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

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1answer
30 views

Derivative of $x(x+2)^3$ in factored form don't see it

Studying for a big test trying to do the derivative of $x(x+2)^3$. I did the product rule and got $3x(x+2)^2 + (x+2)^3$. Pretty standard stuff. I got half credit and was told to simplify it like ...
0
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4answers
29 views

Linear Factorization of Complex Polynomials

I am trying to find a linear factorization of the polynomial $$p(z) = 1 +z+z^2 +z^3 +z^4 +z^5 + z^6 +z^7 +z^8$$ I know what it means by linear factorization in the sense of non-complex polynomials, ...
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0answers
10 views

How to store tables for ECM stage 2

This question about realization ECM stage 2 on GPU. I now that there exists some optimization for the stage 2 of ECM. Namely, let $N$ be a composite number, $q|N$ be a prime, $P=(x_P::z_P)$ be a point ...
1
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2answers
54 views

Quickest way to factorize $\frac{w^2 + 5kw + 4k^2}{w^2+kw}$

I would say 90% of polynomials in my textbook are factorable e.g. $$\frac{w^2 + 5kw + 4k^2}{w^2+kw}$$ This gives $$\frac{(w+k)(w+4k)}{(w+k)w}$$ $$\frac{w+4k}{w}$$ This took me far too long to ...
0
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1answer
30 views

What is the lowest degree of a polynomial with integer coefficients having $\sum_{i=1}^k m_i\sqrt[q_i]{a}$ as a root?

Say you want a polynomial with integer coefficients having a root with value $$v =\sum_{i=1}^k m_i\sqrt[q_i]{a}$$ where $k\ge 1$ and $\forall i: m_i, q_i \in \Bbb{Z}^+$, all the $q_i$ are greater than ...
1
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1answer
53 views

Factor $16x^4-x^2y^2+y^4$

So basically I have the answer to this problem because it's in the book, but I have no clue how the author solved it. It's from Schaum's Precalculus. Thanks. Factor $(16x^4)-(x^2y^2)+(y^4) = (4x^2+...
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2answers
69 views

How to get the partial factor representation of $\frac{1}{x^2+4x+1}$? [closed]

I've been trying to factor this, but I don't see it. What is the partial fractions representation of $$\frac{1}{x^2+4x+1}$$?
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3answers
45 views

alternate forms of $t^3 -1$

I'm looking to understand how $t^3-1$ factorises to $(t-1)(t^2+t+1)$. I know how to find the first factor $(t-1)$, but have trouble finding the second factor $(t^2+t+1)$. I've tried doing long ...
0
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2answers
52 views

How should I start Factoring this?

So im supposted to factor this and I'm not sure where to start, where should I start? $$2x^5 + 3x^4 -10x^3 -15x^2 + 8x + 12$$
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2answers
54 views

How does depressing a polynomial help you factor it?

I've noticed that a lot of the derivations for the cubic and quartic polynomial solutions require you to depress them, but what about in a case by case scenario? How does depressing this cubic, $x^3+...
3
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1answer
94 views

Solving $2(n-1)n(n+1)(n+2)=(m-3)(m+3)$

The question is: Find all pairs $(n,m)\in\mathbb{N}^2$ such that $$2(n-1)n(n+1)(n+2)=(m-3)(m+3)$$ I checked all $n<10000$ and only got $n=1$ and $n=4$ with their corresponding $m$, so I ...
3
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2answers
64 views

How does this factoring work?

$$ (z^2 - 2i ) = (z -1 -i)(z + 1 +i) $$ I see if you multiply out the right-hand side, you obtain the left-hand side, but how does one know to factor like that or this? $$ (z^2 − 3iz − 3 + i) = (z − ...
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1answer
81 views

What is the largest six-digit number with an odd number of positive factors?

What is the largest six-digit number with an odd number of positive factors? So I know the number must be a perfect square, but how do I know six-digit number perfect squares? I'm pretty sure there's ...
2
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2answers
60 views

Prove that a polynomial of odd degree in $ \Bbb R[x]\\$ with no multiples roots must have an odd number of real roots.

The coefficients of the polynomial are in the ring of real numbers. Prove that a polynomial of odd degree in $ \Bbb R[x]\\$ with no multiple roots must have an off number of real roots. I hate to ask ...
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2answers
62 views

Remainder when $x^{2016}+x^{2013}+\cdots+x^6+x^3$ is divided by $x^2+x+1$.

I am currently preparing for a certain quiz show when I encountered this question: What is the remainder when $x^{2016}+x^{2013}+\cdots+x^6+x^3$ is divided by $x^2+x+1$? I know for a fact that $x^3-...
1
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1answer
39 views

Is there an error in this Precalculus handout on Law of Cosins and Herons Formula?

Apologies, I can't include images - not enough rep. An example exercise dealing with the Law of Cosines to prove Heron's formula for the area of a triangle has me befuddled. After substitution ...
0
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2answers
48 views

I need help factoring quadratics.

I need help factoring $x^4-2x^2-3=0$. I do not know how to factor this disguised quadratic.
2
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2answers
52 views

Find a possible f(x)

Do you think that it is possible to find a $f$ such that given a floating point constant $c \gt 0$ and an integer constant $n \gt 0$, then $\forall x_i \gt 1, i=1, 2,...,n$ : $$f\left({1\over x_1}+{...
0
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2answers
59 views

Is a polynomial irreducible modulo a silly large prime

Suppose I have a large ($\approx 2^{64}$) prime $p$, and a polynomial $g(x)$ of degree $d$ that irreducible in $\mathbb{Z}[x]$. If $d << p$, is it likely that $g(x)$ is irreducible in $\mathbb{Z}...
4
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1answer
79 views

Factoring polynomial

I have a polynomial, let's say $p(x)=x^5+x^4+x^2+x+2$ (or any other polynomial with rational coefficients). What is the general recommended way of factoring it into irreducible factors (in $Z[x]$, $Q[...
1
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2answers
162 views

Find a possible f(x) and g(x)

Do you think that it is possible to find a $f$ and a $g$ such that $f(x) \neq x$ and $\forall x \gt 1, y \gt 1$ then $$f\left({1\over 1-{1\over x}-{1\over y}}\right) = {1\over 1-{1\over g(x)}-{1\...
1
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7answers
95 views

Solve $3 = -x^2+4x$ by factoring

I have $3 = -x^2 + 4x$ and I need to solve it by factoring. According to wolframalpha the solution is $x_1 = 1, x_2 = 3$. \begin{align*} 3 & = -x^2 + 4x\\ x^2-4x+3 & = 0 \end{align*} ...
3
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1answer
49 views

What causes the equating-the-coefficients method not to work?

When searching here to find methods for solving quartic polynomials, I came across a question where one of the solutions (at the very end) mentions that the equating coefficients method can fail. http:...
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0answers
74 views

Integer Factorization problem - New Idea

I been thinking about slightly different approach of solving the problem, and I want you to tell me if my idea is reasonable and if it's original(If someone already thought about this, I would be ...
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0answers
27 views

Is Horner Algorithm works or suitable for polynomial with complex coefficients?

I have checked Algorithm of Horner for polynomial factorization with complex coeffecients ,I got it works but i can't show it works in general for any polynomial with complex coeffeciant . My ...
0
votes
4answers
90 views

Factoring $x^4 - 16$

I was following a calculus tutorial that factored the equation $x^4-16$ into $(x^2 +4) (x+2)(x-2)$. Why is the factorization of $x^4-16 = (x^2 + 4)(x+2)(x-2)$ rather than $(x^2 - 4)(x^2 +4)$?
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2answers
94 views

Prove Expression cannot be factored

I am currently working on primes which can be expressed in form of a polynomial. For eg, Find all primes which can be expressed in form $n^4-52n^2+595$ It is very essential to tell whether a ...
0
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1answer
33 views

Express a variable as a function of another [closed]

If we have this formula : $a/(1-$$1\over x$$-$$1\over y$$) = 1/(1-$$1\over xb$$-$$1\over yb$$)$ Is it possible to express $a$ as a function of $b$, independently of $x$ or $y$ ? Thank you
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2answers
32 views

Factor Theorem (Finding values of a and b)

Question: The polynomial $p(x) = 2x^3 - ax^2 + bx + 48$ has $(x-4)$ as a repeated factor, find the values of $a$ and $b$. What I have attempted if $x-4$ is a factor then $x = 4$ is a root ...
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0answers
30 views

Grouping a set of numbers

I have a set of numbers which I don´t know if they belong to the same group (I could also call it factor or treatment, but actually each group is suppose to identify the same biological event). I am ...
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0answers
33 views

Complex Space Factorization

I'm curious as to whether a visualization of the complex space and the area covered through various factorization methods has been put forth before. Each triangle in the mapping below represents a ...
4
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0answers
49 views

Finding how large $p$ needs to be to have $n$ unique factors…

If we take a prime $p$, how large does $p$ have to be so that $p-1$ has at least $n$ factors between $f_1$ and $f_2$? (Note that the factors can be prime or composite) Note that I'm looking more for ...
2
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2answers
108 views

Is there any better way to find n! without just multiplying all the numbers till n? [duplicate]

I was solving some permutation, combination problems where we know that we are to use factorial. So I was thinking if there was any shorter way, maybe a formula, than multiplying all the numbers till $...
0
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2answers
29 views

Factoring under a 4th power

Show that $$\left( a + b \right) ^ 4 = b^4 \left( \frac{a}{b} + 1 \right) ^4$$ Its clear without an exponent $$\left( a + b \right) = b \left( \frac{a}{b} + 1 \right) $$ but I'm not sure why ...
2
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1answer
28 views

Factor Transition Issue

$$ F'(x)=(2ax+b)e^x+(ax^2+bx+c)e^x=(ax^2+(b+2a)x+c+b)e^x $$ Can someone explain this? I can't understand the transition. What method is this?
2
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3answers
47 views

Factorise Algebraic Expression

Background: I came across the following problem in class and my teacher was unable to help. The problem was factorise $x^6 - 1$, if you used the difference of 2 squares then used the sum and ...
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1answer
14 views

factors in polynomial rings with field coefficients

I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following: Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if ...
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1answer
39 views

The appearance of complex numbers when factoring quadratics [closed]

Why does $i$ only get involved in factoring quadratics like $x^2+9$, but not $x^2-9$? Why does the $+$ sign lead to complex numbers?
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1answer
60 views

Is it possible to solve the following equation without using the Rational Root Theorem?

Given $f(x)=x^4+2x^3+2x^2-2x-3$, where $x-1$ is a factor of $f(x)$, how is it possible to solve $f(x)$ without the Rational Root Theorem? Here's my progress: $$f(x)=x^4+2x^3+2x^2-2x-3$$ $$f(x)=(x-1)(...
1
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1answer
25 views

Changing the Form of this Factorisation

I'm brushing up on some high school maths and I'm currently revisiting determinants, specifically the factorisation of determinants. I'm working my way through a problem set and I keep getting stuck ...
2
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0answers
39 views

How Do I Apply the Cantor-Zassenhaus algorithm to $\mathbb{F}_2$?

Recently, I've been trying to implement the Cantor-Zassenhaus algorithm in C++ over $\mathbb{F}_2$. According to this lecture, the algorithm is basically: Input is polynomial $f\in\mathbb{F}_q$ with ...
2
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3answers
56 views

Factorize $2a^3 - b^3 - c^3$

I need to factorize the expression $2a^3 - b^3 - c^3$. I see that one zero is achieved when $a=b=c$, but I can't find the factor(s).
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3answers
52 views

What comes first here? pemdas doesnt really tell me what to do here

So I have this equation: $2x(x+3)(x+3)$ Do I FOIL the $(x+3)$ first or multiply the $2x$ to the first $(x+3)$? Would there be a difference? Isn't multiplication commutative?
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0answers
43 views

Show that $504 \mid n^9 − n^3 $ for any integer $n$ [duplicate]

Not sure how to start this. I know that $504 =2 \times 2 \times2 \times 3 \times 3 \times 7$.
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0answers
18 views

Cracking any linear congruential generator

I have a linear congruential generator $X_{n+1} = (aX_n + b) \bmod 2^k $with given arguments and number $Y$. The problem is to find the smallest $i$ that $X_i = Y$ or tell that there is no such $i$. ...
4
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1answer
79 views

Probability that a number has $m$ indistinct factors

I just discovered Matlab's factor()-function, and I randomly typed in 20081294819, and to my surprise it only had two factors (5099 and 3938281)! I had expected many more factors for such a big number ...
3
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3answers
95 views

Why isn't integer factorization in complexity P, when you can factorize n in O(√n) steps?

It is said that integer factorization is an NP problem. Why isn't it P? You can solve it in $O(\sqrt{n})$ time with trial factorization, and since $\sqrt{n} = n^{1/2}$, to me that looks like a number ...
1
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2answers
20 views

Factorization and Quadratic Non-Residue

Suppose that I can always factor any number modulo $p$ into factors that are smaller than $f(p)$ where $f$ is some function. Does that imply that the least quadratic non-residue is smaller than $f(p)$...
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3answers
34 views

factoring of $ e^{2x}-3e^{x}+2 = 0 $

How does $ e^{2x}-3e^{x}+2 = 0 $ factors to $ (e^x - 1) (e^x - 2 ) $ Because when I try to factor: $ e^{2x}-3e^{x}+2 = 0 $ $ e^{2x}-2e^{x}-1e^{x}+2 = 0 $ But $-2e^{x} * -1e^{x}$ should ...
2
votes
1answer
66 views

Roots of $x^3-x+1$

I am trying to find nice explicit formulas for the roots of the polynomial $x^3-x+1$. Is there some clever way to write down the roots in a reasonably easy way? I found the roots, but my expressions ...