# Tagged Questions

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

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### Computer program for factorization into irreducible polynomials over $\mathbb{Z}_{p^k}$

Hensel's Lemma allows us to factor a polynomial uniquely into basic irreducible factors over $\mathbb{Z}_{p^k}$. Is there a SAGE or Magma command that gives this factorization? Or can anyone help in ...
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### Factoring a polynomial over $\mathbb F_{2^8}$

How do you find the factors of $x^4+x+1$ in $GF(2^8)$ in terms of polynomials? Let me explain, We have primitive irreducible polynomial $p(x)=x^2+x+1$ in $GF(2^2)$ which has root $\alpha^2+\alpha$ in ...
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### AlgebraII factoring polynomials

equation: $2x^2 - 11x - 6$ Using the quadratic formula, I have found the zeros: $x_1 = 6, x_2 = -\frac{1}{2}$ Plug the zeros in: $2x^2 + \frac{1}{2}x - 6x - 6$ This is where I get lost. I factor ...
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### What's the relation between factors of a number and its square root?

For instance, if the square root of a number $N$ is an integer, $N$ is a square number. But for instance $\sqrt{80} = 8.944...$, the fractional part is close to an integer, and indeed $81$ is a square ...
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### A non-UFD where prime=irreducible [duplicate]

It is easy to see that in an atomic domain (where every element factors into irreducibles), we have that all irreducibles are prime iff the domain in question is an UFD. I think it is not true for a ...
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### In $\triangle ABC$ , find the value of $\cos A+\cos B$

The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{(2b)!}$, Then prove that the value of ...
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### Unable to get matched answer using factorization

I have question to solve by factorization. the question is $$(a+b)x^2 + (a+2b+c)x + (b+c) = 0$$ the answer should be $$x = -a, -b.$$ i have done using it \begin{align} (a+b)x^2 + (a+b+b+c)x + ...
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### How to fully factor a polynomial of 4th degree?

How to fully factor this polynomial? $$2x^4+3x^3-32x^2-48x$$ Can anyone describe the full steps to factor it? Thanks for the help.
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### Numbers with special factorisation

We know that any natural number $n$ can be decomposed as $p_1^{k_1}p_2^{k_2}...p_n^{k_n}$. I am looking for numbers which have $k_1=k_2=k_3=....=k_n=1$ i.e. given a number n, identify if it has all ...
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### Solving $3t^2-\frac{12}{3}t+\frac{4}{3}=0$

I need to to solve: $$3t^2-\frac{12}{3}t+\frac{4}{3}=0$$ The solution manual factorizes this to $\dfrac{1}{3}(3t-2)^2$. How can you do this easily?
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### finding poles for a complex rational function

So in working out the details of a trig integration with complex integrals problem, I have ended up with an integrand of $$\frac{z}{z^4+6z^2+1}$$ I need to find the roots of $z^4+6z^2+1$ to use the ...
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### Prove that $n^3 - n$ is divisible by 6 by factoring

I need to prove that $n^3 - n$ is divisible by $6$ by factoring it and by knowing that the product of each consecutive $3$ numbers is divisible by $2$ and $3$. I tried: $n(n^2 - 1)$ Factoring it ...
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### Finding for which value of $a$ are two equations equal(need instructions for method)

I have the equations: $(a - 5x)^2$ and $25x^2 - 5x + a^2$ And I have a list of values for $a$ and for one of them, the two are equal. I just need to know what is the method for solving this - do I ...
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### Why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$

I would like to understand why $(x-5)^2-4$ can be factorised as $(x-5-2)(x-5+2)$ I am particularly concerned with the term, $-4$.
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### Factorizing Given Problem

I have searched through various site's and forums but couldn't find the answer to my problem, $$z^2-\frac{1}{2}z-\frac{1}{4}=0$$ How will you factorize this As I can't find $2$ numbers that give me ...
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### Factoring in $\mathbb{Z}[\sqrt{2}]$

How would one factor a number, say $9+4\sqrt{2}$ in $\mathbb{Z}[\sqrt{2}]$? This is what I've attemped to do: $$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2})$$ $$a_1a_2+a_1b_2\sqrt{2}+a_2b_1\sqrt{2}+2b_1b_2$$ ...
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### Integer factorization complexity

Why isn't the problem of factoring an integer known to be in $P$? Isn't the naive algorithm of trying to divide a number by all the numbers up to its squre root polynomial?
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### Factorize the given equation.

Factorize $$f(t) = t^3 - 11t^2 - 39t - 45$$ Assuming the above polynomial has a rational root, I tested the above equation using $+1$, $-1$, $+2$, $-2$. These did not work out. Then I tried $t =3$. ...
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### Question about factoring/condensing equation rules

I have the equation $x^2 - 6x = 72$ and then $x^2 - 6x - 72 = 0$ that's supposed to turn into $(x-12)(x+6)$. 72/6 = 12. So could just do that with any equation that? Divide the end thing with the ...
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### Calculate sum of all factors of expression

Expression: $$\left(\frac{2x}2\right)^2 \left(\frac{3y}3\right)^3$$ Sum of all factors of above expression is $$2\cdot \left(\dfrac{2x}2\right) + 3\cdot\left(\dfrac {3y}3\right)$$ How ? Can ...
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### Deflating (factoring) a 6th degree polynomial

What is the procedure to factor a 6th degree polynomial of a complex variable? $$P(z)=1+x^2+x^3+x^4+x^5+x^6$$ I do have the correct answer but no idea how to get to it. The answer is: ...