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

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2
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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 ...
0
votes
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!!)
2
votes
1answer
207 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 ...
0
votes
1answer
147 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 ...
0
votes
1answer
22 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?
0
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2answers
408 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 = ...
1
vote
0answers
559 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\} ...
0
votes
1answer
347 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 ...
1
vote
1answer
108 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 ...
1
vote
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 ...
2
votes
1answer
45 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 ...
1
vote
1answer
48 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: ...
1
vote
2answers
175 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
votes
1answer
160 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 ...
4
votes
2answers
486 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: ...
3
votes
1answer
197 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 $R,F2,F5,C$ ...
1
vote
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 ...
3
votes
1answer
121 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 ...
0
votes
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
votes
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
votes
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 ...
3
votes
1answer
87 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 ...
2
votes
3answers
139 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 ${(k+1)}^{k+1}$
1
vote
3answers
75 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
0
votes
2answers
106 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 ...
2
votes
1answer
105 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 ...
0
votes
1answer
57 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
votes
1answer
42 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.
7
votes
7answers
825 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.
1
vote
2answers
254 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.
2
votes
2answers
78 views

The polynomial $x^{{n-1}} + x^{n-2} + x^{{n-3}} +\dots + x +1$ is reducible when $n$ is composite

Is $P(x)=(x^{{n-1}} + x^{n-2} + x^{{n-3}} +\dots + x +1)$ reducible if $n>1$ and $n$ not prime? If $n-1$ is odd, $(x+1)|P(x)$, so if $n$ is even with $n>2$ I can write $$P(x)=(x+1) Q(x)$$ ...
1
vote
2answers
342 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
votes
1answer
250 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
votes
1answer
70 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 ...
1
vote
2answers
356 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 ...
1
vote
1answer
43 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?
1
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2answers
146 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: $$10r^2 - 31r + 15\\ = (10r^2 - 25r) (-6r + 15)\\ = -5r(-2r+5) -3 (2r-5) $$
1
vote
1answer
433 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 ...
0
votes
1answer
74 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
114 views

Number of factors of a big number

How to find the number of factors of $884466000$ without using a calculator?
0
votes
2answers
139 views

Number of Factors of 6

factor of 6 is 1,2,3,6, or factor of 6 is 1,2,3,6,-1,-2,-3,-6 Which one is correct?
5
votes
2answers
662 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
votes
1answer
62 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 ...
1
vote
1answer
51 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? ...
0
votes
1answer
49 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?
8
votes
0answers
121 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
votes
0answers
26 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 $g(v)$ ...
0
votes
3answers
108 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 ...
1
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0answers
490 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 ...
1
vote
0answers
43 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 ...