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

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3
votes
1answer
33 views

Evaluate Derivative $\lim_{x \to 1}\frac{10x-1.86x^2 - 8.14}{x - 1}$

Evaluate Derivative $\lim_{x \to 1}\frac{10x-1.86x^2 - 8.14}{x - 1}$ I've already evaluated the limit using the $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ definition of a limit, but now I'm curious as ...
1
vote
4answers
106 views

Factoring $x^4+x^2+2x+6$

We have to factor $x^4+x^2+2x+6$.Factoring through factor theorem is not helpful in this question as the question does not follow the integral root theorem i.e. the root of this expression is not any ...
1
vote
2answers
42 views

Limit Different if I Factor and Simplify an Equation First.

So I have $$\lim_{t\to -1} \frac {t^2 - t - 2}{t^2 - 1} = \frac 32$$ The solution given was found by factoring and simplifying the equation and then taking the limit of numerator and denominator and ...
3
votes
2answers
60 views

prove $\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$

Show that if $a,b,c,d \geq 0$ and $ab+bc+cd+da=1$ :$$\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$$ yet again it should be solved with Cauchy inequality. thing i have done so far: ...
0
votes
2answers
66 views

how to factorize $x^4+2x^2+4$ to reell coeff.

How do you factor $$x^4+2x^2+4 $$ so it can be written as $$ (x^2+2x+2)(x^2-2x+2) $$
0
votes
0answers
17 views

Can I evaluate polynomials with prime numbers to find possible irreductible factors?

Let $p(x,y)$, $c(x,y)$ and $d(x,y)$ be two variable polynomials with integer coefficients which satisfy $p(x,y)=c(x,y)\cdot d(x,y)$. Given $m, n$ positive prime numbers and given $e(x,y)$ another ...
2
votes
2answers
80 views

Is there a reason for some polynomial quotients to have a remainder equals to zero?

I was helping some highschool students with factorization exercises. They had alternatives to choose the correct factor. Then one of them said to me: We use a calculator and evaluate some prime ...
2
votes
5answers
195 views

Show that this expression is a perfect square?

Show that this expression is a perfect square? $(b^2 + 3a^2 )^2 - 4 ab*(2b^2 - ab - 6a^2)$
0
votes
1answer
52 views

Nonlinear system of equations / factoring two-variable cubic over $\mathbb{R}$

About halfway through a homework problem, I end up with a three-way identity: $$\frac{uw}{v+w} = \frac{uv}{u+w} = \frac{vw}{u+v}$$ (I say I end up with..., but this is the method suggested by my ...
0
votes
3answers
65 views

prove $(a+b+c)^n=a^n+b^n+c^n$ if $(a+b+c)^3=a^3+b^3+c^3$

if $(a+b+c)^3=a^3+b^3+c^3$ and n is odd number,prove that: $$(a+b+c)^n=a^n+b^n+c^n$$ hint of the question was: factor this expression $f(a,b,c)=(a+b+c)^3-(a^3+b^3+c^3)$ after factorization ...
1
vote
3answers
31 views

Factor Cyclic Polynomial

Factor $(a+b)(b+c)(c+a)+abc$. I know this is a cyclic polynomial, but I don't know how to solve problems like this. What should I do?
3
votes
3answers
91 views

Factor $3x^2-11xy+6y^2-xz-4yz-2z^2$

This problem is from my Math Challenge II Algebra class, and it's really confusing. How can you factor something like this? Here's the question again: Factor $3x^2-11xy+6y^2-xz-4yz-2z^2$.
3
votes
2answers
48 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
0
votes
2answers
56 views

give a complete factored form of the polynomial $-6a^5+48a^4+12a$

Give a complete factored form of the polynomial $-6a^5+48a^4+12a$ I have tried solving this equation and I just cant figure it out. Help me, and give me the answer.
1
vote
1answer
28 views

Factoring for simplification?

I need to show that $$\dfrac{\Gamma\left(\alpha_1 + \alpha_2\right)}{\Gamma\left(\alpha_1\right)\Gamma\left(\alpha_2\right)}\left[\dfrac{\tau y^{\tau \alpha_1}\delta^{\tau \alpha_2}}{y\left(y^{\tau} ...
1
vote
3answers
32 views

Factoring when differentiating expressions

I'm having trouble with differentiating a expression. I do it one way, wolfram alpha does it another. Let me show you what I mean. The original expression is this: $$\frac{1}{2u^3}$$ I start by ...
0
votes
3answers
12 views

Setting up word problem for finding length and width

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width. I have not idea where to start. ...
0
votes
1answer
41 views

Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
0
votes
4answers
62 views

How can I factor $x^2 + 2\sqrt{3}\,x + 3$? [closed]

$$x^2 + 2\sqrt{3}\,x + 3$$ Anyone could tell me how may I factor this? Thanks a lot
0
votes
0answers
12 views

When to use factoring by grouping for quadratic equation

There are several ways/methods to perform factoring. I am revising factoring at KhanAcademy, there are factoring by grouping, factoring special product and factoring difference of squares. Although, ...
0
votes
1answer
28 views

remainder is not zero using long division method

Find all zeros of $f(x)=128x^3-48x^2+1$ given that one linear factor occurs twice. let $f(x) $ be equaal to 0 $128x^3-48x^2+1=0,$ $16x^2(8x-3)+1=0,$ trying $x=1/4$ $16/16(2-3)+1=0,$ ...
1
vote
1answer
108 views

Factoring $2x^5+13x^4+50x^3+82x^2+56x+13$

Express $2x^5+13x^4+50x^3+82x^2+56x+13$ as a product of five linear factors. The roots of the polynomial may be real or complex. I had to employ the technique of synthetic division iteratively. ...
0
votes
2answers
58 views

Factoring equation with 4 cubed numbers

The problem is to factor $a^3x - b^3y + b^3x - a^3y$, and the answer is $$(x-y)(a+b)(a^2 - ab + b^2).$$ I got as far as $(x-y)(a^3 - b^3 + b^3 - a^3)$. I mean the above answer fits if it was just ...
6
votes
1answer
50 views

Lattice-Theoretic Interpretation of the Fundamental Theorem of Arithmetic

When equipping $\mathbb{N}^\ast=\mathbb{N}\setminus \{0\}$ with the divisibility relation, it forms a lattice with minimum 1, supremum given by the least common multiple, and infimum given by the ...
0
votes
1answer
54 views

Factoring $x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2$

The subject line pretty much says it all. In my geometry class today, the following equation came up: $$x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2 = 0$$ Specifically, it was in the ...
2
votes
1answer
84 views

What do we know about $\displaystyle \frac{f}{\gcd(f,f')}$ if $f\in\mathbb{F}_{p^d}[X]$?

Let $\mathbb{K}=\mathbb{F}_{p^d}$ and $f\in\mathbb{K}[X]$ be a non-constant polynomial with the factorization $$f=\prod_{i=1}^nf_i^{k_i}$$ where $f_i\in\mathbb{K}[X]$ is irreducible and ...
0
votes
0answers
9 views

Coppersmith method for factorisation

Is anyone familiar with the Coppersmith method? Does anyone know how is the $3\times3$ basis matrix obtained in this case?
2
votes
3answers
85 views

Algebra (not so simple) Factoring

I got stuck on this problem from my Math Challenge II Algebra Class: Factorize the following: $$(x^2+xy+y^2)^2-4xy(x^2+y^2)$$ Hint: Let $u=x+y$ and $v=xy$. Here's what I did: ...
1
vote
1answer
27 views

On the existence of polynomial roots

Assume $F$ is a field, and $f\in F[x]$ is polynomial. To see that $f$ has a root in some extension of $F$, without loss of generality we can assume $f$ is irreducible. Indeed any polynomial $f$ is ...
0
votes
1answer
32 views

Finding greatest common divisor between two polynomials.

I have the following past exam question: Calculate $\operatorname{gcd}(x^3 + 2x^2 + 2,2x^2 + 1)$ in $\mathbb{F}_3$ Now I haven't encountered this sort of gcd before(usually I am trying to solve ...
1
vote
1answer
44 views

Factorial simplification

How can I work with this? $$\frac{(3n)!}{(3(n+1))!}$$ I really don't know how to open this fatorial and then, simplify it. Actually, I have to calculate the limit when $n\to\infty$. Thanks :)
2
votes
0answers
20 views

Finding coefficients of the min polynomial of an $n\times n$

Given an $n\times n$ matrix, for ease assume this matrix is over the $F_m$. What we know about min poly is the the non-zero components of the min polynomial for this case, ie if there is $x^2$, or ...
1
vote
2answers
40 views

Factorising after adding a square

I have been thinking about it for quite some time but am unable to find an answer. Let $a,b,c,d,e$ be any distinct natural numbers. Will the relation : $(x-a)(x-b)+c^2=(x-d)(x-e)$ ever hold? I am ...
-1
votes
6answers
49 views

Polynomial factors involving inequalities

How to factorise the polynomial $p(x) = x^4-2x^3 + 2x - 1$. Hence, solve the inequality $p(x) \gt 0$ ?
3
votes
2answers
71 views

Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
1
vote
3answers
81 views

How would I factor $a^3+b^3+c^3-6abc$

How would I factor the polynomial $a^3+b^3+c^3-6abc$? The values are homogenous, so so must be the factors. I don't know where to go from there.
2
votes
1answer
14 views

Polynomial factorisation on integers modulo n

Is there a known (efficient) algorithm to compute the list of factors of a polynomial modulo $n$ (for any integer $n$)? For example in $\mathbb Z_8$, $X^2+2X$ has a list of 4 factors (multiplicity 1 ...
1
vote
1answer
31 views

Any simpler way to do Pollard's p-1 method?

I found calculating factorization by Pollard's p-1 method is almost impossible if use a conventional scientific calculator. For example, I am trying to factor ...
2
votes
1answer
63 views

Expanding Square Roots, Why No Negative?

I haven't thought through algebra in a while and the last explanation I received of this seemed arbitrary. I hope I can get some clarification here. I understand that $\sqrt{+a} = \pm b$. Here's ...
0
votes
1answer
25 views

Factoring Polynomial Questions

How do you decide whether to use synthetic division or the factor theorem to help you factor a polynomial? Please help me answer.
1
vote
3answers
74 views

Can any factor pairs (where pair is unique) have the same sum?

I have used Stack Overflow but I'm new to this site so I apologize if this is a trivial question. I am creating mathematics software using javascript. I am using a for loop to find all factors for a ...
0
votes
4answers
47 views

Help in factoring $x(y^2+z^2) + y(z^2+x^2) + x(y^2+z^2) + 3xyz$

Some hints on factoring the above expression please
5
votes
5answers
293 views

Factoring in the derivative of a rational function

Given that $$ f(x) = \frac{x}{1+x^2} $$ I have to find $$\frac{f(x) - f(a)}{x-a}$$ So some progressing shows that: $$ \frac{\left(\frac{x}{1+x^2}\right) - \left(\frac{a}{1+a^2}\right)}{x-a} = ...
5
votes
1answer
105 views

Is 292229292292 the longest 29-smooth number made of 2's and 9's?

Is 292229292292 the longest 29-smooth number made of 2's and 9's? The factorization is $2^2 7^8*19*23*29$. Is there a general way to find other numbers of this sort without resorting to brute force ...
1
vote
1answer
25 views

Help with a technique in factoring a polynomial of four terms and two variables

I could not simplify this expression into factors despite the time I put into it: $x^3-64y^3-24xy-8$ I really want to learn how to do this well. Thank you very much for insights/techniques which you ...
0
votes
1answer
19 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
6
votes
1answer
132 views

Factorise $x^4 + 3x^2+ 6x+ 10$

I need to factorise $x^4 + 3x^2 + 6x + 10$ completely over $\mathbb{Q}$. I am not sure how to do this. I can't find any roots of this equation in $\mathbb{Z}$.
1
vote
0answers
20 views

Factorisation algorithm for polynomials in several variables over $\mathbb{Z}$.

What algorithm is used by a CAS to decide how to factor a polynomial in several variables over $\mathbb Z$?
0
votes
1answer
48 views

Is there a name for numbers that have 2 as their greatest common divisor?

Is there a name for numbers that have two as their greatest common divisor? Such as 8 and 130.
9
votes
0answers
150 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...