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

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2answers
104 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
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1answer
104 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
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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
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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.
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7answers
772 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
248 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
77 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 $$P(x)=(x+1) Q(x)$$ ...
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2answers
338 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
246 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
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 ...
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2answers
347 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
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?
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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: $$10r^2 - 31r + 15\\ = (10r^2 - 25r) (-6r + 15)\\ = -5r(-2r+5) -3 (2r-5) $$
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1answer
416 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
71 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
108 views

Number of factors of a big number

How to find the number of factors of $884466000$ without using a calculator?
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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?
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2answers
639 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
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
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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
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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?
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0answers
119 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
24 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
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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 ...
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0answers
474 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
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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 ...
1
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1answer
203 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
1k 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 ...
4
votes
3answers
1k 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 ...
4
votes
2answers
152 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 ...
1
vote
1answer
51 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 ...
0
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2answers
269 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 ...
0
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2answers
41 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?
0
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3answers
40 views

Factorize $(5a + \frac23)^2* (2a - \frac12)^2$

I found this question in my textbook - $$\left(5a + \dfrac23\right)^2\cdot \left(2a - \dfrac12\right)^2$$ I think it is already factorized but is there a way I can factorize it some more ? I tried ...
3
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2answers
66 views

How do you factor this? $x^3 + x - 2$

How do you factor $x^3 + x - 2$? $(x-1)(x^2 + x + 2)$ Note the factored form here. Thanks!
3
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0answers
104 views

Symmetry in the Sum of Factors

The factors of 72 are 1,2,3,4,6,8,9,12,18,24,36,72. The factors can be organized as such $$m=1,2,3,4,6,8$$ $$n=72,36,24,18,12,9$$ Knowing the symmetric properties and that the 12 factors exist. Pair ...
0
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1answer
50 views

Solving or factors the given polynomial.

I have a polynomial and would like to solve it for "r". We can also do factorization if possible but important thing is to find the values of r. We will get possibly three solutions from this ...
0
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2answers
92 views

Finding Factors Efficiently

Let $m$ and $n$ be positive integers. What is the most efficient way to choose factors that solve this equation. Notice that two factors of 2079 must sum to 36. What is a quick way of picking numbers? ...
4
votes
1answer
199 views

Solve $a^3 + b^3 + c^3 = 6abc$

Find solutions for $a^3 + b^3 + c^3 = 6abc$ in $\mathbb{N}$, such that $gcd(a,b,c) = 1$, except for $(1,2,3)$ and its permutations. Using trial and error I found out that if $a,b,c$ are solution ...
4
votes
2answers
74 views

Prove irreducibility of a polynomial

Let $m$ be an integer, squarefree, $m\neq 1$. Prove that $x^3-m$ is irreducible in $\mathbb{Q}[X]$. My thoughts: since $m$ is squarefree, i have the prime factorization $m=p_1\cdots p_k$. Let $p$ be ...
0
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1answer
101 views

Polynomial Factorisation

Consider that we have a polynomial like $$x^3- (a + b +c ) x^2+abx-abc+s$$ Which is multiplication of $$(x-a)(x-b)(x-c)+s$$ Is it possible to reach value= $abc$ knowing the Coefficients and exponents ...
0
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1answer
33 views

factoring a differential quotient

The original function is $$ (y^2+yx)dx+x^2dy = 0 $$ I've arrived at $$\frac{dx}{x} + \frac{du}{u(u+2)} = 0 $$ The text book carries on to factor as such, but I don't understand how they justify it: ...
2
votes
3answers
142 views

Factor $(x+y)^7-(x^7+y^7)$

So I was doing some practice problems to prepare upcoming math contests. This is one of the problems: Factor $(x+y)^7-(x^7+y^7)$ I got zero for $(x+y)^7-(x^7+y^7)$, however, the solutions ...
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vote
3answers
77 views

Expressing a $3\times 3$ determinant as the product of four factors

I am attempting to express the determinant below as a product of four linear factors $$\begin{vmatrix} a & bc & b+c\\ b & ca & c+a\\ c & ab & a+b\\ \end{vmatrix} = ...
0
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3answers
58 views

Factorising a 3 x 3 determinant - What Am I doing Wrong?

$$\begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \\ \end{vmatrix}$$ subtracting the top row from the middle and bottom rows $$ = \begin{vmatrix} 1 & a & a^3 ...
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vote
2answers
55 views

Factorizing a difference of two $n$-th powers

How can be proved that $$a^n-b^n=\displaystyle\prod_{j=1}^{n}(a-\omega^j b)$$ where $\omega=e^{\frac{2\pi i}{n}}$ is a primitive $n$-th root of $1$?
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1answer
112 views

Sum of number of factors of first N numbers [duplicate]

Given a number N ( Value can be large like N < 10^9 ) How can we calculate sum of the number of factors of first N numbers?? Example : For n = 3 Answer: = #f(1) + #f(2) + #f(3) --- { #f(n) ...
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vote
2answers
283 views

Find the square root of $(x^2 + 3x + 7)(x^2 + 5x + 3) + (x − 2)^2$

I want to find the square root of $$(x^2+3x + 7)(x^2+5x+3)+ (x −2)^2$$ First , I would like to know if it is really necessary to expand everything , because I think it is in the given form for a ...
4
votes
4answers
273 views

Factorize : $x^6 − 10x^3 + 27$

I want to factorize $$x^6 − 10x^3 + 27$$ I tried two methods , first I let $y=x^3 $ and converted it into a quadratic but the solutions are not real . The second method I tried was getting it to ...
3
votes
2answers
204 views

Why does prime factorization hold in the set of integers of the form $4k+1$?

I want to prove that in the set $$ S = \{4k+1 : k\text{ is a positive integer}\}$$ (i.e. $S = \{1, 5, 9, 16, \dots \}$) unique prime factorization holds. How do I do that? Edit: a prime in this ...