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

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2
votes
1answer
49 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
vote
0answers
37 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
45 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
48 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
votes
4answers
36 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}\frac{2\...
0
votes
5answers
71 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
1answer
47 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
67 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 $5+5i=(2+i)(2-i)(1+i)$,...
0
votes
1answer
42 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
26 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
42 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
8 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
48 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
108 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
67 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
50 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
82 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
65 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
43 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
62 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
48 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
26 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
90 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
217 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
61 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
56 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 $F(X,...
1
vote
2answers
40 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 the ...
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
557 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 success....
0
votes
1answer
44 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 $$(P-b)\left(V-\frac{a}{V^2}\...
5
votes
1answer
67 views

Decompose $x^4 + x^3 + 1$ into irreducible factors over $\mathbb{Z}_2$

Decompose $x^4 + x^3 + 1$ into irreducible factors over $\mathbb{Z}_2$ I think that the given polynomial is already irreducible in $\mathbb{Z}_2$, therefore the only irreducible factors are $x^4 + x^...
3
votes
2answers
64 views

Finding integers satisfying $m^2 - n^2 = 1111$

We have to find the integers $m$ and $n$ which will satisfy the given condition: $$m^2-n^2=1111.$$ What could be the answer and how? i tried using trial and error and that took a long time.
0
votes
2answers
42 views

factorization of a fourth degree polynomial

What could be the possible factorization of $$2a^4+a^2b^2+ab^3+b^4$$? what term should be added
2
votes
2answers
76 views

Find product of solutions of $x^6=-64$

If the six solutions of $x^6=-64$ are written in the form $a+bi$,where $a$ and $b$ are real, then find the product of those solutions with $a>0$. The answer in my book is given as $4$ but I don'...
2
votes
1answer
54 views

Factoring the quartic $\left(x^{2}+x-1\right)\left(x^{2}+2x-1\right)-2sx\left(2x-1\right)^{2}$

Define $Q{\left(s;x\right)}$ to be the quartic function of $x$ with real parameter $s$ such that $0\le s\le1$ given as $$Q{\left(s;x\right)}=\left(x^{2}+x-1\right)\left(x^{2}+2x-1\right)-2sx\left(2x-...
3
votes
2answers
58 views

Why is the number of divisors of an integer $n$ equal to the number of factors in the factorization of the polynomial $x^n - 1$ over the integers?

Sloane's OEIS A000005 gives the number of divisors of the integer n. A comment (by a very reputable contributor) in this sequence claims that this is also the number of factors in the factorization ...
1
vote
2answers
68 views

Having trouble finding roots/factorizing a cubic equation

I've been trying to find a method that can work for any cubic equation, but I can't seem to find one. Right now, I'm trying to find the roots/ factorize the following equation: $x^3-5x^2+3x+9 = 0$. I'...
7
votes
2answers
139 views

Prove that if $7^n-3^n$ is divisible by $n>1$, then $n$ must be even.

I tried using factorization of $a^n-b^n$ for odd $n$ in an attempt to work through to a situation where the factors are such that they cannot have n as a factor. But I reached nowhere. Here's how I ...
1
vote
0answers
35 views

Factorization of polynomial $f(x,y)$

The motivation is solving the following equations: $$ f(x,y)=0, x=L-kL, y=ks $$ $k$ is the variable, $L$ and $s$ are constants. The plan is: First, to factorize the polynomial as $(a_1x+b_1y+c_1) (...
1
vote
2answers
54 views

Complete the square of three variable quadratic expressions

We know that completing $ax^2+bxy+cz^2$ into forms of $k_{1}(a_{1}x+b_{1}y)^2+k_{2}(a_{2}x+b_{2}y)^2$ is easy and have some fixed routine. But the 3 variable case $$ax^2+by^2+cz^2+dxy+exz+fyx$$does ...
2
votes
2answers
76 views

Find 4 positive integers not exceeding 70,000 such that each have more than 100 divisors

I am looking at problems in Vandendriessche and Lee's Problems in elementary number theory and this is one of their problems: Find $4$ positive integers not exceeding $70000$ such that each have ...
1
vote
1answer
34 views

The form which take continuous functions for which we have $f(x,y)=g(x)h(y)$?

In this question there was raised a question on general sufficient conditions under which we cannot factor the functions and there is an answer which states that: Theorem: Let $f:\mathbb R^2\to\...
1
vote
4answers
61 views

How do you factor $x^2-x-1$?

I know you can't have all integers, but how do you factor this anyway? Wolfram|Alpha gives me $-\frac{1}{4} (1+\sqrt{5}-2 x) (-1+\sqrt{5}+2 x)$. Cymath gives me $(x-\frac{1+\sqrt{5}}{2})(x-\frac{1-\...
3
votes
1answer
63 views

When can a function not be factored?

Are there any general conditions under which a function involving $n$ unknowns cannot be factored into a product of $n$ terms each of which contains only one of the unknowns? For example, $xy$ can be ...
0
votes
2answers
38 views

How is the formula for a finite geometric series found?

I have these two finite geometric series: $S_n$ = $\sum_{k=0}^n ar^k$ r$S_n$ = $\sum_{k=0}^n ar^{k+1}$ And then we substract both series so: $S_n$ - r$S_n$ = a - $ar^{n+1}$ ...
5
votes
0answers
79 views

Does Pollard rho works for Gaussian integers?

Should I expect that the Pollard rho method ...
2
votes
1answer
29 views

Find the smallest $4$ digit number which is a factor of $2005^6 - 1$.

Find the smallest $4$ digit number which is a factor of $2005^6 - 1$. My attempt: We see that $(2005-1)$ is a factor of $2005^6-1$ by the factorization of $2005^6-1$. Therefore, $1002$ is a factor ...
3
votes
2answers
45 views

Gaussian prime factorization.

I have a hard time on factorizing elements from $\mathbb{Z}[i]$, especially $-19+43i$. I know that the primes in $\mathbb{Z}[i]$ are: $1+i$. $p$ from $\mathbb{N}$, $p=4k+3$ , $k$ integer ( $p\equiv ...