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

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2answers
35 views

Limits: Can't understand this worked example

I can't seem to understand the following given example while working with Limits. $$\lim\limits_{x \to \infty}({x\over1+x})^x = \lim\limits_{x\to \infty}({x +1 -1\over1+x})^x = \lim\limits_{x \to \...
10
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2answers
598 views

Sum of factors of a huge number.

I recently appeared in a math olympiad and it had this one question which had me stumped. This was a few weeks back and I have been looking for a way to find its answer ever since, but with no success....
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1answer
44 views

Solve $ax+bx^0+\frac{c}{x}+\frac{d}{x^2}=0$ for $x$

I find myself trying to produce a plot of the van der Waals equation of the form $PV(P)$ to demonstrate the non-ideality of some gases. The van der waals equation is $$(P-b)\left(V-\frac{a}{V^2}\...
5
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1answer
67 views

Decompose $x^4 + x^3 + 1$ into irreducible factors over $\mathbb{Z}_2$

Decompose $x^4 + x^3 + 1$ into irreducible factors over $\mathbb{Z}_2$ I think that the given polynomial is already irreducible in $\mathbb{Z}_2$, therefore the only irreducible factors are $x^4 + x^...
3
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2answers
64 views

Finding integers satisfying $m^2 - n^2 = 1111$

We have to find the integers $m$ and $n$ which will satisfy the given condition: $$m^2-n^2=1111.$$ What could be the answer and how? i tried using trial and error and that took a long time.
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2answers
43 views

factorization of a fourth degree polynomial

What could be the possible factorization of $$2a^4+a^2b^2+ab^3+b^4$$? what term should be added
2
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2answers
76 views

Find product of solutions of $x^6=-64$

If the six solutions of $x^6=-64$ are written in the form $a+bi$,where $a$ and $b$ are real, then find the product of those solutions with $a>0$. The answer in my book is given as $4$ but I don'...
2
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1answer
56 views

Factoring the quartic $\left(x^{2}+x-1\right)\left(x^{2}+2x-1\right)-2sx\left(2x-1\right)^{2}$

Define $Q{\left(s;x\right)}$ to be the quartic function of $x$ with real parameter $s$ such that $0\le s\le1$ given as $$Q{\left(s;x\right)}=\left(x^{2}+x-1\right)\left(x^{2}+2x-1\right)-2sx\left(2x-...
3
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2answers
58 views

Why is the number of divisors of an integer $n$ equal to the number of factors in the factorization of the polynomial $x^n - 1$ over the integers?

Sloane's OEIS A000005 gives the number of divisors of the integer n. A comment (by a very reputable contributor) in this sequence claims that this is also the number of factors in the factorization ...
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2answers
70 views

Having trouble finding roots/factorizing a cubic equation

I've been trying to find a method that can work for any cubic equation, but I can't seem to find one. Right now, I'm trying to find the roots/ factorize the following equation: $x^3-5x^2+3x+9 = 0$. I'...
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2answers
141 views

Prove that if $7^n-3^n$ is divisible by $n>1$, then $n$ must be even.

I tried using factorization of $a^n-b^n$ for odd $n$ in an attempt to work through to a situation where the factors are such that they cannot have n as a factor. But I reached nowhere. Here's how I ...
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0answers
36 views

Factorization of polynomial $f(x,y)$

The motivation is solving the following equations: $$ f(x,y)=0, x=L-kL, y=ks $$ $k$ is the variable, $L$ and $s$ are constants. The plan is: First, to factorize the polynomial as $(a_1x+b_1y+c_1) (...
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2answers
56 views

Complete the square of three variable quadratic expressions

We know that completing $ax^2+bxy+cz^2$ into forms of $k_{1}(a_{1}x+b_{1}y)^2+k_{2}(a_{2}x+b_{2}y)^2$ is easy and have some fixed routine. But the 3 variable case $$ax^2+by^2+cz^2+dxy+exz+fyx$$does ...
2
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2answers
77 views

Find 4 positive integers not exceeding 70,000 such that each have more than 100 divisors

I am looking at problems in Vandendriessche and Lee's Problems in elementary number theory and this is one of their problems: Find $4$ positive integers not exceeding $70000$ such that each have ...
1
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1answer
34 views

The form which take continuous functions for which we have $f(x,y)=g(x)h(y)$?

In this question there was raised a question on general sufficient conditions under which we cannot factor the functions and there is an answer which states that: Theorem: Let $f:\mathbb R^2\to\...
1
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4answers
61 views

How do you factor $x^2-x-1$?

I know you can't have all integers, but how do you factor this anyway? Wolfram|Alpha gives me $-\frac{1}{4} (1+\sqrt{5}-2 x) (-1+\sqrt{5}+2 x)$. Cymath gives me $(x-\frac{1+\sqrt{5}}{2})(x-\frac{1-\...
3
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1answer
63 views

When can a function not be factored?

Are there any general conditions under which a function involving $n$ unknowns cannot be factored into a product of $n$ terms each of which contains only one of the unknowns? For example, $xy$ can be ...
0
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2answers
39 views

How is the formula for a finite geometric series found?

I have these two finite geometric series: $S_n$ = $\sum_{k=0}^n ar^k$ r$S_n$ = $\sum_{k=0}^n ar^{k+1}$ And then we substract both series so: $S_n$ - r$S_n$ = a - $ar^{n+1}$ ...
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0answers
79 views

Does Pollard rho works for Gaussian integers?

Should I expect that the Pollard rho method ...
2
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1answer
30 views

Find the smallest $4$ digit number which is a factor of $2005^6 - 1$.

Find the smallest $4$ digit number which is a factor of $2005^6 - 1$. My attempt: We see that $(2005-1)$ is a factor of $2005^6-1$ by the factorization of $2005^6-1$. Therefore, $1002$ is a factor ...
3
votes
2answers
45 views

Gaussian prime factorization.

I have a hard time on factorizing elements from $\mathbb{Z}[i]$, especially $-19+43i$. I know that the primes in $\mathbb{Z}[i]$ are: $1+i$. $p$ from $\mathbb{N}$, $p=4k+3$ , $k$ integer ( $p\equiv ...
0
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0answers
17 views

Deconvolve/Decompose Components of Equation

I have an equation that has values, R, that depend on both i and j: I want to be able to rewrite this equation so that all Rj are grouped together. In other words, I want to be able to see the ...
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0answers
14 views

Complex filter factorizations with invariant points

Based on this question, using the same $z_0$: $$z_0 = e^{2\pi i / 8}$$ if we modify the sequence from previous question to look like this ($*$ denotes discrete convolution): $$\left(z_0^{[-2k,3k]} * ...
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0answers
19 views

Complex filter factorizations - continued

Continuing from this rather silly trivial question factoring real valued filters into shorter complex ones, hoping this won't be as trivial. If we modify it a bit: $$z_0 = e^{2\pi i / 8}$$ and $$\...
0
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2answers
18 views

Factorisation of a polynomial in $\left(R\left[x\right],+,\cdot\right)$, $\left(Z\left[x\right],+,\cdot\right)$, …

I am given the following question: "Given a random polynomial in $Z\left[x\right]$. We factorise this polynomial in the polynomial rings $\left(R\left[x\right],+,\cdot\right)$, $\left(Z\left[x\right],...
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2answers
20 views

Factoring equations, non quadratic.

I'm taking the MIT opencouseware 6.042, Mathematics for CS. Working with induction proofs. It's been years since I've done this, and I'm not sure how he factored this. Assume p(n) true: $3|(n^3 ...
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1answer
31 views

What is the rule for simplying $ax^2 + bx + c$ for $a \neq 1$

I have a trivial quadratic $-3x^2 + 4x-4/3$ What is the most direct way to show it is equivalent to $-(3x - 2)^2/3$ Confusion: I used the quadratic formula and found the root to be $2/3$, and ...
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1answer
37 views

A polynomial with only one real $x$-intercept without imaginary roots has a root equal to $-c_n/(nc_{n-1})$.

Given coefficients $c_n, c_{n-1}, c_{n-2}, \ldots$ of the polynomial $c_n x^n+c_{n-1}x^{n-1} + \cdots +c_{1}x+c_0,$ prove that for $c_nx^n+c_{n-1}x^{n-1} + \cdots +c_1 x+c_0 = 0,$ $x=-c_n/(nc_{n-1})$. ...
1
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1answer
38 views

Quadratic Sieve Algorithm: Why is $(x − \lfloor \sqrt{n} \rfloor)^2 ≡ n ($mod $p)$?

If someone here understands the Quadratic Sieve Algorithm, I'm having trouble understanding why every prime $p$ in the factor base needs to a prime such that $n$ is a quadratic residue modulo $p$. It ...
1
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1answer
47 views

Find all solutions which satisfy the given conditions $m=9n^3+30n^2-9n$

Let $$ m=9n^3+30n^2-9n $$ where $n \in \mathbb{Q^{+}}$ and $m \in \mathbb{Z} $ . Find all solutions which satisfy the given conditions. I thought oh, there is probably an infinite amount ...
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0answers
13 views

Finding coefficients of some polynomials

Assume we know $f(x)\in\Bbb Z[x]$ of degree at least $4$ and we also know $r\in\Bbb Z$. How do we find $\alpha(x),\beta(x)\in\Bbb Z[x]$ of any degree and $g(x),h(x)\in\Bbb Z[x]$ such that $$f(x)=(g(x)...
0
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1answer
144 views

Factoring the expression $(\sqrt{x^2} -a)^2 + M = 0$

Where, M stands for all other terms in the equation. This is a typical format that you'll see when taking affine sections of an n-torus. I think I figured out how to do it correctly, without violating ...
1
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1answer
61 views

Find a factorization for $P(z)=z^5+z+1$ with $z \in \mathbb{C}$.

Find a factorization for $P(z)=z^5+z+1$ with $z \in \mathbb{C}$. I am a bit confused actually. Is anyone is able to give me a hint to solve the problem involving complex numbers? I think I can use ...
0
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1answer
18 views

Existence of Integer Solution to Quadratic Root

Example Given $91=(6(1)+1)(6(2)+1)=6^2(2)+6(3)+1$. If we wish to find a possible replacement for six we might try solving $2x^2+3x-90=0$. The quadratic equation gives $$x=\frac{-3 \pm 27}{4}$$ Is it ...
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0answers
21 views

Intrinsic Factorization with Modular Extension

The question here is has anyone seen a factorization algorithm similar to this? What is it called? Start with this $XY=N$. Suppose we know one non-trivial factorization of $N$, $X=x$ and $Y=y$. ...
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0answers
25 views

Digit Sums and The Euclidean Algorithm

The numbers that can be cast out for natural base $B$ seem to all be divisors of $B-1$, and length of cycles of differences in sequences of multiples of these divisors seem to be preserved when ...
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0answers
31 views

Factoring Algorithm Using Multilinear Forms

I was wondering if anyone could identify for me a known name for this algorithm, I would appreciate it. I will give an example. Let $N=961$. Then write $(6x+1)(6y+1)=6(160)+1$ Where $0<x\le y$ ...
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5answers
62 views

What does it mean to get rid of $x^{12}$ in the expression $ x^{12} (1-x^4)^6(1-x)^{-6} $ in the following context?

What does the auctor mean when he says that he gets rid of $x^{12}$ in the following context ? (...) For example,consider distributing $23$ toys among $6$ children such that no child gets more ...
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2answers
77 views

Factor $x^2+x+1+i$ in $\mathbb{C}[x]$.

Factor $x^2+x+1+i$ in $\mathbb{C}[x]$. I know the roots are $-i$ and $-1+i$, but I don't know how to go about factoring such polynomial. I tried using the quadratic formula, but I got stuck half way. ...
0
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2answers
56 views

Best Method for Factoring this Expression

I'm wondering if anyone has an opinion on the best way of factoring the following expression. As one can see it is quite complicated. $$(15x^2)(x^3+{4})^{4}(1-2x^{2})^{3}+(12x)(x^{3}+4)^{5}(1-2x^{2})^...
0
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1answer
19 views

I need help factoring 6q(7(6q)^2 +5)

Can someone please show me how to factor $6q(7(6q)^2 +5)$ to show that it is a multiple of 6? I'm working on a division algorithm problem, and I understand concepts of div alg but I really don't have ...
3
votes
5answers
88 views

What is the best way to solve an equation of the form $(f(x))^2-a(f(x))+b=x$?

On a math contest I was told to solve the equation $$(x^2-3x+1)^2-3(x^2-3x+1)+1=x$$ For this particular problem I simplified by letting $$a\equiv x^2-3x+1$$ Then I continued with $$a^2-3a+1-x=0$$...
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0answers
19 views

Is this product of two product factorizations correct?

I am working on an induction proof and would like to know whether this product equality is true: $$\big (\prod_{i=2}^n (\lambda_i-\lambda_1) \prod_{n\ge i > j \ge 1}(\lambda_i - \lambda_j)\big )$$ ...
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2answers
23 views

Factoring 4 terms with a difference of squares as two of the 4 terms

Can someone help show me step by step how to factor: $9x^2 - 24xy + 16y^2 - 81$ I see a difference of squares in the last two terms and am stuck at this stage (did I start wrong in doing the ...
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0answers
38 views

Why is $(3x+2)^n$ $\implies$ $(x-(-\frac{2}{3}))^n$ a valid modification?

Why can one write $(3x+2)^n$ as $(x-(-\frac{2}{3}))^n$? Does it depend on the domain of $x$? Why does the $n$ exponent not cause a problem?
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2answers
44 views

Finite fields: factorization of the trace function over the base field

Let $q$ be a prime power, and $m$ a positive integer. The trace function from $GF(q^m)$ to $GF(q)$ is defined to be the mapping $$Tr : GF(q^m) \rightarrow GF(q) $$ $$Tr(x) = x+x^q+x^{q^2}+\cdots+x^{q^{...
2
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1answer
20 views

Does this factor into a dot product

A hopefully easy question. I have this: $$c^2+(x^2+y^2+z^2)(v_x^2+v_y^2+v_z^2)-2c(x*v_x+y*v_y+z*v_z)$$ And I was wondering if this somehow factors into $$(c-f(\vec{r}, \vec{v}))^2 $$ where $$f(\...
1
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4answers
84 views

Express $-a^3 b + a^3 c+ab^3-a c^3-b^3c+bc^3$ as a product of linear factors.

Express $$-a^3 b + a^3 c+ab^3-a c^3-b^3c+bc^3$$ as a product of linear factors. I have tried rewriting the expression as: $$ab^3-a^3b + a^3c-ac^3 +bc^3-b^3c$$ $$= ab(b^2-a^2)+ac(a^2-c^2)+bc(c^2-...
0
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0answers
96 views

Chudnovsky binary splitting and factoring

In this article, a fast recursive formulation of the Chudnovsky pi formula using binary splitting is given. For $S(a,b)$: $$ m = (a + b) / 2 $$ $$P(a,b) = P(a,m) P(m,b)$$ $$Q(a,b) = Q(a,m) Q(m,b)$$ $...