Questions tagged [factoring]

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

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3
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1answer
33 views

How to factorize $P=(X^2-4X+1)^2+(3X-5)^2$ in $\mathbb R[X]$?

First of all, I searched for roots. I knew that $\exists x\in\mathbb R,\, P(x)=0 \iff \exists x\in\mathbb R,\, x^2-4x+1 = 0 \text{ and } 3x-5 = 0$. However, It is really easy to say that there is no ...
0
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0answers
28 views

Is it always possible to factor an element that is not irreducible into two non-units?

A book I am reading defines irreducible elements of an integral domain $D$ as follows: A nonzero element $a\in D$ is called an irreducible if $a$ is not a unit and, whenever $b$, $c \in D$ with $a=...
0
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2answers
25 views

Does a formula exist that gives a function of only solutions that are multiples of any given integer m up to n multiples?

I feel like this is a stupid question but does a formula exist that gives a continuous function of x such that its only solutions are at integer multiples of m up to n? The domain of the function is ...
0
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3answers
47 views

Factoring out $4x^3+2x^2y-2xy^2-y^3$

This can be factored as follows: $$4x^3+2x^2y-2xy^2-y^3 = (2x^2-y^2)(2x+y)$$ What is a systematic way for finding this factorization?
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2answers
58 views

How to factorize the following expression: $2x^4 + 9x^3+8x^2 +9x+2=0$

How to factorize the following expression: $2x^4 + 9x^3+8x^2 +9x+2=0$ I have tried finding a root through putting values, but I can't find any factor. Please help. PS: I'm an elementary student, the ...
-1
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1answer
55 views

If $a\neq 2$ and $a^3+a^2-a-10 = 0$, then evaluate $a+\frac{5}{a}$ [closed]

If $a\neq 2$ and $a^{3}+a^{2}-a-10 = 0$, then what is the value of $a+\dfrac{5}{a}$? I have plugged equation on WolframAlpha and get the complex values of $a$ but you supposed to do this problem ...
2
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3answers
74 views

Calculation in Routh's theorem

The proof of Routh's theorem concludes with showing$$1-\frac{x}{zx+x+1}-\frac{y}{xy+y+1}-\frac{z}{yz+z+1}=\frac{(xyz-1)^2}{(xz+x+1)(xy+y+1)(yz+z+1)}.$$I seek an elegant "proof from the book" ...
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0answers
33 views

Binomial Denominator Reduction [closed]

I remember having to reduce (or isolate more-so) two terms of a binomial in a denominator of a fraction in order to solve for an integral in my calc 2 class a while back. The fraction was of the form ...
0
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0answers
31 views

Are the polynomials of the form $\sum_{i=1}^p x^{p-i} y^{i-1}$ irreducible if and only if $p$ is prime?

The polynomials in the question are nothing but $\frac{x^p-y^p}{x-y}$ (they are polynomial, if $p\in\mathbb{N}$ and if you suppose $x\neq y$). Playing with Wolfram Alpha it looks that they are ...
0
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2answers
83 views

If positive reals $a$ and $b$ satisfy $a\sqrt{a}+b\sqrt{b}=183, a\sqrt{b}+b\sqrt{a}=182$, find $\frac{9}{5}(a+b)$.

Question from Math Olympiad: Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt {b} = 183$ and $a\sqrt{b} + b\sqrt {a} = 182$. Find $\frac{9}{5}(a+b)$. My approach: $a\sqrt{a} + ...
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0answers
15 views

Need help with some factoring questions

So was attempting to factor 3x^2-7x-6 using criss-cross method when I used the method I got (3x-3)(x+2) however the correct answer was (3x+2)(x-3). Did I do something wrong with the criss-cross method ...
0
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2answers
53 views

Factor $9x^4-37x^2+4$

Factor $9x^4-37x^2+4$ using difference of cubes, sum of cubes, difference of squares, substitution, grouping or common factoring, whichever is simplest method. Usually factoring isn't to difficult ...
1
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2answers
39 views

Question: For what value of $c$ will $P(x)= -2x^3+cx^2-5x+2$ have the same remainder when devided by $x+1$ or $x-2$?

For what value of $c$ will $P(x)= -2x^3+cx^2-5x+2$ have the same remainder when devided by $x+1$ or $x-2$? No idea where to start with this one. Can't use the remainder theorem to find the remainder ...
1
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5answers
84 views

How to fully factorise $2x^4+7x^3+4x^2-4x$?

How to fully factorise $2x^4+7x^3+4x^2-4x$? I'm struggling to factorise polynomials like this one. I'm not sure how to best approach this problem. I've tried using the remainder and factor theorems ...
1
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0answers
38 views

Factoring a polynomial with a degree of 5.

How to factor $x^5+3x^4-19x^3-43x^2-18x-40$? This polynomial was given in our activity. I'm very slow with math. Pls help
0
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0answers
20 views

How to characterize / factor a class of polynomials with constraints?

For context: I am continuing my quest to explore functions which exhibit smooth and non-linear re-weighting properties on $[0,1]$. Let us say I have a polynomial $t\to P(t)$ and a set of conditions $$...
1
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4answers
70 views

Factor $2n^2+3n+1$ as $(n+1)(2n+1)$ [closed]

Sorry if this is a really noob question. I factored this as $(n+1)(n+.5)$ but I need it factored as $(n+1)(2n+1)$ to solve an exercise. I know $.5(2n+1) = (n+.5)$. Yet $.5(n+1)(2n+1) \neq 2n^2+3n+1$. ...
1
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4answers
84 views

Factoring $a^4+b^4+(a-b)^4$

I'm trying to factor $$a^4+b^4+(a-b)^4$$ so the result would be $2(a^2-ab+b^2)^2$ but I can't get that. I rewrite it as: $$a^4+b^4+(a-b)^4=(a^2+b^2)^2-2a^2b^2+(a-b)^4=(a^2-\sqrt2 ab+b^2)(a^2+\sqrt2 ...
1
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1answer
45 views

Question about simplifying $(a+b+c)(ab+bc+ac)=abc$

To simplify this expression, according to math110's answer, we can write LHS as: $$(a+b+c)(ab+bc+ac)=(a+b)(b+c)(a+c)+abc$$ then $abc$ cancels and we get: $(a+b)(b+c)(a+c)=0$ But my question is, how we ...
0
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3answers
64 views

Help with factoring $a^4(b-c)+b^4(c-a)+c^4(a-b)$

I want to factor $a^4(b-c)+b^4(c-a)+c^4(a-b)$. I found a similar question on the site, but still don't know how to factor it. from Macavity's answer I realized it has the factor of $(a-b)(a-c)(b-c)$. ...
1
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1answer
27 views

Question about the factors of a number (apart from 2 consecutive numbers)

Pr. 13. In a class, 25 students were lined up. The teacher wrote a number on the board. • The first student said the number was divisible by 1. • Student number 2 said it was divisible by 2. • Student ...
2
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2answers
51 views

A question related to division [closed]

$2^{21} + 1024^{2} + 16^6$ is divisible by: a) 31 b) 19 c) 13 d) 17 My attempt: I tried using the remainder theorem and the factor theorem. But, nothing. Pls help me in moving forward
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0answers
35 views

Factoring Integers, question on Wiki algorithm

I have questions on the integer factorization algorithm described on Wiki. The Wiki description is heavy English and I translated it so it's more symbolic. My question is if this translation makes ...
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1answer
32 views

How can I prove that the results in the following sequence will always share the same factor?

The following sequence is built of $((A^z)^x -1)/6$ where $z$ increases by $1$ and the conditions for $A$ must be that $(A - 1) \mod 6 = 0$ and for $x$ must be that $x$ is odd and $x > 1$: $((A^1)^...
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3answers
50 views

Can all positive real polynomials be expressed as a sum of polynomial squares?

For $P(x)=y$, where $P(x)$ is some real polynomial strictly above the $x$-axis, can it be expressed as the sum of polynomial squares? *technically a constant may not necessarily be algebraic (e.g $3^...
2
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0answers
17 views

Properties of size function in a general Euclidean domain [duplicate]

In ring theory a given ring $R$ is called a Euclidean domain if there exists a function $\sigma:R -\{0\}\rightarrow \{0,1,2,3...\} $ which satisfies the division algorithm i.e. $ $ if $a,b \in R$ ...
2
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1answer
41 views

Proof, Factor theorem.

Question is : If $f(x)$ is a polynomial with integral coefficients and, suppose that $f(1)$ and $f(2)$ both are odd, then prove that there exists no integer n for which $f(n) = 0.$ My approach : I ...
-1
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3answers
59 views

There are $2^{\omega(a)}$ pairs $(b,c)$ such that $a=bc$ and $\gcd(b,c)=1$

It looks as there are $2^{\omega(a)}$ ordered pairs $(b,c)$ such that $a=bc$ and $\gcd(b,c)=1$, where $\omega(a)$ is the number of different prime factors of $a$. Proof? It's also true that there are ...
1
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0answers
22 views

Factoring functions into analytic parts

Consider this function $$\frac{k^{2}-\xi^{2}}{k^{2}+1}$$ which has singularities at $k=\pm i$, the strips where it is analytic are $$ -1<k^{\prime \prime}<0 \quad \text { or } \quad 0<k^{\...
-1
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1answer
27 views

Factorization $a^x-1 | (a^x)^k - 1$ [duplicate]

$a^x-1 | (a^x)^k - 1$ Is this expression true, and if yes how can I prove it?
0
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1answer
55 views

Removing factor $x$ from a number [closed]

I am required to implement this code function but I have no idea what it is asking for. Can someone explain this to me in simpler terms so that I can try to turn it into code? Given $x$ and $y$, I ...
3
votes
3answers
92 views

$x$ is a real number. $x + \frac{6}{\sqrt x} = 35, x + \frac{1}{x}=$? [closed]

$x$ is a real number. $x + \frac{6}{\sqrt x} = 35$ $x + \frac{1}{x}=$? How can I solve this without using a calculator? Is there even a way to factor the aforementioned equation into the latter?
2
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2answers
70 views

Factorize polynomial of degree 4 given statements

If $$x^4-x^3-13x^2+26x-8 = (x-a)(x-b)(x-c)(x-d)$$ Such that$$cd=-8\\a>b\\c<d$$ What are $a,b,c$ and $d$? Since the problem gave us the polynomial, I thought we can just expand the $(x-a)(x-b)(x-...
0
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1answer
111 views

I have almost no idea how to factor numbers this big. [closed]

$14425638854646469646839767613420413647898432138735230192512819$ is the product of two prime numbers. Each factor is an answer. I have to give both factors as answers. How would I get to them and ...
1
vote
1answer
33 views

Missing factor equal to 0 after factoring

The equation $x^{10}+(13x-1)^{10}=0$ has $10$ complex roots, $r_i$ for integers $1\le i\le5$. Find $\sum^5_{i=1}\frac{1}{r_i\overline{r_i}}.$ My solution was simply to factor the given polynomial ...
0
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4answers
46 views

How to find the zero of a rational function with a numerator that cannot be factored?

I'm a tutor working with an Algebra II student on rational functions, and this problem is stumping both of us: $$\frac{x^3-2x^2-8}{x^2-3x}=0$$ She is supposed to find the zero of the function by hand, ...
1
vote
1answer
30 views

Is it possible to factor a polynomial over a ring which is not an integral domain?

Is it possible to factor a polynomial over a ring which is not an integral domain? I assumed not, but I'm wondering about the following scenario. Let $f \in \mathbb{Z}[x]$ be monic and irreducible but ...
0
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0answers
22 views

Deriving factorization of $a^n-b^n$ for non-euclidian domains

Seeing the answer by Andre Nicolas to this question, it got me thinking: is there a way one can derive the formula for $a^n-b^n$ in a more general setting than an euclidian domain, or at least the ...
0
votes
1answer
34 views

Factoring question on binomial

I'm trying to reduce the following expression. The steps follow an online CAS and I have a question regarding a step. $= (b-1)(\frac {1-b^n} {1-b})$ $= \frac {(1-b^n)(b-1)} {1-b}$ The next steps ...
0
votes
2answers
40 views

Long equation problems

I am given this equation: $$15x^6+31x^5-254x^4-352x^3+329x^2+9x-18=0$$ I know how to solve this question but the problem is it is too long to plug in numbers, what am I advised to do?
0
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1answer
20 views

Multivariate Polynomial Division Question [duplicate]

I am trying to solve the following question: ``Let $F$ be a field and let $F[x, y]$ denote the ring of polynomials in the variable $x$ and $y$ with coefficients in $F$. Suppose $f(x,y)$ belongs to $F[...
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2answers
20 views

Factorize certain polynomials as product of irreducibles

I'm trying to solve this question from my abstract algebra's course. I know it's a very trivial question that can be solved using Ruffini like we are taught before college, but my question is how can ...
2
votes
2answers
47 views

Help me factorise

Please help me factorize $9x^2-4a^2+4ay +y^2$. I took out "-" as a common factor from $(- 4a^2+4ay +y^2)$ but I cannot factorize it further using difference of 2 squares.
0
votes
1answer
23 views

Factoring a multivariate polynomial.

Let $P(x_1,\ldots,x_n)\in \mathbb{C}[x_1,\ldots,x_n]$ a polynomial in complex $n$-variables. There is a method to find out if this polynomial can be written as $$P(x_1,\ldots,x_n)=m(x_1)\ldots m(x_n)$$...
0
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3answers
64 views

I need help understanding factoring better for this problem

Here is the equation: $$\frac{-3(x+h)^2-6(x+h)+4+3x^2+6x-4}{h}$$ I am getting the wrong answer every time. \begin{align*} f & = \frac{-3(x^2+3xh+h^2)-6x+6h+4+3x^2+6x-4}{h}\\ f & = \frac{-3x^2-...
2
votes
1answer
43 views

Factoring Quadratic Formula Equation

I have a Quadratic Equation that I input into the Quadratic Formula to solve for y. $$2xy^{2}+8xy-z=0$$ $$y=\frac{-8x\pm \sqrt{(8x)^2 - 4(2x)(-z)}}{2(2x)}$$ I've been told factoring will take me to: $$...
1
vote
2answers
37 views

Understanding the derivation of Euler's Basel formula.

I was looking at the derivation of Euler's Basel formula and I had a question. Since the zeroes of $\sin(x)$ are $0, ±π, ±2π,...$ , we can factorise $\sin (x)$ : $$\sin(x) = ax\cdot(x-π)\cdot(x+π)\...
0
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0answers
35 views

What does this infinitely high order polynomial expression of $\sin x$ mean?

Since the zeroes of $\sin x$ are $0, ±π, ±2π,...$ , we can factorise sin $x$ : $$\sin(x) = ax.(x-π).(x+π).(x-2π).(x+2π)...$$ I would be really grateful if someone could help me understand the $a$ in ...
-1
votes
1answer
43 views

Condition for polynomial factoring

I encountered a sum , prove that $x²+px+p²$ would be a factor of $(x+p)^n–x^n–p^n$ if $n$ be odd and not a multiple of $3$. I tried breaking $(x+p)^n$ by binomial but it became more complicated and ...
0
votes
2answers
80 views

How can we say later that x can be equal to zero when we took x as a common factor and hence divided by it?

To solve an equation like this: $x^2+x=0$ We can take $x$ as a common factor $\frac{x^2}{x}+\frac{x}{x}=0$ $x(x+1)=0$ However to do such a step we have to say that $x$ isn't equal to zero, because ...

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