# Questions tagged [factoring]

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

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### Question on proof involving functions and factorisation.

Assume that f(xy) = f(x) + f(y) for all positive integers x and y. Show that if the positive integer n has the factorization n = p^a× q^b× r^c , then f(n) = a f(p)+ b f(q)+ c f(r)
1 vote
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### Is there a name for this technique involving breaking a term into multiple terms?

I recently saw a solution to the quadratic equation $x^2-5x-6=0$ that involved re-writing the middle term, $-5x$, into two terms, $x-6x$, so that the expression could be factored and $x$ solved for, ...
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### Factoring Polynomial over GF(8)

It is known that polynomial over GF(2) $x^6+x+1$=$g_1(x)*g_2(x)*g_3(x)$, where $g_i$ is irreducible over GF(8) and deg $g_i(x)$=2. What is the most optimal way to find those polynomials?
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### A Question Regarding the Differences of Terminologies and Theorems Related to Polynomial Division

This will be a long post and there will be a TL;DR at the end. I've recently been re-reading topics on polynomial division to brush up my knowledge on them but sometimes I get a little confused and ...
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### Evaluating integer polynomial at algebraic point

Say one can add and multiply two algebraic numbers. Let there be an (real) algebraic number $A$, defined by a integer polynomial $\mathbb Z[x]$ a two rational endpoints which bracket the given root. ...
• 324
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### Factorising Multivariable Polynomial of an Ellipse

I'm trying to factor: $4xy-8y^2-2x^2+9x=0$ The equation is an ellipse so it should be possible to get this into the form $\frac{(x-a)^2}{p} + \frac{(y-b)^2}{q} - c=0$ I've tried "completing the ...
1 vote
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### Polynomial Solution

If we have this polynomial: P($x$) = $x^3$ - $3x^2$ - 10$x$ + 24 and we want to find the root b: P(b) = 0 in the book, the solution is that b must be a factor of 24, why is that? I couldn't find an ...
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### Factoring the quintic $n^5-16n^4+95n^3-260n^2+324n-144$

I was attempting to solve $n^5-16n^4+95n^3-260n^2+324n-144=0$ but then realised I didn't know how to. How would one go about factoring such a quintic and solve for n? I know that the factored form is ...
90 views

• 10.2k
1 vote
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### Assume that $a=b$. If the function $f(x)$ is monotonously increasing, then (answer 1) $0<a<1$ (answer b).

The given function is $f(x)=x^3-3ax^2+3bx-2$. I am aware that monotonously increasing means to continuously increase, so I tried getting this function's derivative and then setting it to zero, but to ...
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### Factoring $1 + 2\cos(\theta_1) x + 4i\sin(\theta_1) \cos(\theta_2) x^2 + x^3=0$

Is it possible to factorize the cubic $$1 + 2\cos(\theta_1) x + 4i\sin(\theta_1) \cos(\theta_2) x^2 + x^3=0$$ without explicitly using the cubic equation? Given the coefficients, the equation will ...
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### Other ways to factorize $xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$

To factorize $xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$ I used the fact that $x=-y$ and $y=z$ and $x=-z$ make the expression zero. Hence it factorize to $\lambda (x+y)(y-z)(x+z)$ and ...
• 5,676
1 vote
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### Factoring $m^4+2m^3-m^2-2m-8$

Factorize $$m^4+2m^3-m^2-2m-8$$ First I plugged numbers like $m=-3,-2,-1,0,1,2,3$ in the expression and neither of them is a root so I can't use rational roots theorem here to find a factor. I've ...
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• 6,309
1 vote
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### Binary Galois field, test for irreducible polynomial in NASA paper

I am reading Geisel's tutorial$^{\color{red}{\star}}$ on Reed-Solomon codes, in which it is tested whether the polynomial $f(x) = x^4 + x + 1$ is irreducible. From pages 15 and 16, As a first step, ...
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### Factorisation of $X^n-a$ in $\mathbb{F}_p$

Do we know general formula for factorisation of $X^n-a$ in $\mathbb{F}_p$ for any $a\in\mathbb{F}_p$? At least do we know the degree of the factors? Some reflexions: if $a=1$ it is the "...
• 2,516
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### How to differentiate the function, $7x^5\sqrt{3x^4-5x^2}$

$f'(x)= 7x^5(3x^4-5x^2)^{1/2}$ $= 35x^4(3x^4-5x^2)^{-1/2} +1/2(3x^4-5x^2)^{-1/2}(12x^3-10x)(7x^5)$ $= 35x^4(3x^4-5x^2)^{-1/2} + 1/2 (12x^3-10x)(7x^5)$ That is what I have done so far using the product ...
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### Is there a name for the decomposition of $m^1w^1-m^0w^0$ into $(m^1-m^0)w^0+(w^1-w^0)m^0+(m^1-m^0)(w^1-w^0)$?

In the following, what is this kind of decomposition / factoring called? I'd like to read the proof for it or learn more: $$m^1w^1-m^0w^0=(m^1-m^0)w^0+(w^1-w^0)m^0+(m^1-m^0)(w^1-w^0)$$ It doesn't ...
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1 vote
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### Factorize: $1-\frac{1}{6}z^{-1}-\frac{1}{3}z^{-2}$ into $(1-az^{-1})(1-bz^{-1})$

Currently stuck with factorizing $1-\frac{1}{6}z^{-1}-\frac{1}{3}z^{-2}$ into $(1-az^{-1})(1-bz^{-1})$ How do I get there? I would like to know how to do it without guessing. Tried pq, but got stuck ...
• 35
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### If $s = 4m - 3$, then $\sigma(p^s) = (1 + p^{2m-1})(1 + p + \ldots + p^{2m-2})$. Is there a similar factorization for $\sigma(q^t)$ when $t=4n$?

Let $p,q$ be (odd) primes, and let $m,n,s,t$ be positive integers. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. M. A. Nyblom showed that, if $s = 4m - 3$...
• 10.2k
1 vote
### If $ax^2+bx+c$ is irreductible, then exists constants $k_1,k_2,k_3$ such that $ax^2+bx+c = k_1((k_2x+k_3)^2+1).$
I want to prove that: If $ax^2+bx+c$ is irreductible, then exists constants $k_1,k_2,k_3$ such that $$ax^2+bx+c = k_1((k_2x+k_3)^2+1)$$ We note that \begin{align*} &ax^2+bx+c = \frac{1}{4a}(4a^...