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

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3
votes
1answer
76 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
16 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
28 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
62 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
158 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
24 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
84 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 ...
2
votes
1answer
40 views

Factorising a complex polynomial over C

If $f(z)=z^3+7z^2+16z+10$, find all factors of $f(z)$ over $C$. If I had at least one zero or factor I would be able to find the others, but I just don't know how to start.
1
vote
2answers
103 views

Solving inequalities with fractions with unfactorable polynomials

So I've been cracking my head open trying to solve this inequality: $$\frac {x+1}{2-x} \le \frac {x}{3+x}$$ I've been taught you have to put all factors to one side of the inequality (leaving zero ...
0
votes
2answers
106 views

Bezout's Identity for polynomials

Im working out a problem where I find out the GCD of two polynomials using Euclid's Algorithm, and then I need to use Bezout's Identity to make $\gcd(r,s)=ra+sb$ The question gives me $x^5+1$ and ...
0
votes
3answers
64 views

Please help me with a (simple?) “solve for x” problem.

I'm preparing for the GRE and was working through an old textbook (chapter on quadratic equations "completing the square," if that helps) and got stumped on $\displaystyle x^2 +{\frac{5x}{a}} + 6x^2 = ...
1
vote
5answers
51 views

Factor fully $98g^2+112g+32$ by decomposition

By looking at this question I understand it is a complex trinomial so do I just decompose it??I have multiplied 98 by 32 getting 3136, but I'm not quite sure what comes next.
-2
votes
2answers
65 views

Factor fully $625-(y-2)^2$

So far, I have used $(y-2)$ twice (multiplying both) because of the exponent being $2$. But, I need to factor and that's when I get confused. Please help!
0
votes
1answer
101 views

factor to find an algebraic expression for the length and width of the rectangle

the area of the rectangle is defined by $$ 6x^2+13x-28 $$ so far, i have decomposed the expression to get $$(3x-4)(2x+7)$$ but, now i need to find the length and width and that's where I have a bit of ...
2
votes
1answer
43 views

Determine 2 values of $k$ so that $36m^2+8m+k$ can be factored over the integers

So, I really need help with this, thank you very much for helping me. Anyway, I understand that $36m^2+8m+k$ is a complex trinomial and when factoring I should use $a^2+2ab+b^2=(a+b)^2$, but this is ...
1
vote
3answers
111 views

how do you factor $x^2 +kx+40$ over the integer

please please help me, I'm having a lot of troubles. I tried to use a^2+2ab+b^2 formula (like i was told) but that's where get lost. I understand that Factoring uses the opposite operation, but 40 ...
1
vote
2answers
88 views

Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
1
vote
1answer
62 views

Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
2
votes
5answers
112 views

Derivation of factorization of $a^n-b^n$

How does one prove that: $$a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\dots+a^2b^{n-3}+ab^{n-2}+b^{n-1}\right)$$ Better yet, why is $a^n-b^n$ divisible by $a-b$? I would very much appreciate some ...
2
votes
2answers
54 views

Is $\gcd(n, \lfloor\sqrt{n}\rfloor!)$ a solution to the factoring problem?

The factoring problem: Factor $n=pq$ given only $n$ where $p$ and $q$ are primes and $0<p<\sqrt{n}<q<n$ I found that $$\gcd(n, \lfloor\sqrt{n}\rfloor!) = p$$ Would this be considered a ...
1
vote
0answers
37 views

On number of different factorizations over integers of a number field

Let $K$ be a finite field extension of the rational numbers and let $\mathcal{O}_K$ denote its ring of integers. If a rational integer $n$ factors into two distinct ways into irreducible elements in ...
2
votes
2answers
64 views

Show that $P(X) -X$ divides $P(P(X))-X$

Let $P$ be a polynomial in $R[X]$. Then show that $P(X) -X$ divides $P(P(X))-X$
0
votes
1answer
26 views

Why is it the case that the common factors of $x$, $y$ are also common factors of $x + y$

Why is it the case that the common factors of $x$, $y$ are also common factors of $x + y$? For example, $10$, $4$ share factors $2$, $1$. $10 + 4 = 14$ which has factors $2$, $1$ Similarly, $100$, ...
0
votes
1answer
49 views

Struggling to prove that if $n$ is a non zero integer, and $m > 0 \mid n$ then $m \leq |n|$

i need to prove that if $n$ is a non zero integer, and $m > 0$ and $m \mid n$ ($m$ divides $n$), then $m \le |n|$. I feel like i can do it by a combination of proof by contradiction and cases (ie ...
0
votes
2answers
43 views

Factorize $15a^2b^3 - 5a^3b^2 - 10a^4b$

I'm having problems with the factorization of $15a^2b^3 - 5a^3b^2 - 10a^4b$. I get stuck on step 2. 1: $15a^2b^3 - 5a^3b^2 - 10a^4b$ 2: $5a^2b (3b^2 - ab - 2a^2)$ How do I factorize $3b^2 - ab - ...
6
votes
3answers
237 views

Factor $x^4+1$ over $\mathbb{R}$

Factor $x^4+1$ over $\mathbb{R}$ Well, I read this question first wrongly, because the reader is about complex analysis, I did it for $\mathbb{C}$ first. I got. $x^4+1=(x-e^{\pi i/4 })(x-e^{3 ...
2
votes
4answers
158 views

Factoring $x^3 - x^2 - x + 1$

I'm familiar with basic forms of polynomial factoring like completing the square or factoring (e.g. finding that $x^2+x-6=(x+3)(x-2)$, but I'm currently working on integration of rational functions by ...
2
votes
2answers
89 views

Unique number of numbers multiplied together

I'm sure this has been asked before, but how many unique numbers can be made from multiplying $4$ numbers, each between $1$ and $100$? My guess is the all numbers from $1$ to $100^4$ except those ...
0
votes
0answers
29 views

Bounding parameters for Shor's Algorithm

I am working on a program for classical implementation of Shor's algorithm to factor products of small primes, just to better understand how it works. The program will conduct order finding using ...
2
votes
2answers
106 views

Factoring $x^n + y^n$ over the integers

Here's what i know (or think i know) about the factoring. For integer $n> 1 $ 1) If $n$ is a positive power of $2$ then it is irreducible. 2) If $n$ is an odd prime then $$x^n + y^n = (x + ...
0
votes
2answers
57 views

How to factor cubics having no rational roots

$$-8x^3 +8x -3 = 0$$ I've already tried the possible roots of $\pm 1$ and $3$ using the rational roots test, but none of these help break it down into something more workable. How do I solve this ...
0
votes
3answers
37 views

Derivative Confusion

I am confused about something. In derivation we learnt that; a^x = a^x . lna Now the question that comes to mind is what is the difference when we have: a^3 =
1
vote
1answer
107 views

Sums and differences of distinct factors

Given $k, n \in \mathbb{N}$, let $\tau_{k}(n)$ denote the $k$th positive factor of $n$ in strictly increasing order. For example, $\tau_{1}(6) = 1; \tau_{2}(6) = 2; \tau_{3}(6) = 3; \tau_{4}(6) = 6$. ...
2
votes
2answers
121 views

Solve the equations $z^2 + (2 - 2i)z + 2i = 0 $ by completing the square

I tried solving this thing by completing the square and I always end up with something like this $(z^2 + (2 - 2i)z - 2i) + 2i + 2i = 0 $ and it doesn't seem like to me that you can factor the part in ...
0
votes
1answer
40 views

Factoring a matrix out of linear matrix equation

I'm having a bit of trouble following a solution in a textbook, one step in particular. I have the equation $(Z + tV)^{-1}$ where $Z$, $V$ are matrices and $t$ is a scalar. $Z$ is positive definite, ...
2
votes
0answers
67 views

Probability distribution of the (size of the) smallest prime factor

Related question: Expected smallest prime factor Background: Given a toolbox of factorization algorithms (like trial division, ECM, quadratic sieve, GNFS) and a set of large composite numbers, I'm ...
0
votes
2answers
40 views

Mixed Signed Factorising

I need hep factoring mixed signed expressions. I know how to factorise but I'm getting really confused on to which side to put the negetive sign on etc. E.G. $1)$ $x^2 - 10x + 16$ My Steps: $1)$ I ...
1
vote
1answer
849 views

Factoring using the 'Criss-Cross' method

my teacher taught our class how to factorize using the criss cross method, and I did not understand what she tried to communicate to the class. The equation I am trying to solve as an example is $7x^2 ...
14
votes
5answers
1k views

Factoring a hard polynomial

This might seem like a basic question but I want a systematic way to factor the following polynomial: $$n^4+6n^3+11n^2+6n+1.$$ I know the answer but I am having a difficult time factoring this ...
4
votes
3answers
687 views

Factoring the quintic polynomial $x^5+4x^3+x^2+4=0$

I am trying to factor $$x^5+4x^3+x^2+4=0$$ I've used Ruffini's rule to get $$(x+1)(x^4-x^3+5x^2-4x+4)=0$$ But I don't know what to do next. The solution is $(x+1) (x^2+4) (x^2-x+1) = 0$. I've ...
1
vote
2answers
76 views

What is wrong with this equations?

so our math teacher told us how 2 * 2 = 5 today and we were like :O I thought really hard to disprove this but it seems correct, would someone please tell me how is this possible! proving 2 * 2 = 5 ...