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

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0
votes
1answer
49 views

Solving or factors the given polynomial.

I have a polynomial and would like to solve it for "r". We can also do factorization if possible but important thing is to find the values of r. We will get possibly three solutions from this ...
0
votes
2answers
77 views

Finding Factors Efficiently

Let $m$ and $n$ be positive integers. What is the most efficient way to choose factors that solve this equation. Notice that two factors of 2079 must sum to 36. What is a quick way of picking numbers? ...
4
votes
1answer
177 views

Solve $a^3 + b^3 + c^3 = 6abc$

Find solutions for $a^3 + b^3 + c^3 = 6abc$ in $\mathbb{N}$, such that $gcd(a,b,c) = 1$, except for $(1,2,3)$ and its permutations. Using trial and error I found out that if $a,b,c$ are solution ...
4
votes
2answers
63 views

Prove irreducibility of a polynomial

Let $m$ be an integer, squarefree, $m\neq 1$. Prove that $x^3-m$ is irreducible in $\mathbb{Q}[X]$. My thoughts: since $m$ is squarefree, i have the prime factorization $m=p_1\cdots p_k$. Let $p$ be ...
0
votes
1answer
97 views

Polynomial Factorisation

Consider that we have a polynomial like $$x^3- (a + b +c ) x^2+abx-abc+s$$ Which is multiplication of $$(x-a)(x-b)(x-c)+s$$ Is it possible to reach value= $abc$ knowing the Coefficients and exponents ...
0
votes
1answer
32 views

factoring a differential quotient

The original function is $$ (y^2+yx)dx+x^2dy = 0 $$ I've arrived at $$\frac{dx}{x} + \frac{du}{u(u+2)} = 0 $$ The text book carries on to factor as such, but I don't understand how they justify it: ...
-2
votes
3answers
156 views

How do I factor this polynomial $x^5-4x^3+8x^2-32$? [closed]

How do I factor this polynomial? $p(x) = x^5-4x^3+8x^2-32$
2
votes
3answers
128 views

Factor $(x+y)^7-(x^7+y^7)$

So I was doing some practice problems to prepare upcoming math contests. This is one of the problems: Factor $(x+y)^7-(x^7+y^7)$ I got zero for $(x+y)^7-(x^7+y^7)$, however, the solutions ...
1
vote
3answers
74 views

Expressing a $3\times 3$ determinant as the product of four factors

I am attempting to express the determinant below as a product of four linear factors $$\begin{vmatrix} a & bc & b+c\\ b & ca & c+a\\ c & ab & a+b\\ \end{vmatrix} = ...
0
votes
3answers
54 views

Factorising a 3 x 3 determinant - What Am I doing Wrong?

$$\begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \\ \end{vmatrix}$$ subtracting the top row from the middle and bottom rows $$ = \begin{vmatrix} 1 & a & a^3 ...
1
vote
2answers
53 views

Factorizing a difference of two $n$-th powers

How can be proved that $$a^n-b^n=\displaystyle\prod_{j=1}^{n}(a-\omega^j b)$$ where $\omega=e^{\frac{2\pi i}{n}}$ is a primitive $n$-th root of $1$?
0
votes
1answer
96 views

Sum of number of factors of first N numbers [duplicate]

Given a number N ( Value can be large like N < 10^9 ) How can we calculate sum of the number of factors of first N numbers?? Example : For n = 3 Answer: = #f(1) + #f(2) + #f(3) --- { #f(n) ...
1
vote
2answers
280 views

Find the square root of $(x^2 + 3x + 7)(x^2 + 5x + 3) + (x − 2)^2$

I want to find the square root of $$(x^2+3x + 7)(x^2+5x+3)+ (x −2)^2$$ First , I would like to know if it is really necessary to expand everything , because I think it is in the given form for a ...
4
votes
4answers
250 views

Factorize : $x^6 − 10x^3 + 27$

I want to factorize $$x^6 − 10x^3 + 27$$ I tried two methods , first I let $y=x^3 $ and converted it into a quadratic but the solutions are not real . The second method I tried was getting it to ...
3
votes
2answers
199 views

Why does prime factorization hold in the set of integers of the form $4k+1$?

I want to prove that in the set $$ S = \{4k+1 : k\text{ is a positive integer}\}$$ (i.e. $S = \{1, 5, 9, 16, \dots \}$) unique prime factorization holds. How do I do that? Edit: a prime in this ...
2
votes
2answers
202 views

Necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$

I'm trying to find a necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$. What I have now is rather trivial sufficient conditions: if $m\mid n$ then $m=1$ and if ...
2
votes
1answer
88 views

Unexpected data for primes?

I found some unexpected data for primes. Consider $p(n)$ being the product over all primes smaller than or equal to $n$. When factoring $p(n)^a +1$ for $a=1$ or $a=2$ we get the expected amount of ...
1
vote
4answers
92 views

Help in factoring polynomials

Please help in factoring: $x^3 - 13x + 12$ $x^5 - 3x^3 - 4x$ $x^3 - 6x^2 + 5x + 12$ Thank you in advance.
1
vote
1answer
697 views

Fast way to solve a system of linear equations from Givens QR decomposition

I have this system of linear equations: $$ A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix} $$ $$ b= \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix} $$ I ...
0
votes
1answer
25 views

Need help with the steps

Tried this problem several times and still cant get the right answer. Please help! https://webwork2.uncc.edu/webwork2_files/tmp/equations/ba/00ebd18c83856ce9c3b184a9a058a01.png
1
vote
0answers
73 views

Parametric Equation solving over integers

I have a question on my mind I am trying to solve. However I am stuck at a point. If you could help, I would be very pleased. $$\frac {x_2y_2z_2}{x_2y_2+y_2z_2+x_2z_2}=\frac ...
3
votes
1answer
264 views

Solve $x+y+z = x^3 + y^3 + z^3 = 8$ in $\mathbb{Z}$

Solve $x+y+z = x^3 + y^3 + z^3 = 8$ in $\mathbb{Z}$ First I tried to transform this equation, substituting $x = 8-y-z$. So I end up with: $$x^3 + y^3 + z^3 = 8$$ $$(8-y-z)^3 + y^3 + z^3 = 8$$ ...
0
votes
3answers
249 views

Factorising complex equations

I do factorization by just plugging in simple numbers to check whether they are factors, using long division and also using synthetic division. But, how to do it in the case of complex equations. For ...
5
votes
2answers
334 views

When solving a linear differential equation by factoring the operator, how does one guarantee no solutions are lost?

I think the best way to make this question clear is with an example. Lets say we want to solve the differential equation $(\Delta^2 - \lambda^4)\phi=0,$ calculations are greatly simplified if we ...
1
vote
1answer
123 views

Multivariate polynomial divisibility and Gauss's lemma

Let $\mathbb{F}$ be a field and $A(x,y)$ and $B(x,y)$ be polynomials in $\mathbb{F}[x,y]$. We would like to prove that $A(x,y)$ divides $B(x,y)$. Will the following approach work? We can interpret ...
-3
votes
3answers
308 views

Factoring Pre calculus Question

Please help me factor $6x^3-9x^2+2x-3$ by grouping terms.
1
vote
2answers
393 views

Solving inequalities, simplifying radicals, and factoring. (Pre calculus)

(Q.1) Solve for $x$ in $x^3 - 5x > 4x^2$ its a question in pre calculus for dummies workbook, chapter 2. The answer says: then factor the quadratic: $x(x-5)(x+1)>0$. Set your factors equal to ...
6
votes
6answers
922 views

How to factor $2x^2-x-3$?

I know its: $$(x+1)(2x-3)$$ But how do you come to that conclusion?
2
votes
2answers
409 views

Factorising polynomials resulting in surds

I am trying to factorise $x^2-18x+60$. Wolfram Alpha tells me this factorises to $(x-\sqrt21-9)(x+\sqrt21-9)$, but what technique should I be using to find this myself?
3
votes
2answers
80 views

factoring polynomials

I need help with a challenge problem I'm attempting to solve in my math book. The first one is: Factor $4x^2(x-3)^3-6x(x-3)^2+4(x-3)$ I worked through the problem and got ...
1
vote
1answer
96 views

factoring a trigonometric expression

The graph of $2\cos(y)-\cos(3x)-\cos(x)=0$ (here's a link) suggests to me that this expression can be factored on the left, but I haven't uncovered how. Any trig masters out there who can see the ...
5
votes
1answer
136 views

The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin's Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say ...
4
votes
4answers
549 views

Factorise: $2a^4 + a^2b^2 + ab^3 + b^4$

Factorize : $$2a^4 + a^2b^2 + ab^3 + b^4$$ Here is what I did: $$a^4+b^4+2a^2b^2+a^4-a^2b^2+ab^3+b^4$$ $$(a^2+b^2)^2+a^2(a^2-b^2)+b^3(a+b)$$ $$(a^2+b^2)^2+a^2(a+b)(a-b)+b^3(a+b)$$ ...
5
votes
4answers
268 views

Factorize: $a^2(b − c)^3 + b^2(c − a)^3 + c^2(a − b)^3$

I want to factorize $a^2(b − c)^3 + b^2(c − a)^3 + c^2(a − b)^3$ . By inspection , I can see that substituting $b$ for $a$ yields $0$ thus $(a-b)$ is a factor . Similarly $(c-a)$ and $(b-c)$ are ...
1
vote
2answers
108 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
90 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
75 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 ...
75
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
78 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
votes
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
votes
1answer
371 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
177 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
vote
1answer
95 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
189 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
369 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
50 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
135 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
292 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
366 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 ...