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

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1answer
20 views

Factor polynomial into linear factors with complex coefficients.

Question: A polynomial is given. $(a)$ Factor it into linear and irreducible quadratic factors with real coefficients. $(b)$ Factor it completely into linear factors with complex coefficients. $x^3 - ...
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2answers
33 views

Stuck on a simple factoring problem

The answer to this question is probably very obvious but I can't figure it out for some reason: I simply want to factorise: $x^2+5x-2$ I solve $x^2+5x-2 = 0$ i find $x_1 = \dfrac{-5-\sqrt{33}}{2}$ ...
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3answers
39 views

Factoring $12e^{2x} - 32e^x + 16$ [on hold]

Can you help me solve the quadratic equation $12e^{2x}-32e^x+16$ by factoring please?
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0answers
43 views

The irreducibility of $a^{4n}+b^{4n}$ [on hold]

How to prove that $a^{4n}+b^{4n} $, for any natural number $ n $, is irreducible over the rationals?
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1answer
22 views

Probability that a random polynomial over a finite field can be factorized to linear terms.

Suppose that $f\in\mathbb{F}_p[x]$ is a degree $d$ random univariate polynomial with coefficients from a finite field $\mathbb{F}_p$. What is the probability that $f$ can be written as: ...
2
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1answer
43 views

Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? \begin{equation} f\left(a ...
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0answers
12 views

On factoring given $PQ-1$ has small factors.

Suppose we have an RSA number $PQ$ where $PQ-1$ has small factors. Will this give any advantage to factor $PQ$?
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3answers
67 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...
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2answers
64 views

Factoring a degree 4 polynomial without power of 2 term

For my hobby, I'm trying to solve $x$ for $ax^4 + bx^3 + dx + e = 0$. (note there's no $x^2$) I hope there is a simple solution. I'm trying to write it as $(fx + g)(hx^3+i) = 0$ It follows that ...
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2answers
37 views

Factor out (m+1) in the following so that the final answer is $\frac{(2m+1) (m+2) (m+1)} {6}$

Question: $\frac{m (m+1) (2m+1) + 6(m+1)^2}{6}$=$\frac{(2m+3)(m+2)(m+1)}{6}$ I must multiply by 6 on both sides and expand the brackets and collect like terms. I'm I correct? Edit notes: The ...
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1answer
46 views

Factor Completely.

Again, this question if from my final practice exam. Factor Completely. $$81x^4-256y^4$$ I'm able to get this far, How do I know which of the two factors should be factored further. ...
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1answer
59 views

How to factor $a^3 - b^3$?

I know the answer is $(a - b)(a^2 + ab + b^2)$, but how do I arrive there? The example in the book I'm following somehow broke down $a^3 - b^3$ into $a^3 - (a^2)b + (a^2)b - a(b^2) + a(b^2) - b^3$ and ...
0
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1answer
28 views

Interesting 4th order factoring question

$$ A = \frac{(4\cdot2^4 + 1)(4\cdot4^4 + 1)(4\cdot6^4 + 1)}{(4\cdot1^4 + 1)(4\cdot3^4 + 1)(4\cdot7^4 + 1)}$$ What is the value of $ \dfrac{113A}{61}$ ? So i tried factoring this ...
0
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1answer
14 views

Factor out (m+2) in the following equation (m+1)(m+2)+2(m+2)

(m+1)(m+2)+2(m+2) I really needed hints here, I am thinking to start at first two paragraphs and so on. Is my thought correct? hints will be much appreciated.
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4answers
49 views

Taking factors out of this integral?

In the integral: $$\int\frac{-25}{17(2t+3)} + \frac{37}{17(5t-1)} dt$$ Why is the final answer: $$-\frac{25}{34}ln|2t + 3| + \frac{37}{85}ln|5t - 1| + C$$ If you take $-\frac{1}{2}$ as well as the ...
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0answers
11 views

Product of heights of factors smaller than length of a polynomial with integer coefficients

I have the following question. Given a (univariate) polynomial with integer coefficients, I want to prove, if true, that the product of heights of its (irreducible) factors is smaller or equal to its ...
0
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1answer
34 views

Expanding an infinite product of infinite series

Here's a fragment of something I posted in an answer a few months back: \begin{align} & \left( 1 + \frac 1 {a_1} + \frac 1 {a_1^2} + \frac 1 {a_1^3} + \cdots \right) \\ \times {} & \left( ...
1
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1answer
19 views

Find a common factor of multiple matrices

I'm currently facing a problem where I have $N$ matrices $ \{A_1, A_2, \dots, A_N \} $ and I want to find a way to calculate matrices $H$ and $E$ such that $ \exists H \: \exists E\: \forall i \in ...
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1answer
37 views

Factorization of Taylor series.

I know that for a (finite) polynomial $P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_0$ whose zeros are $x_1, x_2, \ldots, x_n$, then we can factorize it as $$P(x) = a_n(x - x_1)(x - x_2) \cdots (x ...
0
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1answer
41 views

Number of solutions for $n^5 + 2 n^4 + n^3 - 3n + 2 $ mod $ 23^2 = 0$, where $0 \leq n < 23^2$ and $n\in \mathbb{N}$

$0 \leq n < 23^2$ and $n\in \mathbb{N}$ For how many $n$ $n^5 + 2 n^4 + n^3 - 3n + 2 $ mod $ 23^2 = 0$
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2answers
479 views

Highschool Exam Question About Cube Factoring

Given; $ a^3 - 3ab^2 = 10 $ and $ b^3 - 3ba^2 = 5$ What is the value of $ a^2 + b^2 $ ?
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4answers
68 views

finding the number of factors $2^{15}\times3^{10}\times5^6$

The number of factors of $2^{15}\times3^{10}\times5^6$ which are either perfect square or perfect cubes(or both) I don't know how to start this even! Plz solve this!
0
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1answer
47 views

Use Fermat factorization to factor $809009\ldots$

Use Fermat factorization to factor $809009\ldots$ So far I have: \begin{align} \sqrt{809009} & = 889.449 \\ & = 890 \\[6pt] \sqrt{890^2 - 809009} & = 130\ldots ∉ \mathbb Z \\[6pt] ...
5
votes
3answers
87 views

What's the best way to compute $\frac{a^4 + b^4 + c^4}{a^2 + b^2 + c^2}$

So, my teacher gave us this to compute yesterday, and I'm completly confused on how should I proceed : $$\frac{1^4 + 2012^4 +2013^4}{1^2 + 2012^2 + 2013^2}$$ I've tried several ways, but most of ...
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3answers
32 views

Factoring a 4th degree trinomial

I am trying to factor $3x^4-8x^3+16$, but I have no idea how to even start. I put into Wolfram Alpha, and it said that the answer was $(x-2)^2 (3 x^2+4 x+4)$. How would you factor something like this ...
2
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3answers
33 views

Find A and B for $A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$

Given:$$A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$$ How does one find A and B ? The answer is: $$A = 3x^{2};B=2x$$ but I can't see how one solves this. I tried subbing in values of x but it didn't lead ...
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0answers
22 views

Linear factors of minimal polynomial dividing $x^r$ - 1

I have a monic minimal polynomial $m(x)$ that divides $x^r - 1$. Apparently $m(x)$ has distinct linear factors over the complex numbers $\mathbb C[x]$. I understand this part, since $\mathbb C[x]$ is ...
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2answers
28 views

Find the cubic polynomial given linear reminders after division by quadratic polynomials?

A cubic polynomial gives remainders $(13x-2)$ and $(-1-7x)$ when divide by $x^2-x-3$ and $x^2-2x+5$ respectively. Find the polynomial. I have written this as: $P(x)=(x^2-x-3)Q(x)+(13x-2)$ ...
0
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1answer
17 views

What is the theory of finding roots of a polynomial equation by looking at the factors of the $a_n$ and $a_0$ term called?

This is commonly taught in high schools in the context of factoring polynomials. I remember this method even has its own wikipedia page (with a proof) but I forget what was the theory called. Could ...
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1answer
24 views

What is the name of this kind of factoring algorithm

I just think about algorithm to find factor of number by doing something like guessing last digit of number and increase digit bit by bit Such as, I want to find factor of 749 Algorithm would begin ...
0
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3answers
23 views

Simplifying with exponents

So somehow I made it all the way to Calc II but struggle when it comes to this basic thing (not exactly the most 100% solid algebra foundation it seems). Unable to simplify this further: $$ ...
1
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1answer
54 views

What will be the $(b^2-a)$

If $a$, $b$ are the real numbers and $$4a^{2}+b^2=4a-(\frac{1}{4b^2})$$ What will be the $b^2-a=$? I tried to be this equation more basic ,but i could not reach the result
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1answer
30 views

finding factors

How can i quickly find the factors of a particular number? Find the number of different factors of 1800 and 3003? This being the question , for 3003 i first found out its prime factors and then i ...
0
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1answer
32 views

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$. How come the answer is left as $\frac{5x}{(x+2)(x-3)} + \frac{4}{(x+2)^2}$. Why don't we go any further?
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1answer
26 views

What does the first x represent in {x, (x+1), (x-3)}?

The question is: "Part of the graph of a polynomial function is shown. Which of the following sets contains only elements that are factors of the polynomial?" The two answer choices left are B. ...
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0answers
9 views

Which polynomial factorization method leads directly to $(1-\alpha_0 z)(1-\alpha_1 z)$

I know how to factor a polynomial $p(z)$ so that it looks like $a_n(z- z_0)\cdots(z-z_n)$, where $z_k$ are its zeros. Now I could squeeze this form into the wanted $(1-\alpha_0 z)\cdots(1-\alpha_n ...
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2answers
38 views

Factor trinomials dividing by the common GCF

I have a doubt with the following problem I found in a book. You have to simplify a polynomial using the GCF. Now, this is the problem I am not able to grasp: $$6x^2-19x-7$$ According to the book, ...
0
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1answer
36 views

Simple question on factoring the difference of 2 perfect squares

(b) (i) Use the identity $A^2-B^2=(A-B)(A+B)$ to factorise the expression $5^{2k}-1$. Do I just put the k as 1 so that the equation is 5^2 and 1^2 Thanks Steve
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1answer
15 views

Solving a characteristic Polynomial of the Hilbert Matrix

I need to find the eigenvalues of the following characteristic polynomial but I can't seem to successfully find the roots of the equation: $P[λ]$ = $λ^5$ - $563/315λ^4$ + $0.3476λ^3$ - $0.0038λ^2$ ...
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0answers
30 views

Factor polynomial series with coefficients of 0 or 1?

Is there any easy way to factor polynomials which have coefficients of only $0$ or $1$ and always have a $+1$ ? For example, factor $9^{25}+ 9^{19}+ 9^{14}+ 9^9 + 9^6 + 9^5 + 1$
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2answers
46 views

Can we use Eisenstein's Irreducibility Criterion to show that $x^4+1$ is not reducible in Q?

As such: Let $a(x)=x^4+1\in\mathbb{Q}\left[x\right]$. Then choose any prime $p$. By Eisenstein's Criterion, we see that $p\nmid 1$, $p\mid 0$ (since all coefficients of intermediate terms are 0), and ...
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1answer
35 views

Irreducible factorisation of polynomial over quotient field

Let $F=\mathbb{Z}_3[x]/<x^2+1>$. Factor $x^4+2$ into irreducibles in $F[x]$. I know that $F$ is a field since $x^2+1$ is irreducible. The usual way to find out that a polynomial is irreducible ...
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0answers
28 views

Finding Factors of a Determinant

Consider the determinant with elements: $a_{11} = ax-by-cz, a_{12}=ay+cz, a_{13}=cx+az$ $a_{21}=ay+bx, a_{22}=by-cz-ax, a_{23}=bz+cy$ $a_{31}=cx+az, a_{32}=bz+cy, a_{33}=cz-ax-by$ Where $a_{ij}$ ...
0
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1answer
49 views

How to factor $2x^4-11x^3-44x^2+149x+84$

I am doing something for math, and I need to factor $$2x^4-11x^3-44x^2+149x+84.$$ How do we factor it?
0
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1answer
69 views

General solution of the differential equation: y' cot x + y = 2

I have to find the general solution of the differential equation:$ y$' $cot$ $x$ + $y$ = $2$. And determine the integration constant using the initial condition $y$(0) = $1$. Additionally presenting ...
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1answer
17 views

Factoring a polynomial to get its zeros

While studying about sums and products of roots of polynomials, I found this on the web: We can take a polynomial, such as: $$f(x) = ax^4 + bx^3 +\dots$$ And then factor it like this: ...
5
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4answers
393 views

Factorization of polynomials with degree higher than 2

I need help to factorize $x^4-x^2+16$. I have tried to take $x^4$ as $(x^2)^2$ and factorize it in the typical way of factorizing a quadratic expression but that did not help. Can someone help me to ...
0
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1answer
43 views

Simple question about finding roots of a polynomial

What am I doing wrong here? This is the denominator of one of my problems and I need to find the roots, so: $6i-z^2+1 \to z=\sqrt{1+6i}$ and $z=-\sqrt{1+6i}$ $\therefore$ ...
0
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0answers
52 views

About an integer factoring algorithm

I have been toying with the following algorithm: ...
1
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1answer
34 views

Separable polynomials are the product of the minimal polynomials of their roots?

I see the following claim in this answer: Since $f$ is separable, it follows that $f(x)$ must be the product of minimal polynomials of [its roots] But, I don't know how we justify this claim. ...