# Tagged Questions

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

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### Factoring binary quadratic form in two second order polynomials

I have a binary quadratic form in $N$ and $D$, $AD^2 + BND + CN^2$, where $A$, $B$, and $C$ are real coefficients and $N$ is a second order polynomial of $x$ with real roots $\lvert r \rvert <1$ ...
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### Are all integral domains in which all irreducible elements are prime G.C.D domains?

I know that in G.C.D domains all irreducible elements are prime. Does the converse of this statement hold? If not, is there a weaker condition than being a G.C.D. domain that is both sufficient and ...
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### Easy method of determining if a polynomial over $\Bbb{Z}$ has any quadratic factors with rational coefficients

There is an easy method of determining whether a monic polynomial $$\sum_0^n a_k x^k$$ with all $a_k \in \Bbb{Z}$ and $a_n = 1$ has any integer roots. At least it is easy if you can factor the ...
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### Regarding the factorization $a^2+3b^2 = cd$.

Let $a,b,c,d$ be positive integers, with $\gcd(c,d)=1$, such that $$a^2+3b^2=cd.$$ By well-known classical results, we have that $c$ and $d$ are both of the form $u^2+3v^2$. QUESTION: Is it valid to ...
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### Universal factoring method or list of methods (trinomials)

I am a student in calculus II. I'm now failing tests solely because I cannot factor; I understand everything else. This is compounded by the fact it seems to exceedingly hard to find anything ...
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### How can we prove that a quadratic equation has at most 2 roots?

A quad equation can be factored into two factors containing $x$, but how can we prove that there no other sets of different factors yielding OTHER VALUES OF $X$?
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### Derivative of $x(x+2)^3$ in factored form don't see it

Studying for a big test trying to do the derivative of $x(x+2)^3$. I did the product rule and got $3x(x+2)^2 + (x+2)^3$. Pretty standard stuff. I got half credit and was told to simplify it like ...
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### Linear Factorization of Complex Polynomials

I am trying to find a linear factorization of the polynomial $$p(z) = 1 +z+z^2 +z^3 +z^4 +z^5 + z^6 +z^7 +z^8$$ I know what it means by linear factorization in the sense of non-complex polynomials, ...
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### How to store tables for ECM stage 2

This question about realization ECM stage 2 on GPU. I now that there exists some optimization for the stage 2 of ECM. Namely, let $N$ be a composite number, $q|N$ be a prime, $P=(x_P::z_P)$ be a point ...
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### Quickest way to factorize $\frac{w^2 + 5kw + 4k^2}{w^2+kw}$

I would say 90% of polynomials in my textbook are factorable e.g. $$\frac{w^2 + 5kw + 4k^2}{w^2+kw}$$ This gives $$\frac{(w+k)(w+4k)}{(w+k)w}$$ $$\frac{w+4k}{w}$$ This took me far too long to ...
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### What is the lowest degree of a polynomial with integer coefficients having $\sum_{i=1}^k m_i\sqrt[q_i]{a}$ as a root?

Say you want a polynomial with integer coefficients having a root with value $$v =\sum_{i=1}^k m_i\sqrt[q_i]{a}$$ where $k\ge 1$ and $\forall i: m_i, q_i \in \Bbb{Z}^+$, all the $q_i$ are greater than ...
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### Solving $2(n-1)n(n+1)(n+2)=(m-3)(m+3)$

The question is: Find all pairs $(n,m)\in\mathbb{N}^2$ such that $$2(n-1)n(n+1)(n+2)=(m-3)(m+3)$$ I checked all $n<10000$ and only got $n=1$ and $n=4$ with their corresponding $m$, so I ...
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