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

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7
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2answers
110 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
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 ...
2
votes
3answers
133 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
67 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
1answer
59 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 ...
1
vote
1answer
81 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
42 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
36 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
360 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
75 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.
1
vote
2answers
158 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
162 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
53 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
178 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
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?
1
vote
2answers
107 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: $$10r^2 - 31r + 15\\ = (10r^2 - 25r) (-6r + 15)\\ = -5r(-2r+5) -3 (2r-5) $$
0
votes
1answer
152 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
44 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.
1
vote
2answers
66 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?
0
votes
2answers
120 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?
5
votes
2answers
293 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
55 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
48 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
41 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?
6
votes
0answers
79 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
20 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
88 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
vote
0answers
228 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
37 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
vote
1answer
74 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 ...
1
vote
2answers
248 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 ...
2
votes
3answers
515 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
97 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
45 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
votes
2answers
113 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
votes
2answers
29 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
votes
3answers
34 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
votes
2answers
61 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
votes
0answers
72 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
votes
1answer
48 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
votes
2answers
55 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
133 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
56 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
votes
1answer
74 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
votes
1answer
27 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
111 views

How do I factor this polynomial $x^5-4x^3+8x^2-32$? [closed]

How do I factor this polynomial? $p(x) = x^5-4x^3+8x^2-32$
2
votes
3answers
95 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 ...
1
vote
3answers
57 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
votes
0answers
245 views

New method derived out of Fermat's factorization method

Let us take two examples: a) $N=943=41*23=(\frac{41+23}{2})^2-(\frac{41-23}{2})^2$ but if we take $B=\frac{N+1}{4}$ then we can represent it as $B={x}^2-({y}^2+y)$ and in our case: ...
0
votes
3answers
46 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 ...