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

learn more… | top users | synonyms

0
votes
3answers
55 views

Setting up word problem for finding length and width

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width. I have not idea where to start. ...
0
votes
1answer
48 views

Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
0
votes
4answers
66 views

How can I factor $x^2 + 2\sqrt{3}\,x + 3$? [closed]

$$x^2 + 2\sqrt{3}\,x + 3$$ Anyone could tell me how may I factor this? Thanks a lot
2
votes
3answers
135 views

When to use factoring by grouping for quadratic equation

There are several ways/methods to perform factoring. I am revising factoring at KhanAcademy, there are factoring by grouping, factoring special product and factoring difference of squares. Although, ...
0
votes
1answer
31 views

remainder is not zero using long division method

Find all zeros of $f(x)=128x^3-48x^2+1$ given that one linear factor occurs twice. let $f(x) $ be equaal to 0 $128x^3-48x^2+1=0,$ $16x^2(8x-3)+1=0,$ trying $x=1/4$ $16/16(2-3)+1=0,$ ...
1
vote
1answer
115 views

Factoring $2x^5+13x^4+50x^3+82x^2+56x+13$

Express $2x^5+13x^4+50x^3+82x^2+56x+13$ as a product of five linear factors. The roots of the polynomial may be real or complex. I had to employ the technique of synthetic division iteratively. ...
0
votes
2answers
64 views

Factoring equation with 4 cubed numbers

The problem is to factor $a^3x - b^3y + b^3x - a^3y$, and the answer is $$(x-y)(a+b)(a^2 - ab + b^2).$$ I got as far as $(x-y)(a^3 - b^3 + b^3 - a^3)$. I mean the above answer fits if it was just ...
6
votes
1answer
63 views

Lattice-Theoretic Interpretation of the Fundamental Theorem of Arithmetic

When equipping $\mathbb{N}^\ast=\mathbb{N}\setminus \{0\}$ with the divisibility relation, it forms a lattice with minimum 1, supremum given by the least common multiple, and infimum given by the ...
0
votes
1answer
60 views

Factoring $x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2$

The subject line pretty much says it all. In my geometry class today, the following equation came up: $$x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2 = 0$$ Specifically, it was in the ...
2
votes
1answer
84 views

What do we know about $\displaystyle \frac{f}{\gcd(f,f')}$ if $f\in\mathbb{F}_{p^d}[X]$?

Let $\mathbb{K}=\mathbb{F}_{p^d}$ and $f\in\mathbb{K}[X]$ be a non-constant polynomial with the factorization $$f=\prod_{i=1}^nf_i^{k_i}$$ where $f_i\in\mathbb{K}[X]$ is irreducible and ...
0
votes
0answers
16 views

Coppersmith method for factorisation

Is anyone familiar with the Coppersmith method? Does anyone know how is the $3\times3$ basis matrix obtained in this case?
2
votes
3answers
102 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
vote
1answer
31 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
42 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
vote
1answer
60 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
vote
0answers
21 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
vote
2answers
40 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
votes
6answers
53 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
73 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
vote
3answers
89 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
24 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
vote
1answer
42 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
75 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
votes
1answer
27 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
vote
3answers
189 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 ...
0
votes
4answers
50 views

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

Some hints on factoring the above expression please
5
votes
5answers
306 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
107 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
vote
1answer
28 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
22 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
143 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
vote
0answers
24 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
votes
1answer
48 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.
9
votes
0answers
237 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
22 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
49 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
32 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 ...
10
votes
3answers
186 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
89 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
18 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 ...
1
vote
1answer
15 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
38 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
54 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
95 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?
1
vote
2answers
104 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
102 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
55 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
67 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
votes
1answer
52 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 ...
1
vote
3answers
61 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 ...