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

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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 ...
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0answers
113 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 ...
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1answer
23 views

Factor out $(m+2)$ in the following equation $(m+1)(m+2)+2(m+2)$

$(m+1)(m+2)+2(m+2)$ I really needed hints here, I am thinking to start at first two paragraphs and so on. Is my thought correct? Hints will be much appreciated.
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3answers
195 views

The ultimate formula to factor them all.

Context I am working on Integer factorization problem, I found a formula for factoring numbers, and I need your help to simplify it. First I will explain how I get there and then I present the ...
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2answers
33 views

Finding value of $m$ such that such that the polynomial is factorized

A polynomial $2x^2+mxy+3y^2-5y-2$ Find the value of $m$ much that $p(xy)$ can be factorized into two linear factors
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2answers
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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 $x_1 = \dfrac{-5-\sqrt{33}}{2}$ ...
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1answer
21 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 - ...
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Factoring $12e^{2x} - 32e^x + 16$ [on hold]

Can you help me solve the quadratic equation $12e^{2x}-32e^x+16$ by factoring please?
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1answer
22 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: ...
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45 views

The irreducibility of $a^{4n}+b^{4n}$ [closed]

How to prove that $a^{4n}+b^{4n} $, for any natural number $ n $, is irreducible over the rationals?
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1answer
43 views

Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? \begin{equation} f\left(a ...
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12 views

On factoring given $PQ-1$ has small factors.

Suppose we have an RSA number $PQ$ where $PQ-1$ has small factors. Will this give any advantage to factor $PQ$?
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2answers
64 views

Factoring a degree 4 polynomial without power of 2 term

For my hobby, I'm trying to solve $x$ for $ax^4 + bx^3 + dx + e = 0$. (note there's no $x^2$) I hope there is a simple solution. I'm trying to write it as $(fx + g)(hx^3+i) = 0$ It follows that ...
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3answers
70 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...
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1answer
31 views

Notation for separating out factors of a number

I have an integer (let's call it $n$), and I want to define it as the product of two values: one that's a pure power of two, and another that is odd. Obviously, these two values are unique for a ...
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2answers
37 views

Factor out (m+1) in the following so that the final answer is $\frac{(2m+1) (m+2) (m+1)} {6}$

Question: $\frac{m (m+1) (2m+1) + 6(m+1)^2}{6}$=$\frac{(2m+3)(m+2)(m+1)}{6}$ I must multiply by 6 on both sides and expand the brackets and collect like terms. I'm I correct? Edit notes: The ...
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1answer
47 views

Factor Completely.

Again, this question if from my final practice exam. Factor Completely. $$81x^4-256y^4$$ I'm able to get this far, How do I know which of the two factors should be factored further. ...
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1answer
59 views

How to factor $a^3 - b^3$?

I know the answer is $(a - b)(a^2 + ab + b^2)$, but how do I arrive there? The example in the book I'm following somehow broke down $a^3 - b^3$ into $a^3 - (a^2)b + (a^2)b - a(b^2) + a(b^2) - b^3$ and ...
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1answer
28 views

Interesting 4th order factoring question

$$ A = \frac{(4\cdot2^4 + 1)(4\cdot4^4 + 1)(4\cdot6^4 + 1)}{(4\cdot1^4 + 1)(4\cdot3^4 + 1)(4\cdot7^4 + 1)}$$ What is the value of $ \dfrac{113A}{61}$ ? So i tried factoring this ...
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4answers
49 views

Taking factors out of this integral?

In the integral: $$\int\frac{-25}{17(2t+3)} + \frac{37}{17(5t-1)} dt$$ Why is the final answer: $$-\frac{25}{34}ln|2t + 3| + \frac{37}{85}ln|5t - 1| + C$$ If you take $-\frac{1}{2}$ as well as the ...
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Product of heights of factors smaller than length of a polynomial with integer coefficients

I have the following question. Given a (univariate) polynomial with integer coefficients, I want to prove, if true, that the product of heights of its (irreducible) factors is smaller or equal to its ...
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1answer
34 views

Expanding an infinite product of infinite series

Here's a fragment of something I posted in an answer a few months back: \begin{align} & \left( 1 + \frac 1 {a_1} + \frac 1 {a_1^2} + \frac 1 {a_1^3} + \cdots \right) \\ \times {} & \left( ...
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1answer
19 views

Find a common factor of multiple matrices

I'm currently facing a problem where I have $N$ matrices $ \{A_1, A_2, \dots, A_N \} $ and I want to find a way to calculate matrices $H$ and $E$ such that $ \exists H \: \exists E\: \forall i \in ...
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1answer
37 views

Factorization of Taylor series.

I know that for a (finite) polynomial $P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_0$ whose zeros are $x_1, x_2, \ldots, x_n$, then we can factorize it as $$P(x) = a_n(x - x_1)(x - x_2) \cdots (x ...
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2answers
743 views

Why perfect square has odd number of factors

can someone please describe me why only the perfect square has odd number of factors.why does other number not has odd numbers of factors? I understand it but don't find any mathmetical proof.Please ...
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1answer
41 views

Number of solutions for $n^5 + 2 n^4 + n^3 - 3n + 2 $ mod $ 23^2 = 0$, where $0 \leq n < 23^2$ and $n\in \mathbb{N}$

$0 \leq n < 23^2$ and $n\in \mathbb{N}$ For how many $n$ $n^5 + 2 n^4 + n^3 - 3n + 2 $ mod $ 23^2 = 0$
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2answers
479 views

Highschool Exam Question About Cube Factoring

Given; $ a^3 - 3ab^2 = 10 $ and $ b^3 - 3ba^2 = 5$ What is the value of $ a^2 + b^2 $ ?
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4answers
70 views

finding the number of factors $2^{15}\times3^{10}\times5^6$

The number of factors of $2^{15}\times3^{10}\times5^6$ which are either perfect square or perfect cubes(or both) I don't know how to start this even! Plz solve this!
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1answer
47 views

Use Fermat factorization to factor $809009\ldots$

Use Fermat factorization to factor $809009\ldots$ So far I have: \begin{align} \sqrt{809009} & = 889.449 \\ & = 890 \\[6pt] \sqrt{890^2 - 809009} & = 130\ldots ∉ \mathbb Z \\[6pt] ...
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3answers
87 views

What's the best way to compute $\frac{a^4 + b^4 + c^4}{a^2 + b^2 + c^2}$

So, my teacher gave us this to compute yesterday, and I'm completly confused on how should I proceed : $$\frac{1^4 + 2012^4 +2013^4}{1^2 + 2012^2 + 2013^2}$$ I've tried several ways, but most of ...
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3answers
32 views

Factoring a 4th degree trinomial

I am trying to factor $3x^4-8x^3+16$, but I have no idea how to even start. I put into Wolfram Alpha, and it said that the answer was $(x-2)^2 (3 x^2+4 x+4)$. How would you factor something like this ...
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3answers
33 views

Find A and B for $A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$

Given:$$A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$$ How does one find A and B ? The answer is: $$A = 3x^{2};B=2x$$ but I can't see how one solves this. I tried subbing in values of x but it didn't lead ...
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5answers
81 views

How to factorize a 4th degree polynomial?

I need help to factorise the following polynomial: $x^4 - 2x^3 + 8x^2 - 14x + 7$ The solution I need to reach is $(x-1)(x^3 - x^2 + 7x - 7)$. I need to factorize to this exactly as it is for ...
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0answers
22 views

Linear factors of minimal polynomial dividing $x^r$ - 1

I have a monic minimal polynomial $m(x)$ that divides $x^r - 1$. Apparently $m(x)$ has distinct linear factors over the complex numbers $\mathbb C[x]$. I understand this part, since $\mathbb C[x]$ is ...
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308 views

proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
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Should this be rephrased into saying no common factors but 1?

This is from Discrete Mathematics and its Applications Example Prove that $\sqrt{2}$ is irrational by giving a proof by contradiction. Solution Let $p$ be the proposition "$\sqrt{2}$ is ...
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2answers
28 views

Find the cubic polynomial given linear reminders after division by quadratic polynomials?

A cubic polynomial gives remainders $(13x-2)$ and $(-1-7x)$ when divide by $x^2-x-3$ and $x^2-2x+5$ respectively. Find the polynomial. I have written this as: $P(x)=(x^2-x-3)Q(x)+(13x-2)$ ...
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1answer
36 views

Simple question on factoring the difference of 2 perfect squares

(b) (i) Use the identity $A^2-B^2=(A-B)(A+B)$ to factorise the expression $5^{2k}-1$. Do I just put the k as 1 so that the equation is 5^2 and 1^2 Thanks Steve
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1answer
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What is the theory of finding roots of a polynomial equation by looking at the factors of the $a_n$ and $a_0$ term called?

This is commonly taught in high schools in the context of factoring polynomials. I remember this method even has its own wikipedia page (with a proof) but I forget what was the theory called. Could ...
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1answer
24 views

What is the name of this kind of factoring algorithm

I just think about algorithm to find factor of number by doing something like guessing last digit of number and increase digit bit by bit Such as, I want to find factor of 749 Algorithm would begin ...
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3answers
119 views

How to factor intricate polynomial $ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $

I would like to know how to factor the following polynomial. $$ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $$ What is the method I should use to factor it? If anyone could help.. Thanks in advance.
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23 views

Simplifying with exponents

So somehow I made it all the way to Calc II but struggle when it comes to this basic thing (not exactly the most 100% solid algebra foundation it seems). Unable to simplify this further: $$ ...
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2answers
117 views

How can we prove that a quadratic equation has at most 2 roots?

A quad equation can be factored into two factors containing $x $, but how can we prove that there no other sets of different factors yielding OTHER VALUES OF $X $?
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1answer
54 views

What will be the $(b^2-a)$

If $a$, $b$ are the real numbers and $$4a^{2}+b^2=4a-(\frac{1}{4b^2})$$ What will be the $b^2-a=$? I tried to be this equation more basic ,but i could not reach the result
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1answer
30 views

finding factors

How can i quickly find the factors of a particular number? Find the number of different factors of 1800 and 3003? This being the question , for 3003 i first found out its prime factors and then i ...
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1answer
32 views

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$. How come the answer is left as $\frac{5x}{(x+2)(x-3)} + \frac{4}{(x+2)^2}$. Why don't we go any further?
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1answer
26 views

What does the first x represent in {x, (x+1), (x-3)}?

The question is: "Part of the graph of a polynomial function is shown. Which of the following sets contains only elements that are factors of the polynomial?" The two answer choices left are B. ...
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0answers
9 views

Which polynomial factorization method leads directly to $(1-\alpha_0 z)(1-\alpha_1 z)$

I know how to factor a polynomial $p(z)$ so that it looks like $a_n(z- z_0)\cdots(z-z_n)$, where $z_k$ are its zeros. Now I could squeeze this form into the wanted $(1-\alpha_0 z)\cdots(1-\alpha_n ...
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2answers
38 views

Factor trinomials dividing by the common GCF

I have a doubt with the following problem I found in a book. You have to simplify a polynomial using the GCF. Now, this is the problem I am not able to grasp: $$6x^2-19x-7$$ According to the book, ...
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1answer
15 views

Solving a characteristic Polynomial of the Hilbert Matrix

I need to find the eigenvalues of the following characteristic polynomial but I can't seem to successfully find the roots of the equation: $P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$ ...