# Tagged Questions

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

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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 ...
1answer
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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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### Average number of linear factors in a monic polynomial of degree $n$ over $\mathbb{F}_p$

Let $p$ be a prime and $P_n$ the set of all monic polynomials with coefficients in $\mathbb{F}_p.$ I am interested in the average number of linear factors of polynomials in $P_n.$ In an exercise in ...
1answer
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### Factor of determinant with identical row

How the following fact applies to determinants (I came across it while solving problems): Consider A is a nxn matrix, the elements of which are real (or complex) polynomials in x. If r rows of the ...
3answers
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### Irreducible factors of x^16 - 1 over GF(3)

Just want to double check my work. I'm trying to list the irreducible factors of $x^{16} − 1$ over $GF (3)$ of degree $1$ and $2$ . Here's what I have: $$x + 1, x + 2, x^2 + x + 2, x^2 + 2x + 2$$ ...
1answer
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### Basic question on Fermat's factorization method

Please excuse me if this is a basic question, or badly phrased, I'm very new to mathematics in general. In Fermat's factorization method - based on the fact that every odd number can be expressed as ...
1answer
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### Factorize matrix determinant

When trying to diagonalize a matrix, say : $$\left(\begin{matrix} 0 & 2 & -1 \\ 3 & -2 & 0 \\ -2 & 2 & 1 \end{matrix}\right)$$ to find the eigenvalues, I have to find ...
3answers
419 views

### Intuitive understanding of the uniqueness of the Fundamental Theorem of Arithmetic.

Basically I am trying to understand why Fundamental Theorem of Arithmetic (FTA) exists, i.e why a natural number cannot be factored primely in two or more different ways. There are two proofs given ...
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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: $$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2})$$ $$a_1a_2+a_1b_2\sqrt{2}+a_2b_1\sqrt{2}+2b_1b_2$$ ...
1answer
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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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### Possible values of $\gcd(a+b, a\times b)$

Main Question: Let $N \in \mathbb{N}$. What are the possible values of $\gcd(a+b, a\times b)$ given that $\gcd(a,b) = N$? Fact 0. If $\gcd(a,b) = N$, then $N \leq \gcd(a+b, a\times b) \leq N^2$...
2answers
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### How to extract factor when expression is with a power

$$f(x) = x^2(2x-3)^3$$ I tried to extract the 2 from the parenthesis. $$f(x) = 2x^2(x-\frac{3}{2})^3$$ But the graphic from this function is different. What should I consider when doing this kind ...
1answer
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### Prime factorization difficulty

From Wikipedia: Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime ...
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### limit of function at $x \rightarrow 2$

ok, so this is a very basic question, i'm trying to find the limit of the following function at $x \rightarrow 2$: $|x^2 + 3x + 2| / (x^2 - 4)$ what i had previously done was simply plug in 2 for ...
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### Limit of square root function at $x \to 6$

I'm trying to find the limit of the following function at $x \to 6$: $$\frac{x^2-36}{\sqrt{x^2-12x+36}}$$ i've simplified it so that it becomes $\dfrac{(x+6)(x-6)}{\sqrt{(x-6)^2}}$, which simplifies ...
1answer
55 views

### Factoring completely using complex cube of unity

How can you completely factor $a^2 + ab + b^2$ and $a^2 - ab + b^2$ completely using $\omega$, the complex root of unity? Is there some general rule for such complex factorisations? Any help would be ...
4answers
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### Show that $(k!)^n$ divides $(kn)!$

Show that $(k!)^n$ divides $(kn)!$ I've tried it but without success. Any help would be great.
1answer
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### Multiplying two fractions with complex numbers

I'm doing $$\frac{6-7i}{1+i}\cdot\frac{1+i}{1+i},$$ and I'm getting the correct value for the numerator (namely, $-1-13i$), but based on the problem answer, I need for the denominator to become $2$. ...
2answers
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### Simplifying an inequality: $4x(x-2) \lt 2(2x-1)(x-3)$

I have: $$4x(x-2) \lt 2(2x-1)(x-3)$$ For the last part, do I multiply both things in $()$ by two then solve them like I normally would? If I solve them and then multiply will it work the same? Is that ...
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### Humankind knows the prime factorization of the first how many consecutive integers?

I am only looking for an approximation. I'm guessing the answer must be somewhere between $10^{20}$ and $10^{50}$. . Edit: Okay so my first initial estimation was pretty poor... I should have ...
3answers
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### Factoring Real and Complex polynomials.

Factor: a) $x^2 + 1 \in \mathbb{R}[x]$ b) $z^3 - i \in \mathbb{C}[x]$ Well I solved for $x^2$ and got $-i$ and $i$, but they aren't from Real. And I couldn't solve for Complex (part b).
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### Basic complex factorisation

Let's say I want to find all the roots of $f(z)=z^8-256$. Factorising it, I find $f(z)=(z-2)(z+2)(z^2+4)(z^4+16)$. $z =\pm2,\,\pm2i$ is only 4 roots. Shouldn't there be another 4?
4answers
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### Solving for x by completing the square in a problem where the solution doesn't seem to have symmetrical answers

So I've been given this problem: $-14x^2 + 45x + 14 = 0$ And I've tried it a number of times but can't seem to solve it. The answer is supposed to be found by completing the square, and the solution ...
2answers
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1answer
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### Polynomial Factoring over a finite field

Ok, so I'm trying to figure out how to factor polynomials over a finite field. My polynomial is x^5 + x^2 + x + 1 and I have to factor over GF(2) I know the answer is (x+1)^2 * (x^3 + x + 1), because ...