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

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0answers
3 views

Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
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3answers
48 views

A problem on polynomial completely

$P(x)=x^3+mx^2+nx+14$ is divisible by $(x+2)$ but leaves a remainder of $-20$ when it is divided by $(x-2)$. Find the values of $m$ and $n$. Hence, factorise the polynomial completely. Now, I get ...
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2answers
513 views

Is 641 the Smallest Factor of any Composite Fermat Number?

Consider the sequence $a_n = 2^{2^n}+1$ of so-called Fermat numbers. It's well known that $a_5$ isn't prime ($a_5 = 641 \cdot 6700417$, this is due to Euler). What I want to know about this sequence ...
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2answers
57 views

Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
3
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2answers
60 views

Factoring $x^{4} +1$, using real factoring to the second degree

Factoring to the second degree using real numbers $$x^{4} +1$$ I know that $ x^{4} +1=(x^{2} + i)(x^{2}-i).\;$ But these are complex but I thought using these in some kind of way? Got no where! ...
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1answer
21 views

Proof the Existence and Uniqueness of Factorization Form of Polynomial with Complex Coefficient

If $p\in P(\Bbb{C})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form $$p(z)=c(z-\lambda_1)....(z-\lambda_m)$$ where ...
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2answers
56 views

Using factoring to solve the equation $(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$

Solve for all values of $r$: $$(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$$ I'm not sure how my thinking isn't really correct here. I know this all seems very elementary and such, ...
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2answers
42 views

Factoring Questions

I have to complete a factoring packet for AP Calculus, and I'm having trouble with three of the questions... Find the missing factor: 1. $2\sqrt{x} + 6x^\frac 32 = 2\sqrt{x}$(_____________) ...
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2answers
55 views

Factoring the sum or difference of two cubes

I'm learning about sums and differences of cubes and I can't understand it very well. I am faced with this problem: $$x^3 - 27$$ I am told to find the sum or difference of the two cubes. I ...
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2answers
35 views

How to factor the quadratic $6x^2-16x=0$? [closed]

How to factor the quadratic $6x^2-16x=0$ ? I need help solving this. I am aware of how to solve this when it is a polynomial, but I do not know how to solve with only two terms?
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1answer
55 views

Help With Factoring Trinomials [closed]

$8x^2+16x-64$ factored completely? I have tried several times and could never get the answer. Could I please get some help?
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1answer
21 views

Factor complex equation

Im having some difficulty in factoring the following complex equation. The image bellow is taken from WolframAlpha, can anyone explain how I can factor this equation. In the task I am told one ...
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4answers
64 views

Factoring the Cubic Equation $2x^3 + x^2 + kx + 6$

$(x + 3)$ is a factor of $2x^3 + x^2 + kx + 6$ Find the value of $k$.
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0answers
16 views

Quadratic Sieve Matrix Reduction

I have read through several other questions asked about this, but I want to be sure. Here are my current steps: Start with $\left[ \begin{array}{} 0 & 0 & 0 & 1\\ 1 & 1 & 1 ...
4
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1answer
79 views

Factoring numbers

I was given the following problem: Prove that 767, 76767, 7676767 ... are all composite. Making a sequence $a(n)$ = {767, 76767, 7676767, ...} you can show that $a(3k+1)$ is divisble by 13, ...
6
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1answer
64 views

When does $(a^p-1)/(a-1)$ have a $p$th power factor?

Today I noted that $(18^3-1)/(18-1) = 343 = 7^3$, and that there are no other solutions to the equation $(a^3-1)/(a-1) = b^3$ with $b \le 100000$. There are, however, many solutions to the equation $$ ...
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2answers
33 views

Is factoring a semiprime easier than matrix multiplication?

I'm currently dealing with complexity estimates of various algorithms and the connected mathematical problems. Up until now, I had in mind that problems such as integer factorization and the discrete ...
2
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2answers
49 views

Question on Factoring

I have very basic Question about factoring, we know that, $$x^2+2xy+y^2 = (x+y)^2$$ $$x^2-2xy+y^2 = (x-y)^2$$ But what will $$x^2-2xy-y^2 = ??$$ $$x^2+2xy-y^2 = ??$$
4
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1answer
28 views

Factoring cubic polynomials with missing terms.

I am working on a linear algebra problem where I have to diagonalize a matrix. The characteristic equation is $-\lambda ^3 - 3 \lambda^2 + 4$. I need to factor this in order to solve part of the ...
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1answer
62 views

Entire function with zeros of even multiplicity is the square of another entire function

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function such that the multiplicity of each of its zeros is even. Must there exist an entire $g$ such that $f(z) = g(z)^{2}$? Progress I ...
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1answer
15 views

Factoring homogeneous polynomials in two variables.

Consider a homogeneous polynomial $F(X,Y)\in\mathbb C[X,Y]$, why we can always write it as: $$F(X,Y)=\prod(a_iX+b_iY)^{r_i}\quad?$$ I can't find a proof of this fact. Many thanks in advance.
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2answers
78 views

Factoring a problem. What is the other factor? [closed]

One of the factors of the polynomial $x^3-5x^2$ is $x+3$. What is the other factor
3
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1answer
37 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 ...
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4answers
206 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 ...
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vote
2answers
45 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
68 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
71 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 $$ (x^2+2x+2)(x^2-2x+2) $$
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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 ...
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2answers
81 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 ...
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5answers
199 views

Show that this expression is a perfect square?

Show that this expression is a perfect square? $(b^2 + 3a^2 )^2 - 4 ab*(2b^2 - ab - 6a^2)$
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1answer
57 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
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3answers
67 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 ...
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3answers
39 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?
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3answers
95 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
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2answers
50 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
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2answers
59 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.
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1answer
29 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} ...
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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 ...
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3answers
17 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. ...
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1answer
44 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 $s(t)$ ...
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4answers
63 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
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3answers
95 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
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1answer
30 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 $x=1/4$ $16/16(2-3)+1=0,$ ...
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1answer
112 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. ...
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2answers
62 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 ...
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1answer
52 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
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1answer
56 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
10 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
88 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: ...