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

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0
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2answers
22 views

Factoring under a 4th power

Show that $$\left( a + b \right) ^ 4 = b^4 \left( \frac{a}{b} + 1 \right) ^4$$ Its clear without an exponent $$\left( a + b \right) = b \left( \frac{a}{b} + 1 \right) $$ but I'm not sure why ...
2
votes
1answer
28 views

Factor Transition Issue

$$ F'(x)=(2ax+b)e^x+(ax^2+bx+c)e^x=(ax^2+(b+2a)x+c+b)e^x $$ Can someone explain this? I can't understand the transition. What method is this?
0
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2answers
74 views

How to factor $x/(x^2+x+5)$? [on hold]

Somebody asked how to factor $\frac{x}{(x^2+x+5)}$ in order to try to solve an integral. He didn't care about the coefficients, he only wanted to know how to factor the expression.
2
votes
3answers
39 views

Factorise Algebraic Expression

Background: I came across the following problem in class and my teacher was unable to help. The problem was factorise $x^6 - 1$, if you used the difference of 2 squares then used the sum and ...
0
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1answer
10 views

factors in polynomial rings with field coefficients

I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following: Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if ...
0
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1answer
27 views

Complex Numbers in Factoring [closed]

Why does "$i$" only get involved in factoring a function when there is a ($+$) in the equation? EX: $x^2 + 9$.
0
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1answer
54 views

Is it possible to solve the following equation without using the Rational Root Theorem?

Given $f(x)=x^4+2x^3+2x^2-2x-3$, where $x-1$ is a factor of $f(x)$, how is it possible to solve $f(x)$ without the Rational Root Theorem? Here's my progress: $$f(x)=x^4+2x^3+2x^2-2x-3$$ ...
1
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1answer
21 views

Changing the Form of this Factorisation

I'm brushing up on some high school maths and I'm currently revisiting determinants, specifically the factorisation of determinants. I'm working my way through a problem set and I keep getting stuck ...
2
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0answers
21 views

How Do I Apply the Cantor-Zassenhaus algorithm to $\mathbb{F}_2$?

Recently, I've been trying to implement the Cantor-Zassenhaus algorithm in C++ over $\mathbb{F}_2$. According to this lecture, the algorithm is basically: Input is polynomial $f\in\mathbb{F}_q$ with ...
2
votes
3answers
53 views

Factorize $2a^3 - b^3 - c^3$

I need to factorize the expression $2a^3 - b^3 - c^3$. I see that one zero is achieved when $a=b=c$, but I can't find the factor(s).
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3answers
51 views

What comes first here? pemdas doesnt really tell me what to do here

So I have this equation: $2x(x+3)(x+3)$ Do I FOIL the $(x+3)$ first or multiply the $2x$ to the first $(x+3)$? Would there be a difference? Isn't multiplication commutative?
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0answers
39 views

Show that $504 \mid n^9 − n^3 $ for any integer $n$ [duplicate]

Not sure how to start this. I know that $504 =2 \times 2 \times2 \times 3 \times 3 \times 7$.
1
vote
0answers
14 views

Cracking any linear congruential generator

I have a linear congruential generator $X_{n+1} = (aX_n + b) \bmod 2^k $with given arguments and number $Y$. The problem is to find the smallest $i$ that $X_i = Y$ or tell that there is no such $i$. ...
4
votes
1answer
58 views

Probability that a number has $m$ indistinct factors

I just discovered Matlab's factor()-function, and I randomly typed in 20081294819, and to my surprise it only had two factors (5099 and 3938281)! I had expected many more factors for such a big number ...
2
votes
3answers
82 views

Why isn't integer factorization in complexity P, when you can factorize n in O(√n) steps?

It is said that integer factorization is an NP problem. Why isn't it P? You can solve it in $O(\sqrt{n})$ time with trial factorization, and since $\sqrt{n} = n^{1/2}$, to me that looks like a number ...
1
vote
2answers
20 views

Factorization and Quadratic Non-Residue

Suppose that I can always factor any number modulo $p$ into factors that are smaller than $f(p)$ where $f$ is some function. Does that imply that the least quadratic non-residue is smaller than ...
0
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3answers
32 views

factoring of $ e^{2x}-3e^{x}+2 = 0 $

How does $ e^{2x}-3e^{x}+2 = 0 $ factors to $ (e^x - 1) (e^x - 2 ) $ Because when I try to factor: $ e^{2x}-3e^{x}+2 = 0 $ $ e^{2x}-2e^{x}-1e^{x}+2 = 0 $ But $-2e^{x} * -1e^{x}$ should ...
2
votes
1answer
60 views

Roots of $x^3-x+1$

I am trying to find nice explicit formulas for the roots of the polynomial $x^3-x+1$. Is there some clever way to write down the roots in a reasonably easy way? I found the roots, but my expressions ...
2
votes
1answer
34 views

The sum of the squares of two factors of a number is a perfect square

Is there any way to get two factors of a number whose sum of the squares is a perfect square. As an example $19354423920$ is a number. which has $4262$, $4541160$ as factors ($19354423920 = 4262 * ...
1
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0answers
33 views

ab = X mod Y (X and Y known)

Is there a better way to determine possibilities for $a \mod Y$ and $b \mod Y$ in the following equation: $ab = X \mod Y$ than by brute force? For example if $ab = 5 \mod 6$ then either $a = 1 \mod ...
0
votes
1answer
37 views

Partial fraction integration with unclear roots

Let's look at a simple example like $\frac{1}{x^3+2x+1}$ here. We know that the denominator has a real root between $0$ and $-1$ (could go closer, but that's not the point). By the concept of slope of ...
0
votes
1answer
42 views

Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$? [closed]

The question is as in the title: Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$?
0
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4answers
30 views

How to evaluate this mixed function limit

$$\lim_{x\rightarrow 0}\frac{\sqrt{1+2x}-\sqrt{1-2x}}{\sin x}$$ What I did was use binomial theorem and the fact that $\lim_{x\rightarrow 0}\dfrac{\sin x}{x} = 1$. $$\lim_{x \rightarrow ...
0
votes
5answers
68 views

How many factors does $36^2$ have

How many factors does $36^2$ have $(A)2 \\ (B)8 \\ (C)24 \\ (D)25 \\ (E)26$ $36^2=2^4\times3^4$ How do i count the number of factors?I don't know.
2
votes
0answers
149 views

Factoring semi-primes, convert algorithm to function [closed]

I found an interesting method of factoring semi-primes when I been searching for ways to predict the mod result of given number. The algorithm This algorithm is ...
2
votes
1answer
24 views

What is meant by “maximal proper factors” of a integer?

I understand what is meant by proper factors, e.g. the proper factors of 36 are 2, 3, 4, 6, 9, 12, & 18. However I've just seen the phrase "maximal proper factors" used in the context of ...
1
vote
2answers
61 views

Prime factorization of Gaussian integers

I want to find $a, b\in\mathbb{Z}[i]$ such that $a(2+3i)+b(5+5i)=1$. I don't know how to do this, but my first thought was to do something with the norm or otherwise factoring ...
0
votes
1answer
35 views

Irreducible polynomials in $\mathbb{C}[X,Y]$

I have the polynomial $X^2+Y^2-1$ in $\mathbb{C}[X,Y]$. Is this irreducible? If not, how do I factorize it? Should I handle this the same as if it were $\mathbb{R}[X,Y]$, or should I do it ...
0
votes
1answer
20 views

How can I find the point of intersection between these functions?

Find the point of intersection between $f(x) = x^3$ and $g(x) = x^{1/3}$ Once I equal each equation to each other, I could factor out the $x$ but the exponent 1/3 is confusing me. Thank you!
0
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3answers
36 views

Solve for $A,B$: $\mathrm{LCM}(A,B)=168$, $\mathrm{HCF(A,B)}=12$

The highest common factor and the lowest common multiple of two numbers $A$ and $B$ are $12$ and $168$ respectively. Find the possible values of $A$ and $B$ with the exception of $12$ and $168$. ...
0
votes
0answers
7 views

Solve $\sum_i^n p_i L^{n-i}s^i(1-x)^{n-i}x^i=\prod_i^{n/2} \left( a_ix^2+ b_ix+c_i\right)$

$p(x)$ is a polynomial with coefficients in terms of S, L. $p_i, S, L$ are rational numbers. $$ p(x)=\sum_i^n p_i L^{n-i}s^i(1-x)^{n-i}x^i=\prod_i^{n/2} \left( a_i(s,L)x^2+ b_i(s,L)x+c_i(s,L) \right) ...
0
votes
1answer
44 views

How to factor polynomials by hand?

Is there a good approach for factoring polynomials by hand (e.g. if you're in an interview situation without access to a computer)? For example $1−4z+5z^2−2z^3$?
1
vote
5answers
96 views

Proving $x^4+2$ cannot be factored into $2$ degree polynomials

My book says that it can't be because if I try to write $x⁴+10x³+15x²+5x+12$ as: $$x^4+2$$ (which is $p(x) mod 5$) then $x⁴+2$ is irreducible because: $$x^4+2 = (ax²+bx+c)(a'x²+b'x+c)$$ is ...
4
votes
3answers
56 views

Find all irreducible polynomials of degree $2$ over $\mathbb{Z}_5$

Obviously if I write all the possible ones and try the roots I'd get a LOT of polynomials $(125)$ and I'd have to test $5$ roots for each of them, which would be a LOT. Is there any idea? I must also ...
1
vote
0answers
47 views

Proving that if $p(x)$ divides $f(x)g(x)$ then $p(x)$ divides $f(x)$ or $g(x)$

I need to somehow prove that if $p(x)$ is irreducible and divides $f(x)g(x)$ then $p(x)$ divides $f(x)$ or $g(x)$. I've been given the hints that I should use the theorems: $p(x)$ is irreducible ...
0
votes
2answers
48 views

Proving some polynomials are irreducible using Eisenstein's criterion

I would like to see if I'm right about these polynomials I tried to prove are irreducible: 1) For the first polynomial I used that if a polynomial is irrational over $\mathbb{Z}_p$ for $p$ prime, and ...
1
vote
1answer
51 views

Geometrically ireducible curve

I know that curve with coefficients in $k$ is geometrically ireducible if it does not factor over algebraic closure of $k$. I have this curve, for example, $$2x^2+2x^2y+2y^2+2xy+3xy^2=1.$$ It's ...
2
votes
1answer
39 views

Factoring integers: What to study?

I want to learn everything related to factoring integers, I have no idea what should I study to learn about this, I want to learn and understand all the currently used algorithms, why a polynomial ...
0
votes
2answers
42 views

How many sets of 8 3-digit consecutive even numbers are possible such that product when divided by 5 gives perfect cube?

The sum of eight three-digit consecutive even number is S.When S is divided by 5, it results in a perfect cube.How many sets of such eight numbers are possible?
0
votes
1answer
26 views

How to find product of factors of a number if the factors are perfect square?

Let M be the set of all the distinct factors of the number N = 6^5 * 5^2 * 10, which are perfect squares. Find the product of the elements contained in the set M. N = 2^6 * 3^5 * 5^3 Even power of ...
1
vote
0answers
25 views

factor $\sum_i^4 p_i x^i$

The polynomial $f(x)= \sum_{i=0}^4 p_i x^i $ whose real coefficients are: $$ p_4 = (L^2 + s^2) ^2 \\ p_3= -4 L^2 (L^2 + s^2)\\ p_2 = 6 L^4 + 2 L^2s^2 \\ ...
1
vote
3answers
73 views

How to factor polynomials?

I am wondering if there is a methodical, algorithmic, brain-dead way to factor polynomials. For example: $x^6 - 14x^{5} + 73x^{4} - 188x^{3} + 256x^{2} - 176x^{1} + 48$ can be written as $(x-1)^2 ...
10
votes
5answers
211 views

How to factor $x^6+x^5-3\,x^4+2\,x^3+3\,x^2+x-1$ by hand?

I know that $x^6+x^5-3\,x^4+2\,x^3+3\,x^2+x-1 = (x^4-x^3+x+1)(x^2+2x-1)$ but I would not know how to do that factoring without a software. Some idea? Thank you!
4
votes
1answer
60 views

Find all $\mathbb{Z}_n$ in which $x^2+2$ divides $x^5-10x+12$

Find all $n\ge2$ such that $x^2+2$ divides $x^5-10x+12$ in $\mathbb{Z}_n$. To begin, I divided $x^5-10x+12$ by $x^2+2$ which gave me: $$x^5-10x+12 = (x^3-2x)(x^2+2)-6x+12$$ So, I guess I need to ...
0
votes
2answers
53 views

Factoring a polynomial of $4$th degree

I was wondering how would you factor $$x^4+4x^3+21x^2-20x+25=0\text{ ?}$$ I cannot find a number that allows the expression to equal zero.
0
votes
2answers
41 views

Factoring a polynomial which shares the same zeros as another

I have this problem: Let $V=V(Y^2-X^2(X+1))\subset \mathbb{A}^2$, the zero set of the polynomial $G(X,Y)=Y^2-X^2(X+1)\in K[X,Y]$ for a field $K$. In my notes it says: If $F(X,Y)\in I(V)$ then ...
1
vote
2answers
35 views

A ball is thrown upward from the top of a tower.If its height is described by $-t ^2+60t+700$,what is the greatest height the ball attains?

A ball is thrown upward from the top of a tower.If its height is described by $-t ^2+60t+700$,what is the greatest height the ball attains ? I've managed to get a solution by realizing that ...
0
votes
2answers
34 views

Limits: Can't understand this worked example

I can't seem to understand the following given example while working with Limits. $$\lim\limits_{x \to \infty}({x\over1+x})^x = \lim\limits_{x\to \infty}({x +1 -1\over1+x})^x = \lim\limits_{x \to ...
10
votes
2answers
444 views

Sum of factors of a huge number.

I recently appeared in a math olympiad and it had this one question which had me stumped. This was a few weeks back and I have been looking for a way to find its answer ever since, but with no ...
0
votes
1answer
43 views

Solve $ax+bx^0+\frac{c}{x}+\frac{d}{x^2}=0$ for $x$

I find myself trying to produce a plot of the van der Waals equation of the form $PV(P)$ to demonstrate the non-ideality of some gases. The van der waals equation is ...