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

learn more… | top users | synonyms

10
votes
3answers
174 views

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

If $\begin{cases}a^4+b^4\in\mathbb Q\\ a^3+b^3\in\mathbb Q\\ a^2+b^2\in\mathbb Q\end{cases}$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$. It is given that $a,b\in\mathbb R$. The proof of ...
2
votes
1answer
84 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
0
votes
1answer
17 views

Find subset of rows whose entries sum to an even number in each column

I am trying to implement Fermat factorization with factor bases. The textbook suggests using row-reduction to find a linearly dependent set of rows. How does one go about finding such a linearly ...
1
vote
1answer
15 views

Using information from a congruence to factor a number

I am being asked to factor $15347$ given that $7331^2 \equiv 1460^2 \pmod{15347}$. I've tried playing around with each of the numbers -- prime factorization, gcd, lcm, etc., but I can't find a ...
3
votes
2answers
33 views

$a,b,c$ are three distinct natural numbers. Then how many ordered triplets $(a,b,c)$ will exist such that L.C.M (a,b,c) = 144.

Let $a,b,c$ be three distinct natural numbers. Then how many ordered triplets $(a,b,c)$ will exist such that L.C.M (a,b,c) = 144. Here's how I proceeded, 144=$(2^4)(3^2)$, so 144 has 15 factors(1 ...
0
votes
2answers
36 views

Cubic factoring question

I'm trying to figure out how a colleague factored an expression. I don't get how: $$a^3+a^2b-(b+1)=(a-1)[a^2+a(b+1)+(b+1)]$$ Multiplying the result I see it's true, but not sure how he got there..is ...
2
votes
2answers
83 views

Can fractions be relatively prime?

Two numbers are relatively prime if they do not share any factors, other than 1. Is it possible for fractions to be relatively prime? To reword this, do fractions even have factors?
1
vote
2answers
79 views

Using telescoping property to prove difference of powers

Ok so I have started working through Apostol calculus and as you can see I am stuck. The problem is that I can not see the telescoping pattern anywhere for following problem. Prove that $$a^n - b^n ...
0
votes
3answers
70 views

How to find sum of factors of $2^{2012}$?

This question really is confusing me and I was wondering if there was a simple way this could be achieved. I've come up with this so far... $\sum_{n=0}^{2012} 2^n$ PS. Please forgive me for my ...
1
vote
1answer
44 views

Efficient factorization of numbers with unique prime factors

I need a factorization algorithm for numbers of the form $n = p_{1}p_{2}\cdots p_{k}$ with $p_i \neq p_j$ for $i \neq j$ and $p_j \in \{p : p \mbox{ is a prime and } p \leq P_s\}$, where $P_s$ is the ...
0
votes
1answer
65 views

Given a polynomial of degree 5, get minimum and maximum without using derivatives

Given a quintic polynomial (in my case, $x^5+2x^4+16x-32$), I am supposed to get its maximum and minimum value for the interval $I=[-2;2]$ without using the derivative of the corresponing polynomial ...
0
votes
0answers
17 views

WordProblem on factors and remainder theorem

Mr.Chaalu while travelling by Ferry queen has travelled the distance one Kilometer more, than the fare he paid per km. Initially he had total amount of Rs.350/- in his wallet. Now he is only left with ...
0
votes
3answers
53 views

Can anyone factor this?

-x^3 + 12x + 16. I am trying to solve for the zeros, but it seems that I have forgotten all my neat little tricks. Not a difference of cubes, or any of the common forumlas. I'm thinking maybe some ...
6
votes
3answers
371 views

Smallest known unfactored composite number?

I'm trying to find examples of "small" numbers which are known to be composite, but for which no prime factors are known. According to this website the number $109!+1$ is a composite number of 177 ...
0
votes
2answers
31 views

Factorise A number in to product of two numbers

I would like to know what is the quickest way to factorise large number ( more than 1000) in to two numbers For Example 2669 2669 is 17 * 157 how can I find this ?
0
votes
1answer
43 views

Trigonometric Functions using factorisation and small angle identities

The function f(x)=sin3x - sin2x + sins is defined for the domain 0 $\le$ x $\le$ $\frac{\pi}{2}$. a) By method of factorisation, show that f(x) = sin2x(2cosx - 1). b) Hence solve the equation f(x) ...
1
vote
2answers
41 views

Convert from decimal to fraction

I know this sounds silly, and it's easy for many situations. But sometimes i have been completely taken back as i don't know how to do it. So please tell me is there any way to convert certain ...
0
votes
2answers
35 views

Factorisation of a polynomial [closed]

I have a polynomial $$t^4-4\lambda t^2-4t^2 $$ I need to give a real value to $\lambda$ such that i get 4 real roots.
0
votes
4answers
61 views

How do I compute the sum of 2 squares

if $x+y=a$ and $xy=b$, what does $x^3+y^3$ equal? I understand that $x^3+y^3=(x+y)(x^2-xy+y^2)$ but I don't see how I can figure out what $x^2$ or $y^2$ equals
1
vote
2answers
47 views

Factor irreducible polynomial in Z[x] and R[x]

I've got a couple of problems from an old exam in abstract algebra that I have difficulty in understanding. 1) Write the polynomial $2x^3 - 10$ as a product of irreducible elements in ...
1
vote
1answer
40 views

Factoring and Simplifying

I'm trying to do this problem, $$(4x + 1)^{15}\cdot\frac{1}{3}(12x - 5)^{-\frac{2}{3}}\cdot 12 + (12x - 5)^{\frac{1}{3}}\cdot15(4x + 1)^{14}\cdot 4$$ I've gotten down to, ...
3
votes
3answers
67 views

Factors of zero?

How many integer factors of 0 are there, and what are they? I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?
1
vote
1answer
23 views

Probability Distribution of Count of Factors for All Numbers

Is the following a known thing? Define "factor count" as the count of factors each number has, then subtract 1. Ignore the number "1" as a factor. For example: Prime numbers have a factor count ...
1
vote
1answer
36 views

How to give irreducible factorization in $\mathbb{Z}_5$?

I have this polynomial: $$f=x^5+2x^3+4x^2+x+4$$ How can i find the irreducible factorization(in $\mathbb{Z}_5$)?I can find the roots easily but thats not enough.
0
votes
1answer
27 views

Prove inequality of following type other than in induction method

I need to prove the following. $$(1^r + 2^r +\ldots + n^r)^n > (n^n)(n!)^r$$ where $r$ being a real number. I tried to solve it through induction method but it got complicated. How to solve this ...
0
votes
1answer
20 views

Factoring for Strong Induction for Fibonacci Sequence

Fibonacci: prove the following theorem: define the Fibonacci sequence $\left\{ a_n\right\}_{n=0}^{\infty}$ by $a_0=a_1=1$ and for integers $k>1$, $a_k=a_{k-1}+a_{k-2}$. Then, for each integer $n$, ...
14
votes
1answer
512 views

Is factoring polynomials easier than factoring integers? [duplicate]

I was reading the book Algebra: Chapter 0 , by Paolo Aluffi, and came across the following assertion, in page 290, Exercise 5.9: It is in fact much harder to factor integers than integers ...
1
vote
2answers
54 views

What are irreducible factors?

What are Irreducible factors? I have to solve this question: Find the irreducible factors of $x^4 + 5x^3 + 8x^2 + 9x + 10$ in ${\bf Z}_{11} [x]$. I couldn't find any websites that explained ...
2
votes
1answer
106 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
1
vote
1answer
24 views

Determine the values of x for which the linear approximation is accurate to within 0.1.

So I've got a function: $$\frac{1}{(1+2x)^4}$$ with its linear approximation: $$1-8x$$ For all values of $x$ where the linear approximation is accurate within $0.1$, then surely we subtract the ...
0
votes
2answers
44 views

How factor with square root

I have the following equation that I'm trying to factor, but I'm stuck at the end. $$\frac{zx^{-4}\sqrt{x}(yz^4)^3}{z^7xy}$$ $$\frac{\frac{1}{x^4}\sqrt{x}(yz^4)^3}{z^6xy}$$ ...
1
vote
3answers
26 views

Need help with basic factoring equation

I'm just trying to brush up on my factoring of quadratic equations. $$\frac1{x+3} + \frac1{x^2 + 5x +6}$$ $$\frac1{x+3} + \frac1{(x+2)(x+3)}$$ $$\frac{(x+2)(x+3) + (x+3)}{(x+2)(x+2)(x+3)}$$ Then ...
0
votes
2answers
24 views

Polynomial identity for a sum

If $$f(x) = \sum_{i=0}^{n}A_i x^i \quad \text{ and } \quad g(x) = \sum_{i=0}^{n}B_i x^i$$ are two degree $n$ polynomials, then we can say that the polynomial $$h(x) = \sum_{k=0}^{2n}C_k x^k \quad ...
-2
votes
3answers
132 views

An intutive way to think about odd and even numbers. [closed]

What is an intuitive way to think about odd and even numbers? And about divisibility also...
3
votes
1answer
75 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
2
votes
1answer
15 views

Factoring a series of Matricies

I have a graduate physics text which has an arbitrary matrix $M$. I am given the following $M^{0} + M^{1} + M^{2}...M^{n-1}$ and then it says this is equal to $(M^{n}-I)(M-I)^{-1}$, where $I$ is the ...
1
vote
3answers
42 views

(Factorization) How can I factorize this?

I'm not sure about how to factorise this. I'd appreciate some help. Thanks! $(12x-y)^2-(4x-3y)^2$.
0
votes
1answer
20 views

Show that a Polynomial has certain factorization

$P(x)$ is a polynomial in $x$ of degree $\leq n-1$. Show that $P(x)$ has $n-1$ distinct roots and thus has the factorization $$k\Pi_{i=2}^n(x-a_i)$$, where the constant $k$ is the coefficient of ...
0
votes
0answers
26 views

Proof for uniqueness for ideal multiplication

I am across the following question here: The uniqueness of a special maximal ideal factorization Let R be a domain, and let I be an ideal that is a product of distinct maximal ideals in two ways, ...
1
vote
0answers
52 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where p is prime integer and f is primitive integer polynomial that is irreducible ...
3
votes
1answer
131 views

Extra help on inequality

Someone very helpfully provided an answer to an inequality. See Hard Olympiad Inequality However I don't get part of their answer. How did they get the last factorization??? Thanks so much for any ...
1
vote
0answers
12 views

By what factor do winning chances increase based on total value?

Say I am entering 24/7 in endless sweepstakes, contests, giveaways, drawings, etc. Assuming each one I enter has no less than 1 in 1,000 chances, but no more than 1 in 1 million (and I enter at least ...
0
votes
3answers
40 views

Basic Algebraic Manipulation

How would I solve for $X$ in this instance? I can't figure out how to get the $X$ variables by themselves and the known values on the other side by themselves. $2(4-X)(4-X)+X = 3$
0
votes
1answer
23 views

How to simplifying and solving this polynomial?

I have a problem with simplifying the polynomial. In the first time, I see that this polynomial is quite simple, but when I'm trying, I realized that this polynomial isn't as easy as I saw. Here is ...
0
votes
1answer
42 views

Factor this equation [closed]

Can someone factor this for me? $(x^{\frac{n}3}-a^{\frac{n}3})$ I am stuck on it. Let n be any natural number.
0
votes
0answers
31 views

Is this factorization true for all $n$ in the natural numbers

I need to know if $x-a=(x^{\frac{n}3}-a^{\frac{n}3})(x^{\frac{n+1}3}+a^{\frac{n}3} x^{\frac{n}3}+a^{\frac{n+1}3})$ Is true. I know its true for $n=1$, is it true for all natural numbers though?
5
votes
3answers
155 views

Prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$

I want to prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$. Eisenstein's criterion doesn't apply here, so I think the simplest method would be to use the Rational Roots Test, right? If I can use ...
0
votes
4answers
23 views

What formula do I use for factoring these?

An elementary question, but I am having a lot of discrepancies identifying the correct formula to use, I can do more complex ones but not the simple ones if that makes sense. a) $8x^3 + 1$ b) $m^2 - ...
1
vote
2answers
38 views

Factoring Fully.

I am completely confused as to what to do, I don't understand how to factor with the brackets. $$42x^7(a+10)+60x^5(a+10)-24x^2(a+10)$$ Also state factoring used... Please and thank you. Steps?!?!
1
vote
2answers
82 views

Factoring $x^4 - x^2 + 1$

I'm interesting in finding the possible quadratic factorization of this polynomial: $x^4 - x^2 + 1$. My first idea was to do long division by $x^2+1$, but I did get a remainder, so I presume this ...