic polynomial (in my case, $x^5+2x^4+16x-32$), I am supposed to get its maximum and minimum value for the interval $I=[-2;2]$ without using the derivative of the corresponing polynomial ...
0
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1answer
59 views

WordProblem on factors and remainder theorem

Mr.Chaalu while travelling by Ferry queen has travelled the distance one Kilometer more, than the fare he paid per km. Initially he had total amount of Rs.350/- in his wallet. Now he is only left with ...
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3answers
64 views

Can anyone factor this?

-x^3 + 12x + 16. I am trying to solve for the zeros, but it seems that I have forgotten all my neat little tricks. Not a difference of cubes, or any of the common forumlas. I'm thinking maybe some ...
6
votes
3answers
522 views

Smallest known unfactored composite number?

I'm trying to find examples of "small" numbers which are known to be composite, but for which no prime factors are known. According to this website the number $109!+1$ is a composite number of 177 ...
0
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2answers
40 views

Factorise A number in to product of two numbers

I would like to know what is the quickest way to factorise large number ( more than 1000) in to two numbers For Example 2669 2669 is 17 * 157 how can I find this ?
0
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1answer
76 views

Trigonometric Functions using factorisation and small angle identities

The function f(x)=sin3x - sin2x + sins is defined for the domain 0 $\le$ x $\le$ $\frac{\pi}{2}$. a) By method of factorisation, show that f(x) = sin2x(2cosx - 1). b) Hence solve the equation f(x) ...
0
votes
2answers
38 views

Factorisation of a polynomial [closed]

I have a polynomial $$t^4-4\lambda t^2-4t^2 $$ I need to give a real value to $\lambda$ such that i get 4 real roots.
11
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5answers
452 views

How did Euler realize $x^4-4x^3+2x^2+4x+4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$?

How did Euler find this factorization? $$\small x^4 − 4x^3 + 2x^2 + 4x + 4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$$ where $\alpha = \sqrt{4+2\sqrt{7}}$ I know that ...
0
votes
4answers
62 views

How do I compute the sum of 2 squares

if $x+y=a$ and $xy=b$, what does $x^3+y^3$ equal? I understand that $x^3+y^3=(x+y)(x^2-xy+y^2)$ but I don't see how I can figure out what $x^2$ or $y^2$ equals
1
vote
2answers
151 views

Factor irreducible polynomial in Z[x] and R[x]

I've got a couple of problems from an old exam in abstract algebra that I have difficulty in understanding. 1) Write the polynomial $2x^3 - 10$ as a product of irreducible elements in ...
1
vote
1answer
45 views

Factoring and Simplifying

I'm trying to do this problem, $$(4x + 1)^{15}\cdot\frac{1}{3}(12x - 5)^{-\frac{2}{3}}\cdot 12 + (12x - 5)^{\frac{1}{3}}\cdot15(4x + 1)^{14}\cdot 4$$ I've gotten down to, ...
3
votes
3answers
192 views

Factors of zero?

How many integer factors of 0 are there, and what are they? I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?
1
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1answer
43 views

Probability distribution of count of factors for all numbers

Is the following known? Define "factor count" as the number of prime factors of the number, minus 1. For example: Prime numbers have a factor count of 1-1 = 0 4 has a factor count of (2 and 2)-1 = ...