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Questions tagged [factoring]

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

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Two reducible polynomials over $\mathbb Q[X]$

I`ve asked it here before, but figured out that there was a minor typo, and as I didnt received any answer I deleted my old question and I am posting again. I apologize if it is elementary, but here ...
Victor Hugo's user avatar
4 votes
2 answers
97 views

ACT practice test, aren't both $3$ and $12$ viable answers? [closed]

The question For which of the following values of $c$ will there be two distinct real solutions to the equation $5x^2+16x+c=0$? and the possible answers are:$\quad$ $\text{F}.\space3\\ \text{G}.\...
Ezra Nielsen's user avatar
7 votes
4 answers
2k views

Is there a generalization of factoring that can be extended to the Real numbers?

I simply mean that factoring integers is well understood, but factoring an irrational or any real number does not seem to make sense, especially taking into account that a large integer many ...
releseabe's user avatar
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0 answers
32 views

Algorithms for factorization of symbolic multivariate polynomials [duplicate]

I've been trying (but utterly failed) to find literature on how CAS implement factorization of symbolic polynomials. Everything I find relentlessly points back to stuff about rings and polynomials ...
UnkemptPanda's user avatar
-1 votes
1 answer
147 views

Factoring $N = pq$ for primes $p,q$ knowing $q\bmod (p-1)$ [closed]

Let $N=pq$ where $p$ and $q$ are both prime numbers. If we know the value $k = (q \bmod (p−1))\, $ can that help us to factor $N$? For example $N= 899=29\times 31$ and $\,k = 31 \bmod 28 = 3$.
Stableu Info's user avatar
0 votes
0 answers
38 views

Factoring bivariate quadratic polynomials.

I want to know if there is any simple method for factoring quadratic polynomials with two variables. Can anyone recommend me a good book from where I can learn about this topic?
Md. Faraaz Nasir Hosain's user avatar
-1 votes
0 answers
19 views

Methods for factoring multivariate polynomials. [duplicate]

Can anyone give me some ways to factorized a polynomial in the form of ax^2+2hxy+by^2+2gx+2fy+c ? Any reference book would also be appreciated.
Md. Faraaz Nasir Hosain's user avatar
1 vote
0 answers
57 views

Why do $n^4+4$ and $(n+2)^4+4$ have a largish common prime factor? [duplicate]

It seems that $n^4+4$ and $(n+2)^4+4$ always have a prime factor other than $2$ or $5$. For example: $94^4+4$ and $96^4+4$ are both divisible by $4513$; $15^4+4$ and $17^4+4$ are both divisible by 257....
rogerl's user avatar
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1 vote
1 answer
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Properties of Nth roots and fractional powers

Context: I'm programming an arbitrary precision math library and created some weird algorithms to calculate a number raised to non-integer powers due to optimizations. From my understanding, raising ...
jared soto's user avatar
0 votes
0 answers
34 views

Irreducible elements, prime elements, prime ideals, and maximal ideals [duplicate]

I'm trying to get the four concepts listed in the title straight in my mind and elucidate the relationship between them. Could someone check the following statements and let me know if they are all ...
Damalone's user avatar
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Can I use the slope of $\sqrt{N}$ to factor semiprimes? I get an equivalence if I try. How can I use related rates?

I need to make two definitions before I get to my question, which is at the bottom. Domain: is the X-axis (variable) Range: is the Y-axis $$f(Domain) = Range$$ I think semiprimes can be factored by ...
steveK's user avatar
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0 answers
61 views

What is range of $P$ values and range of $Q$ values for any given semiprime $N$?

Given $PQ = N$ where $P<Q$ and both $P$ and $Q$ are odd. I determined that the range of values for $P$ is: $\frac{\sqrt{N}}{2} < P \le \sqrt{N}$ and the range of values for $Q$ is: $\sqrt{N} <...
steveK's user avatar
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0 answers
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Using Rational Root Theorem on big numbers for polynomials

One of the things I have tried finding answers to, but have always bothered me is about Rational Root Theorem when being applied to problems such as factoring of polynomials such as: $k^3 + 6k^2 -...
Leonardo's user avatar
0 votes
0 answers
48 views

Division property of outer product? [duplicate]

$A = \{a_1,\cdots,a_m\}$ is a multiset consisting of $m$ non-negative real numbers, with at least one element $a_k > 0$. $B = \{b_1,\cdots,b_n\}$ and $C = \{c_1,\cdots,c_n\}$ are two multisets each ...
graphitump's user avatar
0 votes
1 answer
24 views

Is integral regular ring a UFD?

Let $A$ be a commutative ring such that it is regular and integral. It is known to all that for any prime ideal $\mathfrak{p}$ of $A$, $A_{\mathfrak{p}}$ is a UFD. My question is, whether $A$ itself ...
Vector's user avatar
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1 answer
55 views

Question on a proof of the zeros theorem

The book states the definition of the zeros theorem: Every polynomial of degree $n\geq1$ has exactly $n$ zeros, provided that a zero of multiplicity $k$ is counted $k$ times. Proof $\ $ Let $P$ be a ...
Guilherme Cintra's user avatar
1 vote
0 answers
69 views

Factorize the polynomial $p(x)=x^4+x^3+(1+i)x^2+(1-i)x+3i$. [closed]

I stumbled upon the question that 'Factorize the polynomial $$p(x)=x^4+x^3+(1+i)x^2+(1-i)x+3i$$ It is commonly known that $\mathbb C$ is algebraically closed. So, any polynomial has at least one ...
Fuat Ray's user avatar
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1 vote
0 answers
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How to see that "if the leading coefficient is 1 or -1, then the rational zeros must be factors of the constant term" in a polynomial

If p/q is a rational zero, in lowest terms, of a polynomial P, then to find its rational zero: $$a_n(\frac{p}{q})^n + a_{n-1}(\frac{p}{q})^{n-1} + \cdots + a_1(\frac{p}{q}) + a_0 = 0 \therefore$$ $$\...
Guilherme Cintra's user avatar
0 votes
1 answer
69 views

Reducible/Irreducible Polynomials in Ring Theory

I have this following exercise I've been trying to solve for a while now. We are supposed to study the irreducibility of the polynomial $A=X^4 +1$ in $\mathbb{Z}[X]$ and in $\mathbb{Z}/p \mathbb{Z}$ ...
Seramiti's user avatar
3 votes
1 answer
88 views

Does the existence of zero divisors imply an infinite factorization for some nonzero element?

We know that zero factors infinitely into nonzero nonunits in a ring with zero divisors: $$0=ab=a^2b=a^3b=\ldots$$ and nontrivial idempotents also factor infinitely into nonzero nonunits: $$e=e^2=e^3=...
kalanchloe's user avatar
0 votes
1 answer
57 views

Does it make sense to say a quintic equation has a "repeated quadratic root"?

Regarding the following question from a Further Maths textbook: The equation $2x^5+x^4+36x^3+18x^2+162x+81=0$ has a repeated quadratic root. a) Show that $x=3i$ is a solution of the equation b) Fully ...
Refnom95's user avatar
  • 325
0 votes
0 answers
116 views

find the general solution the recurrence equation $b_n = 3b_{n-1} - b_{n-3}$

here are the steps I have done to try and find the general solution of this relation: $$ b_n = 3b_{n-1} - b_{n-3}\\ = b^n = 3b^{n-1} - b^{n-3}$$ then divide by $b^{n-3}$ to get $$b^3 = 3b^2 - 1$$ then ...
sor3n's user avatar
  • 15
0 votes
2 answers
65 views

For $k\subset F \subset E$ algebraic field extensions, if "all" irreducible polynomials with a root in E factor in $F[x]$ then $F=E$?

Let $k \subseteq F\subseteq E$ be field extensions with $E$ algebraic extension over $k$. Suppose $\forall a \in E\backslash F$ the irreducible polynomial $p(x)$ of $a$ over $k$, factors non-trivially ...
Cezar's user avatar
  • 147
-3 votes
4 answers
112 views

Other methods of solving this question: finding $2y^4-8y^3-5y^2+26y-28$ for $y=1+\sqrt2+\sqrt3$ [closed]

If $y=1+\sqrt2+\sqrt3$ then find the value of $$2y^4-8y^3-5y^2+26y-28$$ There can be many ways to do this question. I would like to know the shorter approaches for this question that are clever to ...
MathStackexchangeIsMarvellous's user avatar
2 votes
1 answer
84 views

GCSE maths level - Is it normal for there to be two possibilities of factorisation for quadratics with a coefficient of x?

The problem and my working It would be useful to know, and its impossible to find anywhere easily on the internet. There also definitely could be a hole in my reasoning too. It was for factorising a ...
ruben alexander's user avatar
0 votes
1 answer
42 views

A Formula for the Norm in the Cyclotomic Field of Degree 5

So you all know and love, the Gaussian integers have a rather neat-looking norm: given $a+bi$, then $N(a+bi)=a^2 + b^2$. When it comes to the cyclotomic integers of degree 3, i.e. the domain $\mathbb{...
StormyTeacup's user avatar
  • 2,032
-2 votes
1 answer
138 views

If $f\in \mathbb{Q}[x,y]$ factors in some field extension of $\mathbb{Q}$, then it also factors in some finite extension of $\mathbb{Q}$? [closed]

If $f(x,y)\in \mathbb{Q}[x,y]$ factors non-trivially (as a product of two non-constant polynomials) in some field extension of $\mathbb{Q}$, then does it also factor non-trivially in some finite field ...
Cezar's user avatar
  • 147
0 votes
0 answers
38 views

Divisibility (algebraic expressions) - can this be generalised?

Consider the expression $a^{3}+b^{3}+c^{3}-3abc$. It is divisible by $(a+b+c)$ Now, consider the expression $a^{3}+b^{3}+c^{3}+d^{3}-3(abc+abd+acd+bcd)$ It is divisible by $(a+b+c+d)$ Can this be ...
Red Five's user avatar
  • 2,792
0 votes
1 answer
42 views

Bound of squarefree part of an integer

I am studying the paper DIOPHANTINE EQUATIONS OF THE FORM $F(X) = G(Y)$ - AN EXPOSITION which discusses the result of Erdos and Selfridge. I am unable to understand the highlighted statement ''Clearly ...
SARTHAK GUPTA's user avatar
0 votes
0 answers
24 views

Difference between polynomials with integer and non integer coefficiants regarding number of roots of polynomial [duplicate]

There is a prior question to show that polynomial of degree 7 with all integer coefficiants has 7 integer values $ P(x1,x2...x7)=+-1$ cant be expressed as product of two polynomials with integer ...
GreyCow's user avatar
  • 33
0 votes
0 answers
51 views

General formula for factoring $ax^2+bx+c$ [duplicate]

I watched a YouTube video on factoring (namely, 100 trinomial factoring (Dedicated to Mr. Hill) by blackpenredpen) and was curious about the math behind the method he first shows around 12:15. Is ...
voltedz's user avatar
0 votes
0 answers
104 views

Coefficients of some polynomials factorization

EDITED If anyone is interested in this question, now there are OEIS draft and repository with detailed data. Let $n,b\in\mathbb{N}$ and $P_{n_b}$ is polynomial whose coefficients are digits of $n$ ...
lesobrod's user avatar
  • 804
0 votes
0 answers
13 views

Factorizing $AMA^T+N=WW^T$ efficiently.

This question is related to this question with a slightly more general setting. A good speedup on this could improve performances of a decision process in multi-objective optimization that I designed. ...
P. Quinton's user avatar
  • 6,076
0 votes
1 answer
140 views

What is the relation between these two descriptions of General Number Field Sieve?

I am reading about General Number Field Sieve (GNFS) algorithm and its implementations, and the literature seems to have two different versions of the algorithm, but I can't find how these two are ...
Sil's user avatar
  • 17k
6 votes
2 answers
226 views

Find all $x \in \mathbb{R}$ such that $1+\sqrt{1-\frac{1}{x}}+\sqrt{x-\frac{1}{x}}=x^2$.

This problem was posed by a friend of mine. He's got it from a social media post. We don't know how to solve it, but there are similar types of questions in this website, whose techniques I've tried ...
asd3weqsda zfewrewgsfd's user avatar
2 votes
1 answer
88 views

How to factorize $2^{3^{k}}+1$

My question: How do we get $2^{3^{k}}+1=(2+1)\left(2^{2}-2+1\right)\left(2^{2 \cdot 3}-2^{3}+1\right) \ldots\left(2^{2 \cdot 3^{k-1}}-2^{3^{k-1}}+1\right)$$(k\ge 0)$ I know $2^{3^{k}}+1=(2+1)(2^{3^{k}-...
Student's user avatar
  • 101
-1 votes
3 answers
95 views

Factoring $x^2 + 4x -3$ step by step.

I have the following equation that I need to factor: $x^2 + 4x -3$. I cannot use the factoring by grouping method as there are no integers that add up to $4$ and give $-3$ when multiplied. What method ...
Paul's user avatar
  • 1
1 vote
2 answers
108 views

Factoring $x^2 - 2 + \frac1{x^2}$ by steps

$$x^2 - 2 + \frac1{x^2}$$ factors out to $$\left(x-\frac1x\right)^2$$ I know this because the answer was given to me in a video. If the answer wasn't given to me, I would have been stuck. I'm not ...
VennDiagram's user avatar
1 vote
1 answer
82 views

Dscriminant for a poynomial of degree higher than 3

I am dealing with a polynomial of degree 5, which I want to prove is positive on a certain interval. I found a statement on Wikipedia that exactly enable me to do this. There is no reference backing ...
Gateau au fromage's user avatar
1 vote
1 answer
77 views

How to find factors of a number that add to a certain sum?

For example, when finding the roots of this quadratic equation: $x^{2}+8x-9=0$ I would write $x^2-x+9x-9=0$ Then write that expression as the product of two linear expressions: $(x+9)(x-1)=0$ Then I ...
Will Fitchet's user avatar
4 votes
1 answer
150 views

Which type of "numbers" will require order 3 factors for its irreducible polynomial factorizations?

The complex numbers are famous for having the property of the fundamental theorem of algebra. All polynomials have at least one root. Together with being factor-able, this means we can factor every ...
mathreadler's user avatar
  • 26.1k
0 votes
1 answer
175 views

Product of two non-negative polynomials has coefficients 0, 1, or 2 [closed]

A 0-1-2 polynomial is a univariate polynomial where all coefficients are either $0$, $1$, or $2$. Is it true that if two real polynomials $P(x), Q(x)$, have their product equal to a 0-1-2 polynomial (...
mick's user avatar
  • 16.4k
0 votes
1 answer
29 views

Induction argument to show that $x-\lambda$ and $x-\bar{\lambda}$ appear the same number of times in the factorization of $p\in P(\mathbb{R})$?

Consider a polynomial $p\in\mathcal{P}(\mathbb{R})$ with a complex root $\lambda$. Since the coefficients of $p$ are real, we know that complex roots occur in conjugate pairs. We can write $p(x)$ as $$...
xoux's user avatar
  • 5,021
0 votes
3 answers
121 views

How to see $2n^2 + 7n +6 = (n+2)(2n+3)$?

In a proof I was reading of the sum of the first $n$ squares formula, the fact is used that $2n^2 + 7n +6 = (n+2)(2n+3)$. I was struggling with the exact logic used in deducing this relation- I see we ...
Princess Mia's user avatar
  • 3,019
1 vote
2 answers
114 views

Exact factorization of $a^m+b^m$ and $a^m-b^m$

Let $a,b$ be two indeterminates. I read here that $a^m + b^m$ is divisible by $(a+b)$ if $m$ is odd and $a^m - b^m$ is divisible by $(a+b)$ if $m$ is even. I am interested in an exact expression $f(a,...
motionart's user avatar
  • 158
0 votes
1 answer
63 views

Factoring bivariate polynomials

Consider the following bivariate polynomial $$P (x,y) = \sum_{m=0}^{2}\sum_{n=0}^{2} a_{m,n} x^{m} y^{n}$$ where the coefficients $a_{m,n}$ can be complex numbers. I was wondering if there are any ...
XZhao's user avatar
  • 3
0 votes
0 answers
76 views

How can this sinh curve be massaged?

Sorry I'm new here so I don't know how to use the math symbols yet but I would really appreciate some help with something I'm stuck on. How can this equation: $$ 𝑦(𝑥)=𝑥 \sinh(\sinh^{-1}(𝑦_0)−\rho \...
Mr_Ryder's user avatar
0 votes
0 answers
54 views

Factorization of $f(x)=x$ in $\Bbb{Z}/n\Bbb{Z}[x]$ where $n$ is the product of $k$ distinct primes (considering $\Bbb{Z}/30\Bbb{Z}[x]$ as example)

Intro This is the final part of the problem 9.4.20 (e), (d) from Dummit and Foote's Abstract Algebra. They formulate it as "Determine all the factorizations of $f(x)=x$ in $\Bbb{Z}/n\Bbb{Z}[x]$ ...
Stanarth's user avatar
  • 116
1 vote
3 answers
163 views

Factoring a year to get a date

Let's factor 2024: $2024=8\times 11\times 23$. So we can make a date: 8/11/23, or August 11, 2023. Or we could also make 11/8/23, November 8, 2023 (if you live in Europe, they will be switched). ...
Valerio's user avatar
  • 340
0 votes
0 answers
59 views

Factoring integers by finding nontrivial divisors

Most integer factoring algorithms work by finding a nontrivial divisor of their input. In order to fully factor a given input, algorithms recursively apply the procedure for finding a nontrivial ...
talbi's user avatar
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