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

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1answer
90 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
53 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
34 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
35 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
30 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
125 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
80 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
190 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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0answers
36 views

Polynomial factorization over an infinite field - is there an algorithm

In my previous questioned I asked how do I factor a polynomial, and I gave an easy example of a polynomial of degree 2. But now I have another question I need to solve. I need to factor ...
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1answer
67 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
31 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
74 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
35 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
66 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 ...
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2answers
113 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 ...
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1answer
79 views

Cholesky decomposition: any theoretical value?

Just read the Wikipedia article on Cholesky decomposition. All the applications listed there were numerical. Are there theoretical arguments to which it is important? For instance, here there is an ...
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3answers
136 views

Factoring out an exponential?

I have the following expression $$\frac{2^{k+1}(k+1)!}{(k+1)^{k+1}}\cdot\frac{k^k}{2^k k!}$$ I get $$\frac{2(k+1)(k^k)}{(k+1)^{k+1}}$$ But how do I factor out the ${(k+1)}^{k+1}$
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3answers
69 views

How do you factor $(10x+24)^2-x^4$?

I tried expanding then decomposition but couldn't find a common factor between two terms
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2answers
75 views

Computing p and q from private key

We are given n (public modulus) where $n=pq$ and $e$ (encryption exponent) using RSA. Then I was able to crack the private key $d$, using Wieners attack. So now, I have $(n,e,d)$. My question: is ...
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1answer
88 views

Number of divisors of a number

Is there any trick to find the number of divisors of any number? For e.g., a quick way to tell the number of divisors of 987655432 (chosen randomly)? EDIT: And it has to be done without prime ...
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1answer
45 views

How to approach factoring problems?

Generally speaking, how should I approach a problem involving factoring? I usually don't have a problem with the more typical forms, but sometimes I just don't know what to do. My calc2 question is ...
3
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1answer
37 views

How to factor these monomials?

This is the original problem: $x^3+x^2y+xy^2+y^3$ Answer: $(x+y)(x^2+y^2)$ I understand that the answer is correct, but I can't figure out how to get to it.
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7answers
452 views

If $a^3 + b^3 +3ab = 1$, find $a+b$

Given that the real numbers $a,b$ satisfy $a^3 + b^3 +3ab = 1$, find $a+b$. I tried to factorize it but unable to do it.
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2answers
92 views

How to find the factors whose sum is minimum

Lets take a number 108. How to find natural numbers a and b such that ab=108 but there sum should be minimum. Please show the solution for number 108.
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2answers
182 views

Getting rid of the denominator of a polynomial

I'm tutoring a high school precalculus student; our current topic is the roots of higher order polynomials. The problem we're solving is: Find a polynomial with the roots $\frac23$, -1, and $(3 + ...
4
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1answer
177 views

How to factor $y = x^5 + 20x^2 + 5$?

How would I factor to solve for x? $x^5 + 20x^2 + 5=0 $? Do I use synthetic division? Is there a faster/easier way? Do I have to keep plugging in numbers to see if they equal to zero? Thanks! I'm ...
2
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1answer
56 views

Using implicit differentiation to solve a function and stuck at factoring out y'.

So here is the question: $$ \tan^{-1}\left(\frac{2x}{y}\right)=\frac{\pi x}{y^2} $$ When I solved it implicitly I got (with much pain in formatting it on this site :P): $$ y^2=\pi ...
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2answers
206 views

Rabin's test for polynomial irreducibility over $\mathbb{F}_2$

I know that $f(x) = x^{169}+x^2+1$ is a reducible polynomial over $\mathbb{F}_2$. I want to show this using Rabin's irreduciblity test. First, I then have to check if $f$ is a divisor of ...
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1answer
41 views

Specific Annual Examination Marks

Steve has recently got his annual exam result.He has got upper than 690 out of 750.His obtained marks has odd number of factors.What is his obtained marks?
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2answers
120 views

Factoring Complex Trinomials

What is the answer for factoring: $$10r^2 - 31r + 15$$ I have tried to solve it. This was my prior attempt: $$10r^2 - 31r + 15\\ = (10r^2 - 25r) (-6r + 15)\\ = -5r(-2r+5) -3 (2r-5) $$
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1answer
198 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
54 views

Polynomial (third degree)

A third degree polynomial $p(x)=0$ when $x=1$ and $x=3$. We also learn that $p(x) \geq 0 $ when $x \geq 1$ and $p(2) =2$. Determine $p(x)$. How should I proceed? I presume no calculus is needed.
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2answers
77 views

Number of factors of a big number

what is the number of the factors of 884466000.how can I do this math without using factoring calculator?
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2answers
121 views

Number of Factors of 6

1.factor of 6 is 1,2,3,6 or 2.factor of 6 is 1,2,3,6,-1,-2,-3,-6 Which one is correct?
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2answers
333 views

Factoring Quadratics: Asterisk Method

I'm teaching my students about factoring quadratics. We've done GCF, difference of two squares, squared binomials, and grouping. One of my colleagues then found this asterisk method on line. It's ...
0
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1answer
58 views

finding factors for gcd

To compute $gcd(25, 11)$, Euclid's algorithm would proceed as follows: $$\underline{25} = 2 \cdot \underline{11}+3$$ $$\underline{11} = 3 \cdot \underline{3}+2$$ $$\underline{3} = 1 \cdot ...
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1answer
49 views

Simple Calculation on Local Rings.

Let $p$ be prime and $\mathbb{Z}_{(p)}$ be the local ring. I already know, that \begin{align} \mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)} \cong \mathbb{Z}/p\mathbb{Z}. \end{align} What ist the explicit map? ...
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1answer
42 views

Factoring $a^m + 1$, an odd prime

Why is it that if $a^m + 1$, an odd prime, with $m = kl$, and $l$ odd. We get: $$a^m + 1 = (a^k + 1)(a^{k(l-1)} - a^{k(l-2)} + \dots + a^k + 1)?$$ What is the name of this property?
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0answers
91 views

Is there an easy way to factor polynomials with two variables?

On a recent precal test, I saw a question involving the following expression: $$(x+1)^2-y^2$$ Which factored out into: $$(x+y+1)(x-y+1)$$ This wasn't very hard, considering that it was already ...
2
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0answers
21 views

Decomposability of a function into two

This is question on the existence of a decomposability of a function $f(v)$ into multiplicative factors $\lambda(v)$ and $g(v)$. The question is non-trivial since the functions $\lambda(v)$ and $g(v)$ ...
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3answers
91 views

How does the author get from one step to another?

I have to apply convolution theorem to find the inverse Laplace transform of a given function. I know that convolution is applied when the given function is multiplication of two functions. The ...
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0answers
272 views

Finding irreducible polynomials and factorization

Need some explanation and checking if my thinking on the solution is correct for the assignment given below: (In these problems you may use without proof which polynomials of degree 2 and 3 are ...
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0answers
38 views

Given a cubic $f(x)$ with specified negative real roots $-a,-b,-c$, what happens when we search for solutions to $f(x)=d$?

Noting Roots of a Certain type of Cubic Equation, what if we have the following simpler form for real $d$: $$(x+a)(x+b)(x+c)=d\tag{1}$$ (With $a,b,c\in \mathbb R^+$.) Clearly, depending on $d$, the ...
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1answer
87 views

Why does completing the square give you the minimum point?

Say we have an equation:$y=$ ${x^2} + 2x + 1$ Completing the square we get: $\eqalign{ & y={x^2} + 2x + 1 \cr & = {(x + 1)^2} - {(1)^2} + 1 \cr & = {(x + 1)^2} \cr} $ The ...
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2answers
316 views

Help Factoring Quadrinomial

I know factoring questions are a dime a dozen but I can't seem to get this one. $-2x^3+2x^2+32x+40$ Factor to obtain the following equation: $-2(x-5)(x+2)^2$ Do I have to use division (I'd prefer ...
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3answers
628 views

Prove that $x^4-x-1$ is irreducible over $\mathbb{Q}$

Prove that $f(x)=x^4-x-1$ is irreducible in $\mathbb{Q}[x]$. All methods I know failed. I can only exclude that $f$ admits a factorization with a factor of degree 3, because in this case $f$ would ...
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2answers
105 views

Factorize $8x^3 + 12x^2 -2x -3$

How do I factorize this - $$8x^3 + 12x^2 -2x -3$$ I tried splitting the middle term but that didn't work , I tried factor theorem with various factors but even that didn't work. What can I do to ...
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1answer
47 views

Factorize $4a^2 - 9b^2 -2a - 3b$

I found this question in my textbook - $$4a^2 - 9b^2 -2a - 3b$$ I am in ninth grade and we have been taught how to factorize using identities, splitting the middle term and by using identities. I ...
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2answers
139 views

equation representing 2 straight lines

Let us assume this equation is given to us we have to factorize it $$12x^2 +7xy-10y^2+13x+45y-3=0$$ By solving we get that this represents two straight lines. But how to factorize it? Is there a ...
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2answers
32 views

Factoring Expressions

I can't seem to factor this expression: $$2(2x^2-x)^2-3(2x^2-x)-9$$ So far, this is what I have done: $$(2x^2-x)(4x^2-2x-3)-9$$ I'm not sure what to do after this though, any hints?