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

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2answers
71 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
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1answer
155 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
84 views

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

Can I calc it in less than $O(\sqrt{n})$ time?
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2answers
54 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
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3answers
116 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\\ ...
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3answers
64 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\\ ...
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1answer
35 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 ...
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1answer
484 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.
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4answers
778 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 + ...
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0answers
101 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
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1answer
344 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
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2answers
342 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
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1answer
59 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.
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2answers
100 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
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2answers
446 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
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1answer
115 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
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1answer
55 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 ...
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6answers
197 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
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3answers
81 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 ...
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4answers
179 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
65 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
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1answer
61 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
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2answers
73 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
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3answers
164 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
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3answers
91 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
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3answers
325 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
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7answers
599 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
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6answers
541 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
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5answers
397 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
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2answers
215 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 | ...
3
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3answers
10k 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?
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3answers
96 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
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2answers
162 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 ...
15
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4answers
730 views

Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$

Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid? I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
2
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0answers
47 views

Find the factorization of the polynomial as a product of irreducible [duplicate]

Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$ Testing with the simplest possible root in this case, $P(1)=0$ Applying the ...
5
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6answers
297 views

Cubing a simple thing

I am trying to expand $\quad (x + 2)^3 $ I am actually not to sure what to do from here, the rules are confusing. To square something is simple, you just foil it. It is easy to memorize and execute. ...
2
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2answers
209 views

Faulty velocity question?

If a ball is thrown vertically upward with a velocity of $160 \text{ ft/s}$, then its height after t seconds is $s = 160t − 16t^2$. a) What is the velocity of the ball when it is $384 \text{ ...
6
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1answer
273 views

Find the value of $x^3-x^{-3}$ given that $x^2+x^{-2} = 83$

If $x>1$ and $x^2+\dfrac {1}{x^2}=83$, find the value of the expression$$x^3-\dfrac {1}{x^3}$$ a) $764$ b) $750$ c) $756$ d) $760$ In this question from given I tried to ...
10
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3answers
471 views

Irreducibility of $x^n-x-1$ over $\mathbb Q$

I want to prove that $p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible. My attempt. GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
0
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1answer
27 views

Relationship between 2 Dimensional Quadratic systems and roots

Given four points $(x_1, y_1) (x_2, y_2) (x_3, y_3) (x_4, y_4)$ How does one construct a system of two equations: $a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$ $b_1x + b_2x^2 + b_3y + b_4y^2 + ...
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5answers
365 views

Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$

Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed: ...
4
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3answers
15k views

Largest prime factor of 600851475143 [duplicate]

I'm trying to use a program to find the largest prime factor of 600851475143. This is for Project Euler here: http://projecteuler.net/problem=3 I first attempted this with the code that goes through ...
0
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4answers
65 views

How to factor $2 - 3p - 3p^2 + 2p^3$ to obtain $ (1-2p) \times (1+p) \times (2-p)$ ?

$$2 - 3p - 3p^2 + 2p^3 = (1-2p) \times (1+p) \times (2-p)$$ I want to factor the left hand side to obtain the right hand side. is there any technique ? Explain step by step, please.
2
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2answers
84 views

How do I factorize equations of the form $x^2 + Bxy + Cy^2 = 0$

Given equation $$ x^2 + Bxy + Cy^2 = 0. $$ I want to factorize it in the form $$ (x + my)(x + ny) = 0. $$ What are the values of $m$ and $n$ in terms of $B$ and $C$? I tried writing the ...
5
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4answers
107 views

In order to factor we must find its zeros?

I am self-learning precalc (Precalculus Demystified) and found the following problem on page 170 : Completely factor the the polynomial. $P(x) = x^3 - 5x^2 + 5x + 3; c = 3$ is a zero. Since $c = 3$ ...
3
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3answers
613 views

Why all odd numbers not ending with 5 divide exactly into a number comprising only 9's?

Help me!!It's really frustrating I can't understand this simple thing.The maths instructor in my video,the renowned Arthur Benjamin,states (clip linked below) the following: ...
2
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3answers
322 views

Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
0
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2answers
59 views

How to simplify or factor this equation

$$1 = x + 2\cdot x$$ How can I simplify this formula for $x$.
3
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3answers
143 views

Solving quadratic equations by completing the square.

Graphing $y=ax^2+ bx + c$ by completing the square Add and subtract the square of half the coefficent of $x$. Group the perfect square trinomial. Write the trinomial as a square of a ...
1
vote
4answers
82 views

I cannot find the last factor of this expression?

I'm supposed to factor $x^8-y^8$ (the exponents are 8 for both if it is too difficult to see) as completely as possible. It is easy to factor this to $(x+y)(x-y)(x^2+y^2)(x^4+y^4)$. However, the book ...