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

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1
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3answers
65 views

How to factorise $(x-1)^2 - (x-5)^2$

My attempt: $a = (x-1)$ $c = (x-5)$ $a^2 - c^2$ which is equal to: $$((x-1) - (x-5))((x-1)+(x-5))$$ But the correct answer is : $8(x-3)$ Can you explain, please?
0
votes
0answers
29 views

factors sum to 1

If I have factors of linear operators say $$(a_1 + A)(a_2 + A)(a_3 + A)\cdots(a_n + A) = 0$$ $A$ being an linear operator(i guess it really doesn't matter its operator or not) why $$\sum_{n} \frac{...
0
votes
5answers
52 views

Formula for factorization of a Quadratic Equation?

To be clear I am looking for an equation to go from $$Ax^2 + Bx + C = 0$$ To $$(Dx + E)(Fx + G) = 0$$ And I need it to be able to be done in a computer as it will be going in my app. Thanks in ...
11
votes
7answers
745 views

Factoring polynomials with a 2nd degree coefficient greater than $1$

I'm learning how to factor polynomials, but I'm having a hard time understanding the approach when the 2nd degree coefficient is greater than $1$. For example, when I begin to factor $12k^4 + 22k^3 - ...
0
votes
1answer
27 views

Factoring Polynomials: How do I express the area and perimeter in factored form?

Our topic is factoring polynomials, and I can't seem to solve this question: Express the area and perimeter of the shaded region in factored form. We've discussed how to solve for the ...
17
votes
4answers
2k views

Have I found all the numbers less than 50,000 with exactly 11 divisors?

The math problem I am trying to solve is to find all positive integers that meet these two conditions: have exactly 11 divisors are less than 50,000 My starting point is a number with exactly 11 ...
1
vote
2answers
37 views

assuring factorization for R[x] when R is a UFD

I wanted to ask, suppose the ring $R$ is a UFD (Unique factorization domain) and I look at $R[x]$, the ring of polynomials over $R$. I wanted to know, how can I assure that when I have some polynomial ...
-5
votes
2answers
44 views

$240x^2y^3 - 180x^3y^4$ - Factor the given. [closed]

$240x^2y^3 - 180x^3y^4$ Question: Please Factor the Given Expression. Thanks. In factoring... Is it similar to the Factor Tree? Instead do we not use the tree? My Answer (attempt 1) $15xy^2 (16xy-...
0
votes
3answers
65 views

Prove there are infinitely many fields in which the polynomial is reducable

Polynomial: $$ t^7+t^2+1$$ I have the solution, but I don't understand the thinking behind it. How are they coming up with the factorization, and specific values of $t$ (shown in solution). The ...
2
votes
0answers
33 views

factoring polynomials in ring of integers modulo powerful number

I am having trouble finding info on how to factor polynomials in ring of integers modulo powerful number. For example: $x^2 - 1$ in $\textbf Z_{8}$. I know by tinkering around that $(x - 1)(x + 1)$...
2
votes
1answer
39 views

$a$ and $b$ are factors of $6^6$ and $a$ is a factor of $b$

How many pairs of ($a$,$b$) of positive integers are there such that $a$ and $b$ are factors of $6^6$ and $a$ is a factor of $b$? What I tried I know $6^6$ an be broken down into $(2)^6 (3)^6$ If $...
0
votes
0answers
44 views

Irreducibility of sums of two polynomials

I'm interested in a special type of polynomial factorization over $\mathbb {Q} $: testing the irreducibility of $f(x)+g(x)$, where $f$ and $g $ are relatively prime and $\text {deg}(f)<\text{deg}(g)...
3
votes
0answers
71 views

Can we continually factorize an expression like $x+y$?

I have a question that, for lack of familiarity or understanding of the relevant fields, I'm not quite sure how to formulate, so I'll just start off with an example and list some questions as I go. As ...
5
votes
6answers
116 views

Factoring out a $7$ from $3^{35}-5$?

Please Note: My main concern now is how to factor $7$ from $3^{35}-5$ using Algebraic techniques, not how to solve the problem itself; the motivation is just for background. Motivation: I was trying ...
3
votes
1answer
25 views

What is the complete (polynomial) factorization of $\sigma(p^k)$, where $p$ is prime with $p \equiv k \equiv 1 \pmod 4$?

The title says it all. What is the complete (polynomial) factorization of $\sigma(p^k)$, where $p$ is prime with $p \equiv k \equiv 1 \pmod 4$? Here, $\sigma = \sigma_{1}$ is the classical sum-...
4
votes
4answers
98 views

Factoring $x^4-11x^2y^2+y^4$

I am brushing up on my precalculus and was wondering how to factor the expression $$ x^4-11x^2y^2+y^4 $$ Thanks for any help!
4
votes
1answer
32 views

On smoothness assumptions in Integer factorization

I have came across a lot of factorization methods and most of them seem to assume smoothness of some numbers. For example When $p-1$ is smooth When $|E(\mathbb{F}_p)|$ is smooth. (Elliptic curve ...
1
vote
1answer
30 views

Find a factorable cubic polynomial with given conditions

I want to write a factorable cubic polynomial in the form $Ax^3+Bx^2+Cx+D$ where $-D = 42C$. $A, B, C,$ and $D$ should be nonzero integers. Is this possible?
1
vote
1answer
35 views

factorize $x^5+ax^3+bx^2+cx+d$ if $d^2+cb^2=abd$

I want to factorize $x^5+ax^3+bx^2+cx+d$ if $d^2+cb^2=abd$ but I don't know how to use the second equality.I tried a lot but I cannot know how to use it for example it is $d^2$ but we have $d$ and if ...
2
votes
0answers
50 views

Factoring $x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$ over $\mathbb{Q}$

For a quntic polynomial to be reducible to the following form over $\mathbb{Q}$: $$x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$$ We need to match the coefficients ($a=B$ obviously, so we ...
2
votes
5answers
100 views

Factorize a third degree polynomial

It's my first time posting here so I'm not used to describing my problem in mathematics. I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but ...
1
vote
4answers
75 views

Why is $-32^{\frac{1}{5}} = 2$

When you factorize $-32$, you get: $-32 = (-16) \cdot 2$ $-16 = (-8) \cdot 2$ $-8 = (-4) \cdot 2$ $-4 = (-2) \cdot 2$ $-32^{\frac{1}{5}} = -2$ The reason I am asking is because you get $-4 = -2 \...
0
votes
1answer
27 views

Factoring GCF from squared quantities

hope you're all well. Quick question: Am I allowed to factor this the way I did here? Thanks for the help, as always! :)
1
vote
2answers
57 views

$f(x+a)$ irreducibility means $f(x)$ irreducibility

Let $a~\in~\mathbb{Z}$ and let $f(x)~\in~\mathbb{Z}\left[x\right]$. Suppose that $f(x+a)$ is irreducible over $\mathbb{Z}$. Prove that $f(x)$ is irreducible over $\mathbb{Z}$. My idea is: $f(x)=u(x)*...
0
votes
0answers
25 views

Factoring polynomials when roots are external to the ring

I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question. It is clear to ...
0
votes
1answer
11 views

Why is the running time of the trial division $O(f \cdot (log N)^2)$?

I saw this being cited in a few paper,but none of them seems to explain why this is the case. Maybe because it is quite trivial, but I am not sure why exactly... Here $f$ is the size of the factor. I ...
2
votes
0answers
34 views

On equivalence of RSA and factoring [duplicate]

Suppose we are given a number "$A$" which is multiple of $\phi(n)$. One can assume factorization to be hard. So you cannot find exact value of $\phi(n)$ from $A$. Clearly using this we can crack ...
-2
votes
1answer
30 views

What is the algorithm to factor something like $2+\frac{1}{x}+x?$ [duplicate]

I came across this in homework but I'm interested in the general example, say $ax+bx^{-1}+c.$
1
vote
2answers
51 views

Factoring $p(x) = x^n -1$ for any natural number $n$

Can I say that from inspection, $(x-1)$ is a factor which implies that $p(1) = 0$. I then used long division which then gave this: $p(x) = (x-1)(\sum \limits _{i=1} ^{n} x^{n-i})$ How would you have ...
1
vote
0answers
15 views

Is it easy to factor if we know $k\phi(PQ)$?

Suppose we know $k\phi(N)=k\phi(PQ)=k(P-1)(Q-1)$ where in $N=PQ$ we have $P,Q$ being similar sized primes and $k\in\Bbb Z$ is unknown can we factor $N$ in polynomial time?
1
vote
1answer
17 views

Prove that $a(x)$ divides $(v(x) - t(x))$

"Let $a(x), b(x) \in \mathbb{R}[x]$, not both the zero polynomial and suppose that gcd[$a(x), b(x)$] = 1. Let $u(x), v(x) \in \mathbb{R}[x]$ be such that $a(x)u(x) + b(x)v(x) = 1$ Let also $s(x)t(x) ...
0
votes
2answers
16 views

Prove that $q(x)$ does not divide $p_k(x)$

Let $n \in \mathbb{N}$ and let $p_1(x), p_2(x), ... *p_n(x)$ be $n$ irreducible polynomials over $\mathbb{R}$. Define the polynomial $p(x) = p_1(x) * p_2(x) *... *p_n(x) + 1 $ where 1 is the constant ...
1
vote
2answers
70 views

Factoring the sequence ${10}^{2n}+10^{n}+1$

While I am waiting for the basketball NBA game between Cleveland Cavaliers and Golden State Warriors to begin I sort of played with the sequence $a_n={10}^{2n}+10^{n}+1$ in a way that I looked for the ...
2
votes
1answer
40 views

Books about multivariate polynomials

I'm looking for a book on multivariate polynomials, preferably a monograph (could also be a chapter inside another book). I'm interested in what can be said about roots, factoring, irreducibility, ...
4
votes
1answer
41 views

How many positive two-digit integers have exactly 8 positive factors?

I solved this problem by listing all two-digit integers and going through each one. Is there an easy way to solve the problem? How many positive two-digit integers have exactly 8 positive factors?
2
votes
1answer
73 views

Finding 8 co-primes $\le 2^n$

We can find 8 co-prime integers $\le 2^n$ for sufficiently large $n$. I'm looking for asymptotic bounds for the minimum distance away from $2^n$ we have to go before finding 8 co-primes. In other ...
0
votes
1answer
36 views

Hard time factoring Normal Distribution based on transformation problem.

My professor gave problems out to practice for our final on Wednesday. This problem is based on the transformation of two random variables. It a 5 part problem, so I will list the necessary portions ...
3
votes
3answers
61 views

What is the sum of the prime factors of $2^{16}-1$?

I know $2^{10}=1024$ and $2^6=64$, but it seems they are not very helpful in solving this problem. There must be a trick to solve the problem in an easy way. What is the sum of the prime factors ...
0
votes
0answers
12 views

Odd factorisation of Indicies?

I'm sorry for the terrible formatting, I'm sorta new so I don't know how to use MathJax very well :( see below for my first attempt :) I came across the indical expression : $$\frac{x-1}{x-x^{1/2}-2}$...
2
votes
2answers
37 views

How to divide the following polynomial and factor it?

The question is $$ (2x^3+3x^2-39x-20) / (x-4) $$ I divided the following and got this as the answer $$ 2x^2+9x+3-8/(x-4))$$ I thought that this was the answer, but when i looked at the answer sheet ...
0
votes
1answer
58 views

Proof that $a^{n}+b^{n}$ is irreducible over $\mathbb Q$

The sum of fourth powers cannot be factored over $\mathbb Q$, since $ a^4+b^4 = (a^2+\sqrt{2}ab+b^2)(a^2-\sqrt{2}ab+b^2)$ And these quadratic factors does not have any real rational factors. How ...
1
vote
2answers
72 views

How to factor the trinomial : $ xy-x+y-1$?

How to factor the trinomial : $ xy-x+y-1$ ? The factorization is $(x+1)(y-1) $ but I don't where it comes from.
0
votes
3answers
50 views

Factorization of the polynomial: $m^2+3m^2n^2-30n^2-10$ [closed]

How can we factor the polynomial $m^2+3m^2n^2-30n^2-10$ ?
1
vote
2answers
51 views

Is $X^5+…+1 \in \mathbb{F_2}[X]$ irreducible?

I am trying to determine if the following polynomials are irreducible in $\mathbb{F_2}[X]$ are irreducible: $f(X)=X^5+X^2+1$ $g(X)=X^5+X^3+1$ There are no linear factors since $f(0)=f(1)=g(...
1
vote
0answers
24 views

Analog of Euler's Factorization method

One of Euler's discoveries was if an integer $n$ can be represented as a sum of two squares in two distinct ways, then one can factor $n$ explicitly. Of course, the method was ineffective as an ...
12
votes
0answers
181 views

Have I discovered an analytic function allowing quick factorization?

So I have this apparently smooth, parametrized function: The function has a single parameter $ m $ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart ...
0
votes
1answer
33 views

Factor polynomial into linear factors with complex coefficients.

Question: A polynomial is given. $(a)$ Factor it into linear and irreducible quadratic factors with real coefficients. $(b)$ Factor it completely into linear factors with complex coefficients. $x^3 - ...
1
vote
2answers
42 views

Stuck on a simple factoring problem

The answer to this question is probably very obvious but I can't figure it out for some reason: I simply want to factorise: $x^2+5x-2$ I solve $x^2+5x-2 = 0$ i find $x_1 = \dfrac{-5-\sqrt{33}}{2}$ ...
-4
votes
3answers
46 views

Factoring $12e^{2x} - 32e^x + 16$ [closed]

Can you help me solve the quadratic equation $12e^{2x}-32e^x+16$ by factoring please?
1
vote
1answer
25 views

Probability that a random polynomial over a finite field can be factorized to linear terms.

Suppose that $f\in\mathbb{F}_p[x]$ is a degree $d$ random univariate polynomial with coefficients from a finite field $\mathbb{F}_p$. What is the probability that $f$ can be written as: $$f(x)=\prod_{...