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

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2answers
47 views

how to make factor of this expression

How do I factor $x^4 + x^2 y^2 + y^4$? And in how many ways can I make the factor? Some methods I know are Mid term Break Substiturion Method Quadratic Equation
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3answers
163 views

Prove that $ x^n - y^n = (x-y) (x^{n-1}+x^{n-2}y\,+ \,\,…\,\,+ y^{n-1})$ [closed]

Prove that $ x^n - y^n = (x-y). (x^{n-1}+x^{n-2}y\,+ \,\,...\,\,+ y^{n-1}) $; $\,\,\,\,\,$$x,y \in \mathbb{R}$
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6answers
91 views

Factoring $s^2+4s+13$

I was looking at an example, and it was factored as follow: $$ s^{2}+4s+13 = (s+2)^{2}+9 $$ How can we do that?
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3answers
76 views

Factoring/approximating an apparently simple formula

Does anyone know if the following formula can be factorized or approximated: $a^3 + b^3 + c^3 + a^2b + ab^2 + a^2c + ac^2 + b^2c + bc^2 + abc$ It looks a lot like $(a + b + c)^3$, except for the ...
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3answers
63 views

Simplifying and Factoring Formulae

I'm am doing an 8th grade math text book, and I came across this simple problem: $8l^3 - 36l^2m + 54lm^2 - 27m^3$ simplifies to? I immediately got to know that it is $(2l - 3m)^3$ , but how do you ...
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1answer
83 views

Factoring Perfect Square Trinomials

How would you factor perfect square trinomials? I have a perfect square trinomial 4x^2 - 20x + 25 = 0, and the answer given to me on the answer key is ...
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2answers
84 views

Prove that the subgroup of the quotient group is cycling and infinitely generated

$$M = \left\{\,\dfrac{m}{13^n}\biggm| m\in \mathbb{Z}, n\in\mathbb{N} \,\right\}, \quad G = M/\mathbb{Z}$$ Prove that any subgroup $H < G$, $H\neq G$ is cyclic and infinitely generated and that ...
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1answer
57 views

Factoring over prime fields

Suppose I have two numbers in Fp that are multiplied together: r*s Is there anything special that needs to be done when prime factorizing since this is a finite field (i.e., is this what is referred ...
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0answers
77 views

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$?

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$ ? For example: $1.$ $a^2+b^2+c^2$ $2.$ $a^2+b^2-c^2$ $3.$ $a^2+b^2+c^2+d^2$ $4.$ ...
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2answers
43 views

Multiples of three and five below n

I am tackling a problem which asks: Find the sum of all the multiples of 3 or 5 below 1000. My reasoning is that since Since $\left\lfloor\frac{1000}{3}\right\rfloor = 333$ and ...
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1answer
82 views

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain?

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain? I have no idea to prove or disprove this... should I prove ...
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4answers
110 views

$\omega^2+\omega+1$divides a polynomial

The question is Show that $f(n)=n^5+n^4+1$ is not prime for $n>4$. The solution is given as Let $\omega$ be the third root of unity. Then $\omega^2+\omega+1=0$. Since ...
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4answers
66 views

Extending Completing the square to expressions of $ (x+y+z)^2$

We all know expressions such as $x^2+14x+49 = (x+7)^2$ because it is easily recognizable as a perfect square. What about a expressions in the form of $x^2+2xy+2yz+2xz+y^2+z^2=(x+y+z)^2$ The question ...
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1answer
53 views

Question about factor of a function

can you explain this question for me? I don't quite understand it. Thank you in advance. $x^2+1$ is a factor of $f(x)$. Which of the following is TRUE? $\text{a)}\qquad f(-1)=0 ...
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2answers
70 views

How to simplify this fraction

It's embarrassing, but I need help solving this one... Need some refresher course for algebra. $$ \frac y{y+\sqrt y} $$
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1answer
33 views

If $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$

Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$. Show that if $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$. I showed that if $a$ is ...
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3answers
138 views

Is there a branch of mathematics that studies the factors of rational numbers?

Is there a branch of mathematics that studies the factors of rational numbers? I am imagining that defining this would work pretty much the same way as defining the factors x of an integer n: $\{x ...
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2answers
35 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
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1answer
37 views

Factors of $x^n+1$ over $\mathbb{Z}[x]$

Is there any equivalent to $x^n-1 = \prod\limits_{d|n} \phi_d$ where $\phi_d$ is the $d$th cyclotomic polynomial but for $x^n+1$? Even better, can we generalize any further?
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4answers
670 views

Factoring a cubic polynomial?

So I have a matrix $$A = \begin{pmatrix} -5 & -6 & 3 \\ 3 & 4 & -3 \\ 0 & 0 & -2 \end{pmatrix} $$ I'm to find the characteristic polynomial and all the eigenvalues of ...
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1answer
48 views

Inversion in factor rings

I have this polynomials: $f = x^{4} + 3x^{3} + x^2 + 3 \in \mathbb{Z}_{5}[x]$, $g = x + 2 \in \mathbb{Z}_{5}[x]$ Does g + (f) have inversion in ring $(\mathbb{Z}_{5}[x]/(f),+,.)$ ? I should found ...
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2answers
867 views

Use a given zero to write P(x) as a product of linear and irreducible quadratic factors

The polynomial in question is: $x^4 - 8x^3 - 19x^2 + 288x - 612$ and the zero is $4 - i$. What I don't understand is how to go from the given zero to factorizing, especially as it's imaginary. ...
2
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2answers
46 views

*Step* in proving that there are infinitely many primes that suffice…

Let $k,n\in \mathbb Z$ with $n=k^2+1$ and let $p$ be an odd prime with $p\mid n$. Prove that $p\equiv1\text{ mod }4$. I found out that $\bar{n}\in\left\{ \bar{1},\bar{2}\right\} $ (denoting ...
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2answers
146 views

Intermediate Problem Solving Patterns involving Prime Factoring

a and b are positive integers such that $a\times b= 500000000,$ where neither a nor b contain any zeros. Find a and b where $a<b.$
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1answer
68 views

If n > 3 and (n + 1) is a square, is there any n that is a prime?

I am looking at properties of squares and came about this property. I am investigating the difference of squares in relation to primes.
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3answers
64 views

Can someone please explain how this was factored?

How was $x^2(x+1)-4(x+1)$ factored into $$(x^2-4)(x+1)?$$ I know this seems very basic but can someone please explain this?
2
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1answer
78 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
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3answers
61 views

Trouble with factoring polonomial to the 3rd degree

I am having trouble factoring this problem: $\displaystyle{-x^{3} + 6x^{2} - 11x + 6}$ I know the answer but i can't figure out how it is done with this. I have tried by grouping and is doesn't seem ...
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6answers
328 views

Find $x$ and $y$ in $2^{x-y} + 1 = 2^x,$ where $x,y$ are integers

I have no idea what to do now. Is there any way to find the integers $x$ and $y$ by factoring? Thank you.
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1answer
73 views

How many integral solutions of $a,\ b,\ c$ are there such that $2^a \cdot 3^b + 9 = c^2 $

How many integral solutions of $a,\ b,\ c$ are there such that $$2^a \cdot 3^b + 9 = c^2.$$ we can get that $$2^a \cdot3^b = (c-3)(c+3) $$ we can make cases if $b \ge 2$ then $c=3k$ then ...
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1answer
65 views

Please help me with factorisation

Is it possible to write $$64x^6-112x^4+56x^2-7$$ in linear factors? If so, what are they? (Finding it really difficult to ask this question!!)
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1answer
209 views

Factorizing $(x-1)(x-3)(x-5)(x-7)-64$

We need to factorize: $$(x-1)(x-3)(x-5)(x-7)-64$$ We can, by the rational root theorem, see that there are no roots of this polynomial.Next observation is that $64=(8)^2$. So this means that if the ...
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1answer
155 views

How long is an arrow in the air?

The height $h$ of an arrow in feet is modeled by $h(t) = -16t^2 + 63t + 4$, where $t$ is the time in seconds since the arrow was shot. How long is the arrow in the air? Could someone explain where to ...
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1answer
23 views

How do I identify repeated irreducible factors?

I thought my solution was correct - but it seems like that's not the case. Can anyone possibly explain to me why I'm wrong?
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2answers
421 views

Spivak Calculus chapter 1, problem 1v

this is my first question here. I am self-studying Spivak's Calculus Fourth Edition and am stuck on Chapter 1 Problem 1v. The question is to prove that: $x^n - y^n = ...
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0answers
576 views

Find the set of cyclotomic cosets of q modulo n

Calculating finite field and factoring $x^n - 1$ over $GF(q)$ first step is to calculate cyclotomic cosets. For example : For $n=9,q=2$ $C_1=\{1,2,4,8,7,5\} = C_4 = C_8 = C_7 = C_5$ $C_3=\{3,6\} ...
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1answer
378 views

What is the mathematical proof behind the shortcut used in this video, Factoring Trinomials with Leading Coefficient not 1 (fast way)?

My teacher found this cool shortcut for factoring. I would like to use, for it saves time, but I feel hesitant using it without knowing the mathematical proof. Can anyone watch the video and explain ...
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1answer
109 views

Frobenius matrix norm vs. 2-norm

From this article about the singular value decomposition: Let $A$ be an $n \times d$ matrix and think of the rows of $A$ as $n$ points in $d$-dimensional space. The Frobenius norm of $A$ is the ...
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1answer
36 views

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $$\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$$ I have a theorem (shown in my text) that says $$\fbox{If $a_1,...,a_m\in\mathbb{N}^*$ then ...
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1answer
46 views

Find the smallest value of $n$ so that the greater potency of $5$ which divides $n!$ is $5^{84}$. What are the other numbers that enjoy this property?

Find the smallest value of $n$ so that the greater potency of $5$ which divides $n!$ is $5^{84}$. What are the other numbers that enjoy this property? I thought I would put together an equation ...
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1answer
49 views

Computations question

a) Determine the prime factorizations of 3850 and 4125 b) Find the value of d = gcd(3850,4125) c) List all the positive divisors of d This is what I have so far. a) 3850: 11, 5, 5, 7, 2 4125: ...
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2answers
179 views

How useful is factoring large numbers - Not for cryptography!

This is a question for my curiosity. Apart from its implications to cryptography. Is factoring large numbers really useful? Are there any examples of where the ability to factor huge numbers, 1K ...
2
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1answer
165 views

Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number.

Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number. Actually, I know a way to solve this, but even if it is very large and ...
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2answers
498 views

Convert Circuit SAT to 3-SAT

I am trying to convert Integer Factorization to $3-SAT$. So far I managed to convert it to Circuit SAT, but I don't know how to make the final step. This is how it look for 3*3 multiplication: ...
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1answer
203 views

Polynomial factorization to irreducible factors with respect to field

I have a question, I think I don't understand this material very well and could use an explanation / some help. Basically we are asked to decompose $x^5-x$ to irreducible factors over $R,F2,F5,C$ ...
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2answers
33 views

If $N$ is not a power of a prime, why does 1 have -1 as a root modulo $N$?

If integer $N$ is not a power of a prime, it is the product of two coprime integer numbers greater than 1. As a consequence of the Chinese remainder theorem, the number 1 has at least four ...
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1answer
123 views

Probability of an ECM factor

Suppose I have a composite number $N$ divisible by some prime $p\le x.$ What is the probability that one iteration of ECM finds $p$, given parameters B1 and B2? Usually people look for factors in ...
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1answer
38 views

$\frac{x^4 - x^3 + ax^2 + bx + c}{x^3 + 2x^2 - 3x + 1}$, remainder $3x^2 - 2x + 1$. Find $(a + b)c$.

Given the polynomials $P(x) = x^4 - x^3 + ax^2 + bx + c\\ Q(x) = x^3 + 2x^2 - 3x + 1\\ R(x) = 3x^2 - 2x + 1$ such that $P(x) = D(x)Q(x) + R(x)$, find $(a + b)c$. I would normally apply little ...
3
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1answer
70 views

Finding $a_n$ such that $x^n+a_1x^{n-1}+\cdots+a_{n-1}+a_n$ cannot be factored when $a_1,\cdots,a_{n-1}$ given

Let $n\ge 4\in\mathbb N$. Suppose that $a_1,a_2,\cdots,a_{n-1}$ are given integers. Then, here is my question. Question : Is the following true for any $(a_1,a_2,\cdots,a_{n-1})$ ? There ...
7
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2answers
127 views

Defining irreducible polynomials recursively: how far can we go?

Fix $n\in\mathbb N$ and a starting polynomial $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_n\ne0$. Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+a_rx^r$ such ...