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

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0
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1answer
14 views

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 $\lvert r \rvert <1$ ...
0
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0answers
20 views

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 ...
2
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0answers
57 views

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 ...
0
votes
1answer
14 views

Substitution of factors of the free term to find factors of a cubic equation

I was taught that finding a factor (and hence a solution) of a cubic equation may be easier if I try if the factors of the free term are roots of the equation. For example, if one has an equation $x^3-...
0
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2answers
123 views

$|x-2|$ as a factor of $|x^n-2^n|$ as a limit of function

I haven't posted in a while but I do have a question on factoring a specific term out of a particular polynomial function and I'm stuck at some point in the process. Please, let's see so that my ...
1
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2answers
41 views

Simple factor of equation

I have this polynomial: $5z^4-12z^3+30z^2-12z+5$ How do I factor it to get the following?: $(5z^2-2z+1)(z^2-2z+5)$ Can someone show me the procedure to perform whenever I encounter with a case like ...
1
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2answers
84 views

Factor 65 into irreducible in $\mathbb{Z}[i]$

Factor 65 into irreducible in $\mathbb{Z}[i]$ I tried to factor 65 in Gaussian integers by Mathematica, and I got $65 = -(1+2i)(2+i)(2+3i)(3+2i)$, but i don't know how to factor it by hand. Could you ...
2
votes
1answer
35 views

How to factor $n$-degree polynomials of this form?

I have come across a very specific form of polynomial, and I was hoping there would be a nice way to factor it - or at least show that it is irreducible. The form is $x^{n+1} + 2k \cdot x^n -1$ where $...
0
votes
1answer
60 views

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 ...
1
vote
4answers
51 views

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 ...
1
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4answers
60 views

How was this factoring of $1-2x-x^2$ achieved?

If I factor $1-2x-x^2$ using the quadratic formula I get $$x=\frac{2\pm \sqrt{4-4(-1)(1)}}{2(-1)}$$ $$x=\frac{2\pm \sqrt{8}}{-2}$$ $$x=-1 \pm \sqrt{2}$$ Let $\alpha = -1 +\sqrt{2}$ and $\beta=-1-\...
0
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0answers
36 views

Strange type of matrix equivalence, $\bf P=Q$. What applications or properties can it have?

Stemming from this question when actually searching for matrix similarities, having found this matrix equivalence: $$\bf A = PBP$$ That is neither transpose nor inversion on either of the $\bf P$s. ...
0
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0answers
27 views

Upper Bound on Number of Factors

Is there a theorem, lemma, or proof somewhere that proves an upper bound for the number of factors that a number can have? If not, would it be fairly trivial to prove that it is $log_2 n$?
-4
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1answer
78 views

Solve the surd equation [closed]

$$ \sqrt{14+ 8\sqrt{3}} = 2\sqrt{2} + b$$ Find b without using factoring b=sqrt{6}
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2answers
44 views
0
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2answers
41 views

Is there a positive integer with $2010$ distinct positive integer factors?

Suppose that, $$f(n)=(\text{ the number of distinct positive integer factors of $n$ including $1$ and $n$ } )$$ Is there a positive integer $m$ such that $f(m)=2010$ ? How can conclude about the ...
12
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5answers
1k views

What is the sum of the reciprocal of all of the factors of a number?

Suppose I have some operation $f(n)$ that is given as $$f(n)=\sum_{k\ge1}\frac1{a_k}$$ Where $a_k$ is the $k$th factor of $n$. For example, $f(100)=\frac11+\frac12+\frac14+\frac15+\frac1{10}+\...
4
votes
2answers
125 views

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 $?
1
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1answer
30 views

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 ...
0
votes
4answers
29 views

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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0answers
11 views

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 ...
1
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2answers
54 views

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 ...
0
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1answer
31 views

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 ...
1
vote
1answer
53 views

Factor $16x^4-x^2y^2+y^4$

So basically I have the answer to this problem because it's in the book, but I have no clue how the author solved it. It's from Schaum's Precalculus. Thanks. Factor $(16x^4)-(x^2y^2)+(y^4) = (4x^2+...
0
votes
2answers
70 views

How to get the partial factor representation of $\frac{1}{x^2+4x+1}$? [closed]

I've been trying to factor this, but I don't see it. What is the partial fractions representation of $$\frac{1}{x^2+4x+1}$$?
0
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3answers
45 views

alternate forms of $t^3 -1$

I'm looking to understand how $t^3-1$ factorises to $(t-1)(t^2+t+1)$. I know how to find the first factor $(t-1)$, but have trouble finding the second factor $(t^2+t+1)$. I've tried doing long ...
0
votes
2answers
52 views

How should I start Factoring this?

So im supposted to factor this and I'm not sure where to start, where should I start? $$2x^5 + 3x^4 -10x^3 -15x^2 + 8x + 12$$
2
votes
2answers
54 views

How does depressing a polynomial help you factor it?

I've noticed that a lot of the derivations for the cubic and quartic polynomial solutions require you to depress them, but what about in a case by case scenario? How does depressing this cubic, $x^3+...
3
votes
1answer
94 views

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 ...
3
votes
2answers
64 views

How does this factoring work?

$$ (z^2 - 2i ) = (z -1 -i)(z + 1 +i) $$ I see if you multiply out the right-hand side, you obtain the left-hand side, but how does one know to factor like that or this? $$ (z^2 − 3iz − 3 + i) = (z − ...
-1
votes
1answer
98 views

What is the largest six-digit number with an odd number of positive factors?

What is the largest six-digit number with an odd number of positive factors? So I know the number must be a perfect square, but how do I know six-digit number perfect squares? I'm pretty sure there's ...
2
votes
2answers
63 views

Prove that a polynomial of odd degree in $ \Bbb R[x]\\$ with no multiples roots must have an odd number of real roots.

The coefficients of the polynomial are in the ring of real numbers. Prove that a polynomial of odd degree in $ \Bbb R[x]\\$ with no multiple roots must have an off number of real roots. I hate to ask ...
5
votes
2answers
62 views

Remainder when $x^{2016}+x^{2013}+\cdots+x^6+x^3$ is divided by $x^2+x+1$.

I am currently preparing for a certain quiz show when I encountered this question: What is the remainder when $x^{2016}+x^{2013}+\cdots+x^6+x^3$ is divided by $x^2+x+1$? I know for a fact that $x^3-...
1
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1answer
39 views

Is there an error in this Precalculus handout on Law of Cosins and Herons Formula?

Apologies, I can't include images - not enough rep. An example exercise dealing with the Law of Cosines to prove Heron's formula for the area of a triangle has me befuddled. After substitution ...
0
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2answers
48 views

I need help factoring quadratics.

I need help factoring $x^4-2x^2-3=0$. I do not know how to factor this disguised quadratic.
2
votes
2answers
52 views

Find a possible f(x)

Do you think that it is possible to find a $f$ such that given a floating point constant $c \gt 0$ and an integer constant $n \gt 0$, then $\forall x_i \gt 1, i=1, 2,...,n$ : $$f\left({1\over x_1}+{...
0
votes
2answers
59 views

Is a polynomial irreducible modulo a silly large prime

Suppose I have a large ($\approx 2^{64}$) prime $p$, and a polynomial $g(x)$ of degree $d$ that irreducible in $\mathbb{Z}[x]$. If $d << p$, is it likely that $g(x)$ is irreducible in $\mathbb{Z}...
4
votes
1answer
80 views

Factoring polynomial

I have a polynomial, let's say $p(x)=x^5+x^4+x^2+x+2$ (or any other polynomial with rational coefficients). What is the general recommended way of factoring it into irreducible factors (in $Z[x]$, $Q[...
1
vote
2answers
162 views

Find a possible f(x) and g(x)

Do you think that it is possible to find a $f$ and a $g$ such that $f(x) \neq x$ and $\forall x \gt 1, y \gt 1$ then $$f\left({1\over 1-{1\over x}-{1\over y}}\right) = {1\over 1-{1\over g(x)}-{1\...
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7answers
95 views

Solve $3 = -x^2+4x$ by factoring

I have $3 = -x^2 + 4x$ and I need to solve it by factoring. According to wolframalpha the solution is $x_1 = 1, x_2 = 3$. \begin{align*} 3 & = -x^2 + 4x\\ x^2-4x+3 & = 0 \end{align*} ...
3
votes
1answer
50 views

What causes the equating-the-coefficients method not to work?

When searching here to find methods for solving quartic polynomials, I came across a question where one of the solutions (at the very end) mentions that the equating coefficients method can fail. http:...
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vote
0answers
77 views

Integer Factorization problem - New Idea

I been thinking about slightly different approach of solving the problem, and I want you to tell me if my idea is reasonable and if it's original(If someone already thought about this, I would be ...
0
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0answers
27 views

Is Horner Algorithm works or suitable for polynomial with complex coefficients?

I have checked Algorithm of Horner for polynomial factorization with complex coeffecients ,I got it works but i can't show it works in general for any polynomial with complex coeffeciant . My ...
0
votes
4answers
90 views

Factoring $x^4 - 16$

I was following a calculus tutorial that factored the equation $x^4-16$ into $(x^2 +4) (x+2)(x-2)$. Why is the factorization of $x^4-16 = (x^2 + 4)(x+2)(x-2)$ rather than $(x^2 - 4)(x^2 +4)$?
0
votes
2answers
100 views

Prove Expression cannot be factored

I am currently working on primes which can be expressed in form of a polynomial. For eg, Find all primes which can be expressed in form $n^4-52n^2+595$ It is very essential to tell whether a ...
0
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1answer
35 views

Express a variable as a function of another [closed]

If we have this formula : $a/(1-$$1\over x$$-$$1\over y$$) = 1/(1-$$1\over xb$$-$$1\over yb$$)$ Is it possible to express $a$ as a function of $b$, independently of $x$ or $y$ ? Thank you
0
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2answers
32 views

Factor Theorem (Finding values of a and b)

Question: The polynomial $p(x) = 2x^3 - ax^2 + bx + 48$ has $(x-4)$ as a repeated factor, find the values of $a$ and $b$. What I have attempted if $x-4$ is a factor then $x = 4$ is a root ...
1
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0answers
33 views

Grouping a set of numbers

I have a set of numbers which I don´t know if they belong to the same group (I could also call it factor or treatment, but actually each group is suppose to identify the same biological event). I am ...
0
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0answers
33 views

Complex Space Factorization

I'm curious as to whether a visualization of the complex space and the area covered through various factorization methods has been put forth before. Each triangle in the mapping below represents a ...
4
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0answers
49 views

Finding how large $p$ needs to be to have $n$ unique factors…

If we take a prime $p$, how large does $p$ have to be so that $p-1$ has at least $n$ factors between $f_1$ and $f_2$? (Note that the factors can be prime or composite) Note that I'm looking more for ...