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

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1answer
28 views

Which factors determine whether a set of variables are suitable for factor analysis?

Which factors determine whether a set of variables are suitable for factor analysis? I am looking as much for an explanation of the question as a tentative answer to it. So grateful for any help on ...
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2answers
81 views

Factorization of a polynomial

I need to find the roots of this polynomial $$2x^2-x^4-x=0.$$ I know that it is necessary the factorization to obtain $$-x(x-1)(x^2+x-1)=0.$$ I asked to factorize my polynomial to Mathematica. The ...
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2answers
171 views

Prime factorization: easiest way?

For prime factorization, is there another way of doing it, distinct from dividing the number by a series of primes (starting by the smallest)? Couldn't we also pick the same series of primes and ...
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1answer
29 views

Factoring difference of squares?

So I have a quick question on factoring when something is to a power other than 2 or 3. I've got this problem: $a^4-16$ and I think that I should use the difference of squares so that I would get: ...
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2answers
63 views

How to factor $a^2-2a$

The part of the problem I'm doing has me factoring this: $a^2-2a$ and I'm at a loss on how to factor it. Would I be right in saying: $(a-1)^2$ Okay so I just ran across this part now: $a^4-16$ I'm ...
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4answers
297 views

Determine the largest power of 10 that is a factor of $50!\,$?

How would one find the largest power of 10 which is a factor of $50!\,{}$?
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0answers
81 views

What is the line of thinking to get $[(n^2+3n+1)^2-5n(n+1)^2]$ from $(n^4+n^3+n^2+n+1)$?

There is this one little tiny step along the working that I don't quite understand, but I think it is better if I write the whole problem and solution for clarity. Problem: Factorise $5^{1995}-1$ ...
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1answer
85 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
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2answers
103 views

GCD of the already GCD

Say $a$ and $b$ are integers. $\gcd(a,b)$ is then $d$. Now if $a$ equals $dm$ for some integer $m$ and b equals $dn$ for some integer n, how come the gcd of this m and n is always 1?
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4answers
64 views

Recognizing the proper polynomial factorization to solve an indeterminate limit

I had to solve the $\lim_{x \to 3} \frac{x^3-3x^2-x+3}{x^2-x-6}$ that is indeed an indeterminate form ($\frac{0}{0}$). The approach I adopted was to factor the polinomials so that I can deviate from ...
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1answer
546 views

Factoring $(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$

How to prove the following equality? $$(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$$ I did it $$\begin{aligned} a^2b + a^2c + ab^2 + cb^2 + bc^2 + ac^2 + 2abc &=a^2(b + c) + bc(b + c) + a(b^2 + ...
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2answers
208 views

Fast method for solving modular exponential function with semi-prime modulus

Assume we have a semi-prime $N = pq.$ Assume $N$ is not divisible by $2$ or $5$. We want to solve the equation $$10^x \equiv 1 \mod N, \ \ \ x>0.$$ One solution is enough. Is there any fast method ...
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6answers
182 views

How to factorize $2x^2+5x+3$?

I'm doing pre-calculus course at coursera.org and I'm in trouble with this solution $$2x^2 +5x +3 = (2x+3)(x+1)$$ By trial, using ac-method I got stuck: $$ ac = (2)(3) = 6\\ 6 + ? = 5 \Rightarrow~ ? ...
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4answers
66 views

Factoring with rational exponents

I'm not quite sure how to do this question. Every way that I tried doing it didn't yield an answer that is equivalent to the original question. $$(2x+1)^{2/3}-4(2x+1)^{-1/3}$$ When I tried doing ...
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1answer
43 views

A question on HCF and Equations. The question is given in the picture below.

Please also mention how you arrived at the answer.
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3answers
971 views

Proof of $a^n+b^n$ divisible by a+b when n is odd

I read somewhere that $(a^n - b^n)$ It is always divisible by a-b. When n is even it is also divisible by a+b. When n is odd it is not divisible by a+b. and $(a^n + b^n)$ ...
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1answer
87 views

Efficient way to find lowest divisor of an integer.

I have followed the given way to find the lowest divisor of an integer, Let us assume n is the given integer. Check n is ...
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4answers
127 views

How to determine if $2+x+y$ is a factor of $4-(x+y)^2$?

I know it is a factor but how could have I determined that it was? Feel free to link whatever concept is needed than solve it. Studying for clep and it's one of the practice problems. When I expand it ...
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2answers
85 views

Question about Polynomial Factor Theorem

I was reading the solution to an algebra problem but got stuck at one part. Problem is here: (http://math.la.asu.edu/~ifulman/mat194/problem-solving.pdf) Example 4.2.6 -- page 140 of the PDF (the book ...
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2answers
432 views

How can I factor the polynomial $x^7-1$ in GF(2)?

The result is $(x+1)(x^3+x+1)(x^3+x^2+1)$, but I don't understand how I can calculate it.
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1answer
161 views

If discriminant of f is a perfect square, then we can factor f into linear factors

Let $f$ be a binary quadratic form with integer coeficients, $f(x,y)=ax^2+bxy+cy^2$. I'm trying to prove that if $d=b^2-4ac$ is a perfect square $d=k^2$, then we can factor ...
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1answer
94 views

Which positive integers can be written in the following form?

I was investigating a generalisation of this problem and found that it reduced to finding where the expression $$\frac{p(p+2m+1)}{2}$$ is an integer, where $p\ge 2$ and $m \ge 0$. Since exactly one of ...
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1answer
98 views

$gcd(a,b)$ in a UFD subring is not a greatest common divisor in the ring

Give a counterexample that $R$ is a unique factorization domain but not a principal ideal domain, $S$ is a ring containing $R$, such that $a,b\in R$, $gcd(a,b)$ in $R$ is not a greatest common ...
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2answers
45 views

how to make factor of this expression

How do I factor $x^4 + x^2 y^2 + y^4$? And in how many ways can I make the factor? Some methods I know are Mid term Break Substiturion Method Quadratic Equation
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3answers
152 views

Prove that $ x^n - y^n = (x-y) (x^{n-1}+x^{n-2}y\,+ \,\,…\,\,+ y^{n-1})$ [closed]

Prove that $ x^n - y^n = (x-y). (x^{n-1}+x^{n-2}y\,+ \,\,...\,\,+ y^{n-1}) $; $\,\,\,\,\,$$x,y \in \mathbb{R}$
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6answers
91 views

Factoring $s^2+4s+13$

I was looking at an example, and it was factored as follow: $$ s^{2}+4s+13 = (s+2)^{2}+9 $$ How can we do that?
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3answers
76 views

Factoring/approximating an apparently simple formula

Does anyone know if the following formula can be factorized or approximated: $a^3 + b^3 + c^3 + a^2b + ab^2 + a^2c + ac^2 + b^2c + bc^2 + abc$ It looks a lot like $(a + b + c)^3$, except for the ...
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3answers
59 views

Simplifying and Factoring Formulae

I'm am doing an 8th grade math text book, and I came across this simple problem: $8l^3 - 36l^2m + 54lm^2 - 27m^3$ simplifies to? I immediately got to know that it is $(2l - 3m)^3$ , but how do you ...
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1answer
76 views

Factoring Perfect Square Trinomials

How would you factor perfect square trinomials? I have a perfect square trinomial 4x^2 - 20x + 25 = 0, and the answer given to me on the answer key is ...
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2answers
79 views

Prove that the subgroup of the quotient group is cycling and infinitely generated

$$M = \left\{\,\dfrac{m}{13^n}\biggm| m\in \mathbb{Z}, n\in\mathbb{N} \,\right\}, \quad G = M/\mathbb{Z}$$ Prove that any subgroup $H < G$, $H\neq G$ is cyclic and infinitely generated and that ...
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1answer
55 views

Factoring over prime fields

Suppose I have two numbers in Fp that are multiplied together: r*s Is there anything special that needs to be done when prime factorizing since this is a finite field (i.e., is this what is referred ...
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0answers
73 views

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$?

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$ ? For example: $1.$ $a^2+b^2+c^2$ $2.$ $a^2+b^2-c^2$ $3.$ $a^2+b^2+c^2+d^2$ $4.$ ...
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2answers
41 views

Multiples of three and five below n

I am tackling a problem which asks: Find the sum of all the multiples of 3 or 5 below 1000. My reasoning is that since Since $\left\lfloor\frac{1000}{3}\right\rfloor = 333$ and ...
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1answer
75 views

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain?

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain? I have no idea to prove or disprove this... should I prove ...
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4answers
105 views

$\omega^2+\omega+1$divides a polynomial

The question is Show that $f(n)=n^5+n^4+1$ is not prime for $n>4$. The solution is given as Let $\omega$ be the third root of unity. Then $\omega^2+\omega+1=0$. Since ...
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1answer
51 views

Question about factor of a function

can you explain this question for me? I don't quite understand it. Thank you in advance. $x^2+1$ is a factor of $f(x)$. Which of the following is TRUE? $\text{a)}\qquad f(-1)=0 ...
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2answers
66 views

How to simplify this fraction

It's embarrassing, but I need help solving this one... Need some refresher course for algebra. $$ \frac y{y+\sqrt y} $$
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1answer
33 views

If $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$

Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$. Show that if $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$. I showed that if $a$ is ...
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3answers
124 views

Is there a branch of mathematics that studies the factors of rational numbers?

Is there a branch of mathematics that studies the factors of rational numbers? I am imagining that defining this would work pretty much the same way as defining the factors x of an integer n: $\{x ...
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2answers
35 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
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1answer
36 views

Factors of $x^n+1$ over $\mathbb{Z}[x]$

Is there any equivalent to $x^n-1 = \prod\limits_{d|n} \phi_d$ where $\phi_d$ is the $d$th cyclotomic polynomial but for $x^n+1$? Even better, can we generalize any further?
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4answers
510 views

Factoring a cubic polynomial?

So I have a matrix $$A = \begin{pmatrix} -5 & -6 & 3 \\ 3 & 4 & -3 \\ 0 & 0 & -2 \end{pmatrix} $$ I'm to find the characteristic polynomial and all the eigenvalues of ...
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1answer
47 views

Inversion in factor rings

I have this polynomials: $f = x^{4} + 3x^{3} + x^2 + 3 \in \mathbb{Z}_{5}[x]$, $g = x + 2 \in \mathbb{Z}_{5}[x]$ Does g + (f) have inversion in ring $(\mathbb{Z}_{5}[x]/(f),+,.)$ ? I should found ...
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2answers
737 views

Use a given zero to write P(x) as a product of linear and irreducible quadratic factors

The polynomial in question is: $x^4 - 8x^3 - 19x^2 + 288x - 612$ and the zero is $4 - i$. What I don't understand is how to go from the given zero to factorizing, especially as it's imaginary. ...
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2answers
46 views

*Step* in proving that there are infinitely many primes that suffice…

Let $k,n\in \mathbb Z$ with $n=k^2+1$ and let $p$ be an odd prime with $p\mid n$. Prove that $p\equiv1\text{ mod }4$. I found out that $\bar{n}\in\left\{ \bar{1},\bar{2}\right\} $ (denoting ...
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134 views

Intermediate Problem Solving Patterns involving Prime Factoring

a and b are positive integers such that $a\times b= 500000000,$ where neither a nor b contain any zeros. Find a and b where $a<b.$
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1answer
67 views

If n > 3 and (n + 1) is a square, is there any n that is a prime?

I am looking at properties of squares and came about this property. I am investigating the difference of squares in relation to primes.
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3answers
62 views

Can someone please explain how this was factored?

How was $x^2(x+1)-4(x+1)$ factored into $$(x^2-4)(x+1)?$$ I know this seems very basic but can someone please explain this?
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1answer
70 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
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3answers
58 views

Trouble with factoring polonomial to the 3rd degree

I am having trouble factoring this problem: $\displaystyle{-x^{3} + 6x^{2} - 11x + 6}$ I know the answer but i can't figure out how it is done with this. I have tried by grouping and is doesn't seem ...