# Tagged Questions

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

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### Factorisation of the polynomial $a^6+8a^3+27$

I would like to factorize $a^6+8a^3+27$. I got different answers but one of the answers is $$(a^2-a+3)(a^4+a^3-2a^2+3a+9)$$ Can someone tell me how to get this answer? Thanks.
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### rank revealing qr factorization selection of exactly k rows

I have been trying to implement the algorithm presented in the following paper: An Improved Approximation Algorithm for the Column Subset Selection Problem There is a part of the algorithm which ...
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### How to factorise $(x-1)^2 - (x-5)^2$

My attempt: $a = (x-1)$ $c = (x-5)$ $a^2 - c^2$ which is equal to: $$((x-1) - (x-5))((x-1)+(x-5))$$ But the correct answer is : $8(x-3)$ Can you explain, please?
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If I have factors of linear operators say $$(a_1 + A)(a_2 + A)(a_3 + A)\cdots(a_n + A) = 0$$ $A$ being an linear operator(i guess it really doesn't matter its operator or not) why $$\sum_{n} \frac{... 5answers 53 views ### Formula for factorization of a Quadratic Equation? To be clear I am looking for an equation to go from$$Ax^2 + Bx + C = 0$$To$$(Dx + E)(Fx + G) = 0$$And I need it to be able to be done in a computer as it will be going in my app. Thanks in ... 7answers 748 views ### Factoring polynomials with a 2nd degree coefficient greater than 1 I'm learning how to factor polynomials, but I'm having a hard time understanding the approach when the 2nd degree coefficient is greater than 1. For example, when I begin to factor 12k^4 + 22k^3 - ... 1answer 29 views ### Factoring Polynomials: How do I express the area and perimeter in factored form? Our topic is factoring polynomials, and I can't seem to solve this question: Express the area and perimeter of the shaded region in factored form. We've discussed how to solve for the ... 4answers 2k views ### Have I found all the numbers less than 50,000 with exactly 11 divisors? The math problem I am trying to solve is to find all positive integers that meet these two conditions: have exactly 11 divisors are less than 50,000 My starting point is a number with exactly 11 ... 2answers 38 views ### assuring factorization for R[x] when R is a UFD I wanted to ask, suppose the ring R is a UFD (Unique factorization domain) and I look at R[x], the ring of polynomials over R. I wanted to know, how can I assure that when I have some polynomial ... 2answers 45 views ### 240x^2y^3 - 180x^3y^4 - Factor the given. [closed] 240x^2y^3 - 180x^3y^4 Question: Please Factor the Given Expression. Thanks. In factoring... Is it similar to the Factor Tree? Instead do we not use the tree? My Answer (attempt 1) 15xy^2 (16xy-... 3answers 65 views ### Prove there are infinitely many fields in which the polynomial is reducable Polynomial:$$ t^7+t^2+1$$I have the solution, but I don't understand the thinking behind it. How are they coming up with the factorization, and specific values of t (shown in solution). The ... 0answers 35 views ### factoring polynomials in ring of integers modulo powerful number I am having trouble finding info on how to factor polynomials in ring of integers modulo powerful number. For example: x^2 - 1 in \textbf Z_{8}. I know by tinkering around that (x - 1)(x + 1)... 1answer 42 views ### a and b are factors of 6^6 and a is a factor of b How many pairs of (a,b) of positive integers are there such that a and b are factors of 6^6 and a is a factor of b? What I tried I know 6^6 an be broken down into (2)^6 (3)^6 If ... 0answers 44 views ### Irreducibility of sums of two polynomials I'm interested in a special type of polynomial factorization over \mathbb {Q} : testing the irreducibility of f(x)+g(x), where f and g  are relatively prime and \text {deg}(f)<\text{deg}(g)... 0answers 71 views ### Can we continually factorize an expression like x+y? I have a question that, for lack of familiarity or understanding of the relevant fields, I'm not quite sure how to formulate, so I'll just start off with an example and list some questions as I go. As ... 6answers 117 views ### Factoring out a 7 from 3^{35}-5? Please Note: My main concern now is how to factor 7 from 3^{35}-5 using Algebraic techniques, not how to solve the problem itself; the motivation is just for background. Motivation: I was trying ... 1answer 26 views ### What is the complete (polynomial) factorization of \sigma(p^k), where p is prime with p \equiv k \equiv 1 \pmod 4? The title says it all. What is the complete (polynomial) factorization of \sigma(p^k), where p is prime with p \equiv k \equiv 1 \pmod 4? Here, \sigma = \sigma_{1} is the classical sum-... 4answers 98 views ### Factoring x^4-11x^2y^2+y^4 I am brushing up on my precalculus and was wondering how to factor the expression$$ x^4-11x^2y^2+y^4 $$Thanks for any help! 1answer 34 views ### On smoothness assumptions in Integer factorization I have came across a lot of factorization methods and most of them seem to assume smoothness of some numbers. For example When p-1 is smooth When |E(\mathbb{F}_p)| is smooth. (Elliptic curve ... 1answer 30 views ### Find a factorable cubic polynomial with given conditions I want to write a factorable cubic polynomial in the form Ax^3+Bx^2+Cx+D where -D = 42C. A, B, C, and D should be nonzero integers. Is this possible? 1answer 35 views ### factorize x^5+ax^3+bx^2+cx+d if d^2+cb^2=abd I want to factorize x^5+ax^3+bx^2+cx+d if d^2+cb^2=abd but I don't know how to use the second equality.I tried a lot but I cannot know how to use it for example it is d^2 but we have d and if ... 0answers 50 views ### Factoring x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q) over \mathbb{Q} For a quntic polynomial to be reducible to the following form over \mathbb{Q}:$$x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$$We need to match the coefficients (a=B obviously, so we ... 5answers 100 views ### Factorize a third degree polynomial It's my first time posting here so I'm not used to describing my problem in mathematics. I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but ... 4answers 75 views ### Why is -32^{\frac{1}{5}} = 2 When you factorize -32, you get: -32 = (-16) \cdot 2 -16 = (-8) \cdot 2 -8 = (-4) \cdot 2 -4 = (-2) \cdot 2 -32^{\frac{1}{5}} = -2 The reason I am asking is because you get -4 = -2 \... 1answer 27 views ### Factoring GCF from squared quantities hope you're all well. Quick question: Am I allowed to factor this the way I did here? Thanks for the help, as always! :) 2answers 59 views ### f(x+a) irreducibility means f(x) irreducibility Let a~\in~\mathbb{Z} and let f(x)~\in~\mathbb{Z}\left[x\right]. Suppose that f(x+a) is irreducible over \mathbb{Z}. Prove that f(x) is irreducible over \mathbb{Z}. My idea is: f(x)=u(x)*... 0answers 26 views ### Factoring polynomials when roots are external to the ring I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question. It is clear to ... 1answer 11 views ### Why is the running time of the trial division O(f \cdot (log N)^2)? I saw this being cited in a few paper,but none of them seems to explain why this is the case. Maybe because it is quite trivial, but I am not sure why exactly... Here f is the size of the factor. I ... 0answers 35 views ### On equivalence of RSA and factoring [duplicate] Suppose we are given a number "A" which is multiple of \phi(n). One can assume factorization to be hard. So you cannot find exact value of \phi(n) from A. Clearly using this we can crack ... 1answer 31 views ### What is the algorithm to factor something like 2+\frac{1}{x}+x? [duplicate] I came across this in homework but I'm interested in the general example, say ax+bx^{-1}+c. 2answers 51 views ### Factoring p(x) = x^n -1 for any natural number n Can I say that from inspection, (x-1) is a factor which implies that p(1) = 0. I then used long division which then gave this: p(x) = (x-1)(\sum \limits _{i=1} ^{n} x^{n-i}) How would you have ... 0answers 15 views ### Is it easy to factor if we know k\phi(PQ)? Suppose we know k\phi(N)=k\phi(PQ)=k(P-1)(Q-1) where in N=PQ we have P,Q being similar sized primes and k\in\Bbb Z is unknown can we factor N in polynomial time? 1answer 17 views ### Prove that a(x) divides (v(x) - t(x)) "Let a(x), b(x) \in \mathbb{R}[x], not both the zero polynomial and suppose that gcd[a(x), b(x)] = 1. Let u(x), v(x) \in \mathbb{R}[x] be such that a(x)u(x) + b(x)v(x) = 1 Let also s(x)t(x) ... 2answers 16 views ### Prove that q(x) does not divide p_k(x) Let n \in \mathbb{N} and let p_1(x), p_2(x), ... *p_n(x) be n irreducible polynomials over \mathbb{R}. Define the polynomial p(x) = p_1(x) * p_2(x) *... *p_n(x) + 1  where 1 is the constant ... 2answers 70 views ### Factoring the sequence {10}^{2n}+10^{n}+1 While I am waiting for the basketball NBA game between Cleveland Cavaliers and Golden State Warriors to begin I sort of played with the sequence a_n={10}^{2n}+10^{n}+1 in a way that I looked for the ... 1answer 40 views ### Books about multivariate polynomials I'm looking for a book on multivariate polynomials, preferably a monograph (could also be a chapter inside another book). I'm interested in what can be said about roots, factoring, irreducibility, ... 1answer 41 views ### How many positive two-digit integers have exactly 8 positive factors? I solved this problem by listing all two-digit integers and going through each one. Is there an easy way to solve the problem? How many positive two-digit integers have exactly 8 positive factors? 1answer 73 views ### Finding 8 co-primes \le 2^n We can find 8 co-prime integers \le 2^n for sufficiently large n. I'm looking for asymptotic bounds for the minimum distance away from 2^n we have to go before finding 8 co-primes. In other ... 1answer 37 views ### Hard time factoring Normal Distribution based on transformation problem. My professor gave problems out to practice for our final on Wednesday. This problem is based on the transformation of two random variables. It a 5 part problem, so I will list the necessary portions ... 3answers 61 views ### What is the sum of the prime factors of 2^{16}-1? I know 2^{10}=1024 and 2^6=64, but it seems they are not very helpful in solving this problem. There must be a trick to solve the problem in an easy way. What is the sum of the prime factors ... 0answers 12 views ### Odd factorisation of Indicies? I'm sorry for the terrible formatting, I'm sorta new so I don't know how to use MathJax very well :( see below for my first attempt :) I came across the indical expression :$$\frac{x-1}{x-x^{1/2}-2}$... 2answers 37 views ### How to divide the following polynomial and factor it? The question is $$(2x^3+3x^2-39x-20) / (x-4)$$ I divided the following and got this as the answer $$2x^2+9x+3-8/(x-4))$$ I thought that this was the answer, but when i looked at the answer sheet ... 1answer 59 views ### Proof that$a^{n}+b^{n}$is irreducible over$\mathbb Q$The sum of fourth powers cannot be factored over$\mathbb Q$, since$ a^4+b^4 = (a^2+\sqrt{2}ab+b^2)(a^2-\sqrt{2}ab+b^2)$And these quadratic factors does not have any real rational factors. How ... 2answers 73 views ### How to factor the trinomial :$ xy-x+y-1$? How to factor the trinomial :$ xy-x+y-1$? The factorization is$(x+1)(y-1) $but I don't where it comes from. 3answers 50 views ### Factorization of the polynomial:$m^2+3m^2n^2-30n^2-10$[closed] How can we factor the polynomial$m^2+3m^2n^2-30n^2-10$? 2answers 51 views ### Is$X^5+…+1 \in \mathbb{F_2}[X]$irreducible? I am trying to determine if the following polynomials are irreducible in$\mathbb{F_2}[X]$are irreducible:$f(X)=X^5+X^2+1g(X)=X^5+X^3+1$There are no linear factors since$f(0)=f(1)=g(...
One of Euler's discoveries was if an integer $n$ can be represented as a sum of two squares in two distinct ways, then one can factor $n$ explicitly. Of course, the method was ineffective as an ...
So I have this apparently smooth, parametrized function: The function has a single parameter $m$ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart ...