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)
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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. ...
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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 ...
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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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2 answers
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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 ...
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1 answer
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Derivatives and double roots in $\mathbb{Z}/p^n\mathbb{Z}$

Is there a notion of double roots of polynomials in the ring $\mathbb{Z}/p^n\mathbb{Z}$? By that I mean if for a polynomial $P(x)$ we have that $p^n|P(a) $ and $p^n|P'(a)$, then $(x-a)^2$ divides $P(x)...
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find all integers m such that the number $m^5+m+1$ is prime [duplicate]

I really don't know how to start this problem. The problem should be related to factoring the equation to derive an answer, but I don't know how.
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1 answer
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Scary integral function [duplicate]

If $f,g,h,\phi$ are polynomials in $x$, and $$p(x)= \left (\int_1^xf(t)h(t)dt\right) \left(\int_1^xg(t)\phi(t)dt\right)-\left(\int_1^xf(t)\phi(t)dt\right) \left(\int_1^xg(t)h(t)dt\right) $$ is ...
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Please check if my solution is correct [duplicate]

Let $f(x)=a x^{2}+b x+c$ be a quadratic polynomial with integral coefficients, where $a \neq 0$. Show that (i) if $f(x)$ is factorisable into linear factors with integral coefficients, then there are ...
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How to rearrange equation to cubic polynomial form?

Summarize the problem I have these 2 equations and i need a cubic polynomial in terms of Z so i can run numerical routines on it. How can i be sure that's possible to factor that way and second how ...
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Factor $a x^n + b x + c = 0$

I wish to factor the polynomial equation $$a x^n + b x + c = 0$$ When $a = b$ and $n=5$, we have the Bring-Jerrard normal form of the quintic $x^5 + x + c = 0$. Using the Lagrange Inversion theorem, I ...
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3 answers
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$(x-\lambda)$ and $(x-\overline{\lambda})$ appear the same number of times in the factorization of $p$ Linear Algebra Done Right 3rd Edition Axler

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 4.15 Polynomials with real coefficients have zeros in pairs Suppose $p\in\mathcal{P}(\mathbb{C})$ is a polynomial with ...
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Is the polynomial $p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$?

My initial question in the present post is pretty basic: Is the (integer) polynomial $$p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$? When $k=...
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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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4 answers
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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 ...
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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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3 votes
1 answer
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How to find factors of a matrix determinant

I'm working on some problems for fun and came across this problem I'm stuck with. Here's the question: Use row operations to show that $ x + \omega y + \omega^2 z $ is a factor of $ \Delta $, where $ \...
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1 answer
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Find the intersection point between these two equations

We have $$ f(x) = 12\sqrt x $$ and $$ g(x) = x^2 - 7x + 12 $$ I need to find where they intersect. So far I've reduced the expression to $$ 12\sqrt x = x^2 - 7x + 12 $$ $$ 12 \sqrt x = (x-4)(x-3) $$ ...
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1 answer
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Can a number have infinite number of factors if we include rational numbers

Can a number such as 10 have infinite factors if we include multiplying two rational numbers or a rational number and integer or any other combinations? Google says factors of 10 are 1×10, 2×5 and the ...
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1 answer
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How many factors of $2400$ are not factors of $3600$? [closed]

I solved this question by writing all the factors and then just selecting the factors as per the question requirement. But I want to know is there any other way to solve this? Please help !!! Thanks ...
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What do you call rings that have unique factorizations?

For example, integers, gaussian integers, and polynomials all have unique factorizations. What are these rings (or this property) referred to as? Or is unique factorization a ubiquitous property that ...
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Factoring and simplifying a function with a 2nd degree polynomial as its numerator and 3rd degree polynomial as its denominator

I have the function $$ P(x)=\frac{\frac{l^2\beta}{6v^2}x^2+\frac{2l^2\beta^2}{6v^2}x+\beta}{\frac{l^2}{6v^2}x^3+\left(\frac{l^2\beta}{2v^2}+\frac{l}{2v}\right)x^2+\left(\frac{l^2\beta^2}{3v^2}+1+\frac{...
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1 answer
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Factorising a two-variable equation

I have an equation: $-\frac{\cos(kL_1)}{\sin(kL_1)} + 2\frac{1-\cos(kL_2)}{\sin(kL_2)} =0 $ that I would like to factor to separate my variables as such: $[f(kL_2)][f(kL_1)] =0$, I've been trying for ...
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1 answer
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How to factor $X^4+5X^3-2X^2-2$ into its irreducible form over $\Bbb{Z}_{11}$ [closed]

The polynomial $X^4+5X^3-2X^2-2$ has no roots in $\Bbb{Z}_{11}$ so I am unsure as to how I am meant to factorise in such a scenario when I cannot use the factor theorem. How am I meant to progress? ...
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4 votes
2 answers
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Does the pattern of identities for $(x+y+z)^{2n-1}$ continue past $2 n - 1 = 5$?

We have the following identities. \begin{align} (x+y+z)^3&=x^3+y^3+z^3+3(x+y)(y+z)(z+x)\\ (x+y+z)^5&=x^5+y^5+z^5+5(x+y)(y+z)(z+x)(x^2+y^2+z^2+xy+yz+zx) \end{align} I'm wondering is there any ...
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-3 votes
1 answer
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How to factor out $x$ from $(x^n+1)^\frac{1}{n}$ [closed]

How do we factor x out of $(x^n+1)^\frac{1}{n}$
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Could integer factorization be approximable in polynomial time? [closed]

This would mean that there is an algorithm that can take a semi prime, and return a number close to a factor of the semi prime, by some definition of the world close(this could also involve P-adic ...
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2 votes
1 answer
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Better Method for factoring involving Characteristic Polynomial

I am in a linear algebra class right now, and I am reviewing for diagonalization. With my final coming up I am looking to cut down time spent on finding the eigenvalues of $A$ said matrix $A$. An ...
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2 answers
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Factoring $x^3-6x^2+11x-6$ without using Rational Roots Theorem

To factor $P(x)=x^3-6x^2+11x-6$ one way is using Rational Roots Theorem and recognizing that $x=1$ makes $P(x)$ zero. But I want to factor without using it. I tried, $$x^3-6x^2+11x-6=x^2(x-6)+11(x-6)+...
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0 answers
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Tips factoring formula to form of $\frac{1}{f(x)}=\frac{1}{g(x)}+\frac{1}{h(x)}+...?$

$$\frac{1}{x^4(x^2+1)}=\frac{1}{x^2+1}-\frac{1}{x^2}+\frac{1}{x^4}$$ I'm having trouble transforming this kind of equation. The goal is to transform the orignal formula to mutiple formulas and keep ...
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2 votes
1 answer
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Numbers for Testing Integer Factoring Algorithms

I'm looking for a list of numbers with which to test an integer factorization algorithm (for a computer). Something that has numbers harder than the ones I could easily come up with. Do any resources ...
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0 answers
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Proof of $f(x)\in I\Leftrightarrow f(i)=0\Leftrightarrow f(x)=q(x)(x^2+1) (q(x)\in\mathbb{R}[x])$.

I am reading "Introduction to Algebra" (in Japanese) by Makoto Ishida. The following example is in this book: Example 2: Let $g$ be a mapping from $\mathbb{R}[x]$ to $\mathbb{C}$ such that $...
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1 vote
1 answer
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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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2 votes
1 answer
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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 "...
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5 answers
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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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0 answers
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Author calls factors $(x+1-\sqrt2)(x+1+\sqrt2)$ "rational so far as $x$ is concerned". What does this mean?

I'm reading Elementary Algebra by Rouse W. W. Ball, and factorization strategies for quadratics are being discussed. In the book, we are trying to factor $x^2+2x-1$, so we complete the square $$x^2+2x-...
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Maximal degree of irreducible polynomials

This is a question I have thought about for a while. We know that every polynomial $p \in \mathbb C[z]$ can be written as a product of monomials $$p(z) = a \displaystyle\prod_{i=1}^n(z-z_i).$$ Now for ...
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2 votes
1 answer
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Equation with polynomial with integer coeficients.

Let $p>3$ be a prime number. Prove that there doesn't exist a pair of polynomials $(f,g)\in{\mathbb{Z}[X]\times\mathbb{Z}[X]}$ such that: $X^{2p}+pX^{p+1}-1=[(X+1)^p+p\cdot f(X)]\cdot[(X-1)^p+p\...
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1 vote
0 answers
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Showing that a function has at least three roots

I am working on the following problem: Show that the function $f : \mathbb{R} \mapsto \mathbb{R}, f(x) = 2^{x} - 4x^{2} + x + 3$ has at least three roots. So I have to show where $f(x) = 0$ but I ...
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-2 votes
2 answers
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Formula for $x^n+1$ [closed]

Is there any formula for the factorization of $x^n+1?$ for arbitrary positive real number $x$ and a positive integer $n$? Or can it be written as a sum of products of powers of $x$? Edit: I tried to ...
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0 answers
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separation of variables - light factoring, not for solving differential equations

I haven't found much discussion about separation of variables without actually factoring. Basically, I would like to separate an expression into factors which may be composite, but each factor will ...
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1 vote
2 answers
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How does $\frac{k^2(k+1)^2}{4} + (k+1)^3$ become $\frac{(k+1)^2(k+2)^2}{4}$?

As part of an induction proof, the authors of a beginner combinatorics text reduce/factor a polynomial as follows, but do not show the minutiae of their algebraic steps: $$\frac{k^2(k+1)^2}{4} + (k+1)^...
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0 votes
1 answer
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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
2 answers
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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 ...
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1 answer
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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 $s = 4m - 3$...
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1 vote
1 answer
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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^...
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