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

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1
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2answers
113 views

How do I factor equations involving $e^x$?

I was reviewing some of my notes from Calculus 1 so that I can prepare for Calculus 2 this fall, and I ran into one problem where I don't understand how the factoring works. $$\lim_{x\to\infty} ...
26
votes
2answers
2k views

Factorize $(x+1)(x+2)(x+3)(x+6)- 3x^2$

I'm preparing for an exam and was solving a few sample questions when I got this question - Factorize : $$(x+1)(x+2)(x+3)(x+6)- 3x^2$$ I don't really know where to start, but I expanded everything to ...
2
votes
1answer
93 views

Greatest common divisor of $n!$ and $ H_n n!$

Let $H_n$ be the $n$th harmonic number, ie. $H_n=1+\frac{1}{2}+\frac{1}{3}+ \cdots+\frac{1}{n} .$ I would like to get the value of $\gcd(n!,H_n n!)$, where $\gcd$ is the greatest common divisor, ie, ...
2
votes
2answers
85 views

Nonunits in a Noetherian Domain have an Irreducible Factor

I think I've proven the following statement without using the fact that it is a domain: Prove every nonunit in a Noetherian domain has an irreducible factor. Proof: Suppose we have a ring which ...
77
votes
11answers
4k views

Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and ...
3
votes
2answers
82 views

Factorization of a linear combination of matrices

I'm trying to understand the determinant from Axler Sheldon's paper and there is one point in the very beginning that I don't understand :S (Link below to the paper) ...
15
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9answers
2k views

Is $83^{27} +1 $ a prime number?

I'm having problems with exercises on proving whether or not a given number is prime. Is $83^{27} + 1$ prime?
5
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1answer
374 views

Isomorphism of formal power series factorrings over polynomials

This problem is taken from the Hartshorne's book Algebraic Geometry, Chapter 1, Section 5, Problem 14(a). Two polynomials $f(x,y)$ and $g(x,y)$ are written in the form $$f(x,y) = f_{r}(x,y) + ...
3
votes
2answers
178 views

Why can't $x^k+5x^{k-1}+3$ be factored?

I have a polynomial $P(x)=x^k+5x^{k-1}+3$, where $k\in\mathbb{Z}$ and $k>1$. Now I have to show that you can't factor $P(x)$ into two polynomials with degree $\ge1$ and only integer coefficients. ...
1
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1answer
98 views

how to interpret theorem about polynomial factorization over modulo ring?

polynomial $X^n+a_1X^{n-1}+...+a_n \in \Bbb Z_2[X]$ doesn't have linear factors $\iff a_n(1+\sum a_i) \neq 0$. How then $f(X)=X+1$ can has no linear factors? Doesn't the condition expands to ...
2
votes
4answers
195 views

How to factor $8xy^3+8x^2-8x^3y-8y^2$

How can I factor $8xy^3+8x^2-8x^3y-8y^2$ or the different form $2x(4y^3+4x)-2y(4x^3+4y)$ Is there any general methods that work? A possible solution should be $8(x^2-y^2)(1-xy)$ But please do not ...
1
vote
1answer
394 views

Cholesky/LU decomposition from matrix and its inverse?

Usually, we have a matrix $A$ and want to calculate the $LU$ (or sometimes Cholesky, depending on $A$'s properties) decomposition. This is often the hard part. Now, if we have the $LU$ decomposition ...
2
votes
1answer
52 views

(CHECK) Cardinality of Terms in the Expansion of a Product of Multinomials

QUESTION: How many terms are there in the expansion of $$(x+y)(a+b+c)(e+f+g)(h+i)$$ I'd like some help with this one, but I'd also like to discuss a method of generalization on the problem, ...
1
vote
1answer
140 views

Simplifying the expression $(\sqrt{5}+\sqrt{7})/(\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21})$

Alrite guys, this question might sound stupid, but I can't find a way to simplify this complicated expression: $$\frac{\sqrt{5}+\sqrt{7}}{\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}}$$ I can't take the ...
3
votes
3answers
326 views

Strategies for Factoring Expressions with Four Terms

I'm trying to come up with a general strategy for factoring expressions with four terms on the basis of the symmetries of the expressions. One thought I had was the following: count up the number of ...
7
votes
3answers
466 views

Is $t^4+7$ reducible over $\mathbb{Z}_{17}$?

Is $f=t^4+7$ reducible over $\mathbb{Z}_{17}$? Attempt: I checked that $f$ has not roots in $\mathbb{Z}_{17}$, so the only possible factorization is with quadratic factors. Assuming ...
2
votes
2answers
102 views

Factoring a given polynomial

I am trying to factor the polynomial $$(a-1)x^2 + a^2xy+(a+1)y^2.$$ The problem previous to it in the book uses the method of factoring a polynomial of the form $$ax^2 + bx +c$$ by inspection, and ...
1
vote
2answers
75 views

Factorizations of $x^2+x$ in $\mathbb Z_6[x]$

So I was looking through my old algebra book and found a question that I can't seem to answer. Find two Factorizations of $x^2+x$ as the product of nonconstant polynomials that are not associates of ...
2
votes
1answer
187 views

When factoring polynomials does not result in repeated factors

I found the following statement in the book introduction to finite fields and their applications: Let $x^n-1 = f_1(x)f_2(x)\dots f_m(x)$ be the decomposition of $x^n-1$ into monic irreducible ...
1
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1answer
85 views

How many divisors of $n$ are less than or equal to $m$?

Can I calc it in less than $O(\sqrt{n})$ time?
0
votes
2answers
57 views

Can I change the order of two terms when factoring: $x^2(x^2-4-3x)$ to $x^2(x^2-3x-4)$?

I'm doing homework and I'm stuck on this assignment: $$x^4 - 4x^2 - 3x^3$$ I figured this would equal $$x^2(x^2-4-3x)$$ Now I know if I would change the order to $$x^2(x^2-3x-4)$$ I can factorise ...
1
vote
3answers
120 views

Factor Equation

Help me with this, Question: factor $x^3y-x^3z+y^3z-xy^3+xz^3-yz^3$. Solution: $$\begin{eqnarray}&=&x^3y-x^3z+y^3z-xy^3+xz^3-yz^3\\ ...
1
vote
3answers
66 views

Factor Equations

Please check my answer in factoring this equations: Question 1. Factor $(x+1)^4+(x+3)^4-272$. Solution: $$\begin{eqnarray}&=&(x+1)^4+(x+3)^4-272\\&=&(x+1)^4+(x+3)^4-272+16-16\\ ...
0
votes
1answer
37 views

Trying to isolate X in this formula

I have this formula: Position in an array = $x + y$ * Self.width + layer * Self.width * Self.height If I know the position how can I find $x$ based on the position only with this formula? How to ...
0
votes
1answer
526 views

Factor $x^6 +5x^3 +8$

I wanted to know, how can I factor $x^6 +5x^3 +8$, I have no idea. Is there any method to know if a polynomial is factored. Just some advice will do. Help appreciated. Thanks.
15
votes
4answers
803 views

Factor $x^4 - 11x^2y^2 + y^4$

This is an exercise from Schaum's Outline of Precalculus. It doesn't give a worked solution, just the answer. The question is: Factor $x^4 - 11x^2y^2 + y^4$ The answer is: $(x^2 - 3xy -y^2)(x^2 + ...
2
votes
0answers
107 views

Quadratic Diophantine Equations in Polynomial Time

Considering the problem of finding lattice points $(x_1, x_2 ... x_n)$ that satisfy a quadratic law: $F(x_1, x_2... x_n) = 0$ such that $F(x_1, x_2... x_n)$ is a second degree polynomial It is ...
2
votes
1answer
404 views

Patterns in $GF(2)$ Polynomial division.

I am testing Prime polynomials in $GF(2)$ and have noticed a pattern that I hope will help. There's a calculator here if you want to familiarise yourself with polynomials over $GF(2)$. I am testing ...
1
vote
2answers
515 views

Test to see if a degree $\leq4$ polynomial is factorable

I'm in the middle of a programming project and we'd like to have tests to determine if polynomials in $\mathbb{Z}[x]$ of degrees up to 4 are factorable over $\mathbb{Q}$. A test that computes the ...
0
votes
1answer
61 views

how do you find the highest common factor of two multivariate polynomials?

How do you find the highest common factor of two multivariate polynomials? I am happy to get answers that are only useful for polynomials over the real numbers, as that is what I am dealing with.
1
vote
2answers
106 views

Factoring any single-variable polynomial in $\mathbb C$

The fundamental theorem of algebra says $$ \forall p(x):\mathbb C \to \mathbb C,\ p(x) = a\prod_{n=0}^m\big(b_nx+c_n\big) $$ where $p(x)$ is a single-variable polynomial, and $\{a;m\}\cup\{\forall ...
2
votes
2answers
559 views

Primality test square root of n

I was reading about primality test and at the wikipedia page it said that we just have to test the divisors of $n$ from $2$ to $\sqrt n$, but look at this number: $$7551935939 = 35099 \cdot 215161$$ ...
2
votes
1answer
121 views

Factoring a third degree polynomial

I'm trying to find all solutions for $36x^3-127x+91=0$ with $x \in \mathbb{R}$. So, I tried to factor this polynomial. It can be written in the following way: $$ (ax^2+bx+c)\cdot(dx+e)\quad ...
5
votes
1answer
58 views

Balanced factors

A non-square number cannot be factored to two identical factors. However, not all non-squares are equal: some of them can be factored to relatively close factors (for example, $6=2*3$), while others ...
5
votes
6answers
211 views

Factoring Quadratic Trinomials

I'm currently doing some homework, but I'm COMPLETELY stuck on one problem. I need to factor the following trinomial: $$5x^2+7xy+2y^2$$ How can I solve this problem? I have no idea what to do ...
0
votes
3answers
84 views

Factorize polynomial in $\mathbb R[x]$ and $\mathbb C[x]$

Factorize the polynomial $x^7-7x^6-x^5+7x^4+x^3-7^2-x+7$ So, I have to factorize this in $\Bbb R[x]$ and $\Bbb C[x]$, but when I'm trying to apply the Ruffini schema, I don't know how to put the ...
4
votes
4answers
192 views

I found out that $p^n$ only has the factors ${p^{n-1}, p^{n-2}, \ldots p^0=1}$, is there a reason why?

So I've known this for a while, and only finally thought to ask about it.. so, any prime number ($p$) to a power $n$ has the factors $\{p^{n-1},\ p^{n-2},\ ...\ p^1,\ p^0 = 1\}$ So, e.g., $5^4 = ...
1
vote
1answer
69 views

Divisibility and factors [duplicate]

1) Can factors be negative? Please prove your opinion. 2)If prime factorization is given to you, how will you find out how many composite factors are there? Not the factors, just how many. For 2), my ...
1
vote
1answer
63 views

How to isolate j?

can anyone explain me how to isolate the j variable please? $$q = \frac{1 - (1 + j)^{-n}}{j} p $$ TIA
1
vote
2answers
74 views

$\operatorname{\mathcal{Jac}}\left( \mathbb{Q}[x] / (x^8-1) \right)$

$\DeclareMathOperator{\Jac}{\mathcal{Jac}}$ Using the fact that $R := \mathbb{Q}[x]/(x^8-1)$ is a Jacobson ring and thus its Jacobson radical is equal to its Nilradical, I already computed that $\Jac ...
3
votes
3answers
178 views

Simplified form of $\left(6-\frac{2}{x}\right)\div\left(9-\frac{1}{x^2}\right)$.

Tried this one a couple of times but can't seem to figure it out. I am trying to simplify the expression: $$\left(6-\frac{2}{x}\right)\div\left(9-\frac{1}{x^2}\right)$$ So my attempt at this is: ...
6
votes
3answers
93 views

How do I factor this?

How do I factor $p^2+8pq+16q^2-9r^2$? I know how to group the first two terms, but I dont know what to do with the other half. Can someone help me with this problem?
5
votes
3answers
348 views

Algebraic expression in its most simplified form

I am trying to simplify the algebraic expression: $$\bigg(x-\dfrac{4}{(x-3)}\bigg)\div \bigg(x+\dfrac{2+6x}{(x-3)}\bigg)$$ I am having trouble though. My current thoughts are: ...
8
votes
7answers
619 views

Solve $\sqrt{x+4}-\sqrt{x+1}=1$ for $x$

Can someone give me some hints on how to start solving $\sqrt{x+4}-\sqrt{x+1}=1$ for x? Like I tried to factor it expand it, or even multiplying both sides by its conjugate but nothing comes up ...
3
votes
6answers
591 views

Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$

Where do I start to solve a equation for x like the one below? $$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$ After squaring it, it's too complicated; but there's nothing to factor or to ...
5
votes
5answers
412 views

How to show $x^4 - 1296 = (x^3-6x^2+36x-216)(x+6)$

How to get this result: $x^4-1296 = (x^3-6x^2+36x-216)(x+6)$? It is part of a question about finding limits at mooculus.
3
votes
2answers
221 views

Does this polynomial factorize further?

I just did a national exam and this question was in it; I am convinced this does not work: Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorize this cubic fully. My attempt 1 | ...
4
votes
4answers
13k views

How to factor a four term polynomial without grouping?

$$2x^3 + 9x^2 +7x -6$$ This equation doesn't factor by grouping, and other than that I have no idea how to solve this problem. Will someone please help?
1
vote
3answers
98 views

find out the value of $\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$

If $(x-3)^2+(y-5)^2+(z-4)^2=0$,then find out the value of $$\dfrac {x^2}{9}+\dfrac {y^2}{25}+\dfrac {z^2}{16}$$ just give hint to start solution.
14
votes
2answers
164 views

Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$

How do I simplify: $$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$ Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...